From b4d2810e6d631a17f54784ec3c978b53a3ee77ca Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pelletier?= Date: Mon, 3 Feb 2014 18:48:33 -0500 Subject: [PATCH] =?UTF-8?q?modifications=20et=20cr=C3=A9ation=20de=20la=20?= =?UTF-8?q?copie=20pour=20le=20d=C3=A9p=C3=B4t=20final=20avec=20les=20r?= =?UTF-8?q?=C3=A8gles=20de=20la=20FESP?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- depotfinal/908144032.pdf | Bin 0 -> 1035172 bytes depotfinal/908144032/.Rhistory | 0 depotfinal/908144032/908144032.pdf | Bin 0 -> 1035172 bytes depotfinal/908144032/908144032.tex | 111 + .../908144032/908144032.tex~ | 2 +- depotfinal/908144032/abstract.tex | 11 + depotfinal/908144032/annexe1.tex | 571 ++++ depotfinal/908144032/annexe2.tex | 145 + depotfinal/908144032/annexe3.tex | 31 + depotfinal/908144032/by-sa.pdf | Bin 0 -> 6761 bytes depotfinal/908144032/by.pdf | Bin 0 -> 1605 bytes depotfinal/908144032/chapitre1.tex | 834 +++++ 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nd?=;ami_eqE+_^ClLCXoSZmXqs^~MCtV@SD5eO3-Q_lYYaaK`x literal 0 HcmV?d00001 diff --git a/depotfinal/908144032/908144032.tex b/depotfinal/908144032/908144032.tex new file mode 100644 index 0000000..abb8871 --- /dev/null +++ b/depotfinal/908144032/908144032.tex @@ -0,0 +1,111 @@ +%% GABARIT POUR MÉMOIRE STANDARD +%% +%% Consulter la documentation de la classe ulthese pour une +%% description détaillée de la classe, de ce gabarit et des options +%% disponibles. +%% +%% [Ne pas hésiter à supprimer les commentaires après les avoir lus.] +%% +%% Déclaration de la classe avec les langues les plus courantes. Le +%% français sera la langue par défaut du document. +\documentclass[11pt,english,francais]{ulthese} \usepackage{scrtime} +\usepackage{natbib} +\usepackage{tikz} \usepackage{multirow} +%% Encodage utilisé pour les caractères accentués dans les fichiers +%% source du document. Les gabarits sont encodés en UTF-8. +\usepackage[utf8]{inputenc} +%% Charger ici les autres paquetages nécessaires pour le document. +%% Quelques exemples; décommenter au besoin. +\usepackage{amsmath} % recommandé pour les mathématiques +\numberwithin{equation}{section} % pour numéroter les équations selon les sections +\usepackage{amsfonts} \usepackage{amsthm} \usepackage{thmtools} +% \usepackage{mathpazo} % texte en Palatino +% \usepackage{mathptmx} % texte en Times +%% Gestion des hyperliens dans le document. S'assurer que hyperref est +%% le dernier paquetage chargé. +\usepackage{hyperref} \hypersetup{colorlinks,allcolors=ULlinkcolor} +\usepackage{cleveref} +\usepackage[hypcap]{caption} % permet de faire le lien au dessus des figures au lieu d en dessous +\usepackage[off]{auto-pst-pdf} +%% Pour utiliser Computer Modern comme police sans empattements +%% (serif) plutôt que Helvetica, décommenter la ligne ci-dessous. +\renewcommand{\sfdefault}{cmss} + +%% Options de mise en forme du mode français de babel. Consulter la +%% documentation du paquetage babel pour les options disponibles. +\frenchbsetup{% + CompactItemize=false, % ne pas compacter les listes + ThinSpaceInFrenchNumbers=true % espace fine dans les nombres +} \DeclareMathOperator{\sgn}{sgn} +%% Style de la bibliographie. +\bibliographystyle{plainnat-fr} +%% Déclarations de la page titre. Remplacer les éléments entre < >. +%% Supprimer les caractères < >. Couper un long titre ou un long +%% sous-titre manuellement avec \\. Terminer une éventuelle deuxième +%% ligne de titre ou de sous-titre avec \vspace*{-\baselineskip}. +\titre{Modélisation des rendements financiers \\ à l'aide de la + distribution de Laplace \\ asymétrique généralisée + \vspace*{-\baselineskip}} +% \titre{Ceci est un exemple de long titre \\ +% avec saut de ligne manuel \vspace*{-\baselineskip}} +% \soustitre{Sous-titre le cas échéant} +% \soustitre{Ceci est un exemple de long-titre \\ +% avec saut de ligne manuel \vspace*{-\baselineskip}} +\auteur{François Pelletier} \programme{Maîtrise en actuariat} +\annee{2014} +\MSc % ou l'un de \LLM, \MA, \MMus, \MServSoc, \MScGeogr, \MATDR +\setcounter{tocdepth}{4} +\begin{document} +\newtheorem{theo}{Théorème}[subsection] +\newtheorem{hypothese}{Hypothèse}[chapter] + +\frontmatter % pages liminaires + +\pagetitre % production de la page titre + +\include{resume} % résumé français +\include{abstract} % résumé anglais +\cleardoublepage + +\tableofcontents % production de la TdM +\cleardoublepage + +\listoftables % production de la liste des tableaux +\cleardoublepage + +\listoffigures % production de la liste des figures +\cleardoublepage + +%\dedicace{Dédicace si désiré} +%\cleardoublepage + +\epigraphe{Dieu ne se soucie pas de nos difficultés mathématiques. Il intègre empiriquement.}{Albert Einstein} +\cleardoublepage + +\include{remerciements} % remerciements + +\mainmatter % corps du document + +\include{introduction} % introduction +\include{chapitre1} % chapitre 1 +\include{chapitre2} % chapitre 2, etc. +\include{chapitre3} +\include{chapitre4} +\include{chapitre5} +\include{chapitre6} +\include{chapitre7} +\include{chapitre8} +\include{chapitre9} +\include{conclusion} % conclusion + +\appendix % annexes le cas échéant + +\include{annexe1} % Annexe 1 +\include{annexe2} % Annexe 2 +\include{annexe3} % Annexe 3 + +\bibliography{memoire} % production de la bibliographie + +\include{deed} + +\end{document} diff --git a/memoire/gabarit-maitrise.tex b/depotfinal/908144032/908144032.tex~ similarity index 99% rename from memoire/gabarit-maitrise.tex rename to depotfinal/908144032/908144032.tex~ index 69658dc..61e811c 100644 --- a/memoire/gabarit-maitrise.tex +++ b/depotfinal/908144032/908144032.tex~ @@ -38,7 +38,7 @@ ThinSpaceInFrenchNumbers=true % espace fine dans les nombres } \DeclareMathOperator{\sgn}{sgn} %% Style de la bibliographie. -\bibliographystyle{plainnatmod} +\bibliographystyle{plainnat-fr} %% Déclarations de la page titre. Remplacer les éléments entre < >. %% Supprimer les caractères < >. Couper un long titre ou un long %% sous-titre manuellement avec \\. Terminer une éventuelle deuxième diff --git a/depotfinal/908144032/abstract.tex b/depotfinal/908144032/abstract.tex new file mode 100644 index 0000000..b6cc1c9 --- /dev/null +++ b/depotfinal/908144032/abstract.tex @@ -0,0 +1,11 @@ +\chapter*{Abstract} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Abstract} % inclure dans TdM + +\begin{otherlanguage*}{english} + Classical models in finance are based on a set of hypotheses that are not always empirically verifiable. The main objective behind this master's thesis is to show that the generalized asymmetric Laplace distribution is an interesting alternative to those models. To support this idea, we focus on some of its properties to develop parametric estimation, approximation and testing methods, then we work out some principles of derivatives pricing. Finally, we have a numerical example to illustrate these concepts. +\end{otherlanguage*} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/annexe1.tex b/depotfinal/908144032/annexe1.tex new file mode 100644 index 0000000..d65e140 --- /dev/null +++ b/depotfinal/908144032/annexe1.tex @@ -0,0 +1,571 @@ +\chapter{Éléments de théorie des probabilités} + +\section{Définitions de base} +\label{sec:introvariablealeatoire} + +Une \textbf{variable aléatoire} est une variable dont la valeur est +déterminée par une expérience aléatoire. On associe une probabilité à +chacune des valeurs possibles. On appelle \textbf{évènement} tout +ensemble de réalisations possible de l'expérience +aléatoire. L'entièreté de tous les évènements se nomme +l'\textbf{ensemble fondamental} et est notée $\Omega$. + +Une \textbf{probabilité} associée à un évènement $A$ doit satisfaire +un ensemble de trois axiomes \citep{dodge2004statistique}. +\begin{enumerate} +\item Une probabilité est une valeur comprise entre 0 et 1: + \begin{equation} + \label{eq:condition1probabilite} + 0 \leq P(A) \leq 1. + \end{equation} + +\item La probabilité de l'évènement correspondant à l'ensemble + fondamental est 1: + \begin{equation} + \label{eq:condition2probabilite} + P(\Omega) = 1. + \end{equation} + +\item Pour chaque séquence d'évènements mutuellement exclusifs + $A_1,A_2,\ldots$: + \begin{equation} + \label{eq:condition3probabilite} + P \left[ \bigcup_{i=1}^{\infty} A_i \right] = \sum_{i=1}^{\infty} P(A_i). + \end{equation} +\end{enumerate} + +Le respect de ces conditions devra notamment être vérifié lors de +l'utilisation de méthodes numériques. + +Toujours selon \cite{dodge2004statistique}, on définit: +\begin{enumerate} +\item La \textbf{densité}, notée $f_X(x)$, permet de déterminer la + probabilité qu'une variable aléatoire $X$ prenne une valeur dans un + intervalle fixé $\left[ a,b \right]$. En intégrant cette fonction + sur l'intervalle $\left[ a,b \right]$, on obtient cette probabilité: + \begin{equation} + \label{eq:densitedefinition} + P(a \leq X \leq b) = \int_{a}^{b} f_X(x) dx. + \end{equation} +\item La \textbf{fonction de répartition}, notée $F_X(x)$, d’une + variable aléatoire est définie comme la probabilité que celle-ci + prenne une valeur inférieure ou égale à un certain nombre $b \in + \mathbb{R}$: + \begin{equation} + \label{eq:repartitiondefinition} + F_X(b) = P(X \leq b) = \int_{-\infty}^{b} f_X(x) dx. + \end{equation} +\end{enumerate} + + +\section{Transformées d'une variable aléatoire} +\label{sec:transformees} + +Les transformées d'une variable aléatoire résultent de l'application +d'un opérateur intégral, ou à noyau, sur la fonction de densité. + +Certaines d'entre elles permettent de déterminer entièrement leur +distribution. Parmi celles-ci, on retrouve la fonction caractéristique +et les fonctions génératrices des moments et des cumulants, qui sont +les plus couramment utilisées. + +Certaines transformées permettent de modifier la distribution d'une +variable aléatoire. Parmi celles-ci, la transformée d'Esscher, qui est +employée en actuariat et en finance, est aussi étroitement liée au +concept d'utilité en sciences économiques. + +\subsection{La fonction caractéristique} +\label{sec:fonctioncaract} + +\subsubsection{Transformée de Fourier} +\label{sec:transfourier} + +La transformée de Fourier est une opération qui transforme une +fonction $g(\xi)$ en une autre $f(x)$ par l'intégration, sur son +domaine, du produit $\mathrm{e}^{-i x\xi}\,g(\xi)$: +\begin{equation} + \label{eq:transfourier} + \mathcal{F}(g):x\mapsto f(x) = \int_{-\infty}^{\infty} \mathrm{e}^{-ix\xi}\,g(\xi)\,\mathrm{d}\xi. +\end{equation} + +On définit aussi la transformée de Fourier inverse +\index{Transformée!de Fourier!inverse}, qui permet de retrouver la +fonction initiale $g(\xi)$ à partir de la transformée $f(x)$: +\begin{equation} + \label{eq:transinvfourier} + g(\xi)=\mathcal{F}^{-1}(f(x)) = {1 \over 2\pi}\, \int_{-\infty}^{\infty} \mathrm{e}^{ix \xi}\,f(x)\, \mathrm{d}x. +\end{equation} + +\subsubsection{Définition} +\label{sec:deffncaract} + +La fonction caractéristique \index{Fonction!caractéristique} est +définie comme étant la transformée de Fourier inverse de la densité +\index{Fonction!de densité}, dans le cas continu, ou de masse de +probabilité \index{Fonction!de masse de probabilité}, dans le cas +discret. C'est donc une application directe de +\eqref{eq:transinvfourier} qui s'exprime par l'intégrale de +Riemann-Stieltjes: +\begin{equation} + \label{eq:deffncaract} + \phi_X(s) = \int_{-\infty}^{\infty} e^{isx} dF_X(x). +\end{equation} + +Cette intégrale est toujours convergente, comme le démontre +\cite{stuart1987kendall}. + +\subsubsection{Les moments} +\label{sec:momentsfncaract} +On définit le moment d'ordre $r$ d'une distribution de probabilités +comme étant la quantité représentée par l'espérance de la puissance +$r$ d'une variable aléatoire: +\begin{equation} + \label{eq:defmoments} + E[X^r] = \int_{-\infty}^{\infty} x^r dF_X(x). +\end{equation} + +La fonction caractéristique\index{Fonction!caractéristique}, +lorsqu'elle est différenciable, permettra de générer les moments de la +distribution en utilisant la propriété suivante +\citep{lukacs1960characteristic}: +\begin{equation*} + \frac{d^r\phi_X(s)}{ds^r} = i^r \int_{-\infty}^{\infty} e^{isx} x^r dF_X(x). +\end{equation*} + +En posant $s=0$ par la suite, on obtient, pour les différentes +dérivées, l'équation suivante: +\begin{equation*} + \left[ \frac{d^r\phi_X(s)}{ds^r} \right]_{s=0} = i^r\,E[X^r]. +\end{equation*} + +Ce qui nous permet de définir les différents moments de la +distribution à partir de la fonction caractéristique: +\begin{equation} + \label{eq:fncaractmoments} + E[X^r] = (-i)^r\,\left[ \frac{d^r\phi_X(s)}{ds^r} \right]_{s=0}. +\end{equation} + +\subsection{Inversion de la fonction caractéristique} +\label{sec:inversioncaract} + +\subsubsection{La densité} +\label{sec:densitefncaract} + +On obtient la densité de la variable aléatoire $X$ en calculant la +transformée de Fourier \eqref{eq:transfourier} de la fonction +caractéristique suivante: +\begin{equation} + \label{eq:caractdensite} + f_X(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-isx} \phi_X(s) ds. +\end{equation} + +On peut utiliser l'intégration numérique si l'intégrale n'a pas de +solution analytique, la méthode de la transformée de Fourier rapide +(section \ref{sec:methodeFFT}) ou encore la méthode du point de selle. + +\subsubsection{La fonction de répartition} +\label{sec:repartitionfncaract} + +On peut obtenir directement l'expression de la fonction de +répartition, sans passer par l'intégration de la densité de +probabilité, en utilisant le théorème de \cite{gil1951note}: +\begin{equation} + \label{eq:gilpelaez} + F_X(x) = \frac{1}{2} + \frac{1}{2\pi} \int_{0}^{\infty} \frac{e^{isx}\phi_X(-s)-e^{-isx}\phi_X(s)}{is} ds. +\end{equation} + +On peut exprimer cette intégrale sous la forme suivante lorsque la +densité $f(x)$ est strictement continue \citep[p.66]{epps2007pricing}: +\begin{equation} + \label{eq:gilpelaez2} + F_X(x) = \frac{1}{2} - \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{e^{-isx}}{is} \phi(s) ds. +\end{equation} +Cette forme est moins appropriée pour l'intégration numérique, mais +sera utile dans plusieurs calculs. On préfèrera la ramener à la forme +suivante avec le théorème 2 de \cite{wendel1961non}: +\begin{equation} + \label{eq:inversionfncaract} + F_X(x) = \frac{1}{2} - \frac{1}{\pi}\int_{0}^{\infty} \frac{Im\left[e^{-isx}\phi_X(s)\right]}{s} ds. +\end{equation} + +Cette fonction peut être difficile à intégrer de manière efficace +numériquement, surtout lorsqu'on l'évalue en des points situés aux +extrémités de la distribution. On peut alors privilégier l'égalité +donnée par le théorème 4 de \cite{shephard1991characteristic}, qui +utilise le noyau de Fejér afin de réduire l'erreur d'intégration, pour +obtenir le résultat suivant: +\begin{equation} + \label{eq:shepherd} + F_X(x) = \frac{1}{2} - \frac{1}{2\pi} \lim_{n\rightarrow\infty} \int_{0}^{n} \underbrace{\left[ 1-\frac{s}{n} \right]}_{\text{Noyau de Fejér}} \left[ \frac{e^{isx}\phi_X(-s)-e^{-isx}\phi_X(s)}{is} \right] ds. +\end{equation} + +De même, en utilisant le raisonnement qui permet de passer de la +définition de la transformée \eqref{eq:gilpelaez} à celle de son +inverse \eqref{eq:inversionfncaract}, on obtient, pour le résultat +\eqref{eq:shepherd}, l'expression suivante: +\begin{equation} + \label{eq:approxinvfncaract} + F_X(x) = \frac{1}{2} - \frac{1}{\pi} \lim_{n\rightarrow\infty} \int_{0}^{n} \left[ 1-\frac{s}{n} \right] \frac{Im\left[e^{-isx}\phi_X(s)\right]}{s}ds. +\end{equation} + +On peut alors obtenir une approximation en fixant la borne +d'intégration supérieure $n$ qui définit la précision +désirée. Certains logiciels d'intégration numérique prennent en charge +les bornes infinies. + +La fonction caractéristique permet d'identifier la distribution d'une +somme de variables aléatoires indépendantes $Z = X_1, \ldots, X_n$, +appelée produit de convolution et noté $Z = X_1 * \ldots * X_n$: +\begin{equation} + \label{eq:convocaract} + \phi_{Z}(s) = \phi_{X_1+\ldots+X_n}(s) = \prod_{i=1}^n \phi_{X_i}(s). +\end{equation} + +Lorsque les variables aléatoires sommées sont aussi identiquement +distribuées, la fonction caractéristique de $Z$ est la $n^e$ puissance +de celle de $X$: +\begin{equation} + \label{eq:convocaractIID} + \phi_{Z}(s) = \phi_{X_1+\ldots+X_n}(s) = \left[\phi_{X}(s)\right]^n. +\end{equation} + +Cette fonction est donc une solution de rechange intéressante à +utiliser lorsque aucune forme analytique pour la densité ou la +fonction de répartition pour une distribution donnée n'existe. + +\subsection{La fonction génératrice des moments} +\label{sec:fgm} +La fonction génératrice des moments $M_X(s)$ +\index{Fonction!génératrice des moments} est définie comme étant la +transformée de Laplace inverse \index{Transformée!de Laplace!inverse} +de la densité: +\begin{equation} + \label{eq:deffngenmom} + M_X(s) = \int_{-\infty}^{\infty} e^{sx} dF_X(x). +\end{equation} + +Cette intégrale, contrairement à \eqref{eq:deffncaract}, ne converge +pas toujours, ce qui signifie que certaines distributions n'ont pas de +fonction génératrice des moments. Comme son nom l'indique, cette +fonction permet de générer les moments de la distribution de la +variable $X$, ce qui se fait essentiellement de la même manière +qu'avec la définition \eqref{eq:fncaractmoments}. Par contre, on n'a +pas à éliminer le terme complexe, ce qui a pour avantage de simplifier +les calculs: +\begin{equation} + \label{eq:fgmmoments} + E[X^r] = \left[ \frac{d^r M_X(s)}{ds^r} \right]_{s=0}. +\end{equation} + +La fonction génératrice des moments peut aussi être obtenue à partir +de la fonction caractéristique : +\begin{equation} + \label{eq:fncaractfgm} + M_X(s) = \phi(-is). +\end{equation} + +Ceci permet d'en déduire qu'elles possèdent des caractéristiques +communes, notamment le produit de convolution. + +\subsection{La fonction génératrice des cumulants} +\label{sec:fgc} +La fonction génératrice des cumulants $K_X(s)$ est définie comme le +logarithme de la fonction génératrice des moments: +\begin{equation} + \label{eq:fgc} + K_X(s) = \ln{M_X(s)}. +\end{equation} + +Cette fonction est utilisée pour générer les cumulants, des quantités +étroitement liées aux moments, qui peuvent à leur tour être utilisés +dans le cadre de méthodes d'estimation paramétrique. Parmi celles-ci, +on retrouve la méthode des cumulants, dont l'objectif est de former un +système d'équations où les valeurs empiriques et théoriques de ces +quantités sont égales. Les valeurs empiriques peuvent être obtenues à +partir des moments échantillonnaux \citep{stuart1987kendall}: +\begin{subequations}\label{eq:cumulantsempiriques} + \begin{align} + \hat{K}_1 &= \hat{m}_1 \label{eq:cumulantsempiriques1}\\ + \hat{K}_2 &= \hat{m}_2' - \hat{m}_1^2 \label{eq:cumulantsempiriques2}\\ + \hat{K}_3 &= \hat{m}_3' - 3\hat{m}_1\hat{m}_2' + 2\hat{m}_1^2 \label{eq:cumulantsempiriques3}\\ + \hat{K}_4 &= \hat{m}_4' - 3\hat{m}_2'^2 - 4\hat{m}_1\hat{m}_3' + 12\hat{m}_1^2\hat{m}_2' - + 6\hat{m}_1^4 \label{eq:cumulantsempiriques4}. + \end{align} +\end{subequations} + +On remarquera aussi la forme de la première dérivée de cette dernière, +qui sera utilisée pour estimer la densité et la fonction de +répartition avec la méthode du point de selle: +\begin{equation} + \label{eq:derivfgc} + \frac{dK_X(s)}{ds} = K^{\prime}_X(s) = \frac{M^{\prime}_X(s)}{M_X(s)}. +\end{equation} + +\subsection{La transformée d'Esscher} +\label{sec:transesscher} + +La transformée d'Esscher est utilisée pour transformer une fonction de +densité en une autre à l'aide d'un coefficient exponentiel $h$: +\begin{equation} + \label{eq:esschertransform} + f(x;h)=\frac{e^{hx}f(x)}{\int_{-\infty}^\infty e^{hx} f(x) dx}. +\end{equation} + +Lorsque l'on ne dispose pas d'une forme analytique de la densité, on +peut exprimer la transformée d'Esscher par sa fonction génératrice des +moments: +\begin{equation} + \label{eq:esscherMx} + M_X(s;h) = \frac{M_X(s+h)}{M_X(h)}. +\end{equation} + +\section{La transformée de Fourier rapide} +\label{sec:methodeFFT} + +Étant donné l'importance de la transformée de Fourier et de son +inverse en sciences physiques, plusieurs algorithmes ont été +développés afin d'en effectuer l'intégration numérique. Ces +algorithmes se regroupent sous l'appellation de transformée de Fourier +rapide. + +L'existence de ces algorithmes favorise l'utilisation de la fonction +caractéristique et de ses propriétés, notamment pour l'agrégation de +risques en actuariat, ou encore le calcul de probabilités pour des +distributions n'ayant pas de forme explicite pour la fonction de +répartition. + +Ces algorithmes numériques sont utilisés respectivement pour calculer +de manière optimisée des sommes de la forme +\begin{align} + \hat{f}(x_u) &= \sum_{j=1}^{N} \phi(\zeta_j)\,e^{-{2\pi i \over N} (j-1)(u-1) } \qquad u = 1,\dots,n. \label{eq:sommefft} \\ + \hat{\phi}(\zeta_j) &= \sum_{u=1}^{N} f(x_u)\,e^{{2\pi i \over N} + (j-1)(u-1) } \qquad j = 1,\dots,n. \label{eq:sommeifft} +\end{align} + +Par exemple, pour évaluer l'intégrale \eqref{eq:caractdensite} afin +d'obtenir la valeur de la densité sur un certain domaine +$\left[a,b\right]$, on devra discrétiser celle-ci sur nombre $N$ de +points de discrétisation.On définit alors les différents paramètres de +ces deux algorithmes: +\begin{itemize} +\item $\eta = \frac{b-a}{N}$, le pas de discrétisation pour la + variable aléatoire $u$ +\item $\zeta = \eta(j-1)$, la variable de transformation +\item $\lambda = \frac{2\pi}{N\eta}$, le pas de discrétisation pour la + variable de transformation $\zeta$ +\item $c=\frac{N\lambda}{2}$, la borne supérieure d'intégration +\end{itemize} + +On obtient alors une sommation de la forme \eqref{eq:sommefft}: +\begin{subequations}\label{eq:sommefftdensite} + \begin{align} + f_{FFT}(x_u) &\approx \frac{1}{2\pi} \sum_{j=1}^N e^{-i\lambda\zeta_j(u-1)} e^{-ic\zeta_j} \phi(-c+\lambda\,(j-1)) \eta, \label{eq:sommefftdensite1}\\ + &\approx \frac{1}{2\pi} \sum_{j=1}^N + e^{-i\frac{2\pi}{N}(j-1)(u-1)} e^{-ic\zeta_j} + \phi(-c+\lambda\,(j-1)) \eta.\label{eq:sommefftdensite2} + \end{align} +\end{subequations} + +Plusieurs logiciels d'analyse numérique possèdent une implémentation +de la méthode de la transformée de Fourier rapide. Essentiellement, +c'est une fonction $f_{FFT}(X_u)$ qui prend comme argument un vecteur +de valeurs, appelé signal, et qui en retourne un autre, de même +longueur, le spectre de fréquences. En statistique, la fonction +caractéristique empirique constitue le signal et la densité, le +spectre de fréquences. Le signal $X_u$ est défini comme suit, à partir +des constantes définies précédemment: +\begin{align} + \label{eq:signalfftfonction} + X_u &= e^{-i\lambda\zeta_j(u-1)} \phi\left(-c+(j-1)\lambda\right) + \eta. +\end{align} + +La somme \eqref{eq:sommefftdensite1} devient alors +\begin{align} + \label{eq:fftfonction} + f_{FFT}(X_u) &\approx \frac{1}{2\pi} \sum_{j=1}^N X_u e^{-ic\zeta_j}. +\end{align} + +On recouvre la densité aux points $x_u=a+(j-1)\eta, j=1,\ldots,N$ en +appliquant la transformation suivante au vecteur $f_{FFT}(X_u)$: +\begin{align} + \label{eq:densitefftfonction} + f(x_u) = \frac{1}{N} e^{icx_u} f_{FFT}(X_u). +\end{align} + +\section{Processus de Lévy} +\label{sec:processuslevy} + +\subsection{Définition et propriétés} +\label{sec:defproplevy} + +Le processus de Lévy, tel que présenté par \cite{barndorff2001levy}, +est une classe générale de processus stochastiques regroupant entre +autres les processus de Poisson composés et les processus de Wiener. +Il est défini comme continu à droite, limité à gauche (càdlàg), ayant +pour point de départ l'origine et ayant des incréments indépendants et +homogènes. Il est infiniment divisible, tout processus de Lévy peut +ainsi être considéré comme une convolution de plusieurs autres. Cette +propriété est très intéressante dans un contexte de rendements +financiers, car la même distribution pourra être utilisée avec +n'importe quel intervalle d'échantillonnage des prix. + +\subsubsection{Représentation de Lévy-Khintchine} +\label{sec:levykhintchine} + +Le processus de Lévy est généralement représenté par sa fonction +caractéristique sous la forme de Lévy-Khintchine: +\begin{align} + \mathbb{E}\Big[e^{i\theta X(t)} \Big] &= \exp \Bigg( + \underbrace{ait\theta}_{\text{composante de dérive}} \nonumber\\ + &\qquad - + \underbrace{\frac{1}{2}\sigma^2t\theta^2}_{\text{composante de + diffusion}} \nonumber\\ &\qquad + \underbrace{t + \int_{\mathbb{R}\backslash\{0\}} \big( e^{i\theta x}-1 -i\theta x \mathbf{I}_{|x|<1}\big)\,\nu(dx)}_{\text{composante de saut}} \Bigg)\label{eq:levykhintchine}\\ + &= \exp \bigg(-t\Psi(\theta) \bigg). +\end{align} + +$\Psi(\theta)$ est appelé l'exposant caractéristique de l'incrément de +longueur 1 $(X(t+1)-X(t))$. Un processus de Lévy est souvent décrit +par son triplet générateur $(a,\sigma^2,\nu)$. Cette description +permettra de classer certains processus de Lévy selon deux catégories: +\begin{itemize} +\item Si $\sigma^2=0$, $\lbrace X(t) \rbrace$ est un processus de + sauts +\item Si $a=0 \mbox{ et } \sigma^2=0$, alors le processus $\lbrace + X(t) \rbrace$ est un processus de sauts purs +\end{itemize} + +L'élément $\nu(dx)$ de la composante de saut est appelé la mesure de +Lévy. Un exemple de processus de Lévy représenté sous la forme de +Lévy-Khintchine qui ne fait pas partie de ces deux catégories est +défini par le modèle de Press \eqref{eq:fncaractpress}. + +\subsubsection{Représentation de Lévy-Itô} +\label{sec:levyito} +La décomposition de Lévy-Itô est une représentation alternative +décrivant la trajectoire d'une réalisation du processus. Cette +dernière a une interprétation intéressante en finance décrite par +\cite{applebaum2004levy}: +\begin{align} + \label{eq:levyitodecomposition} + X(t) &= \underbrace{at + B_{\sigma^2}(t)}_{\text{mouvement brownien}} \nonumber\\ &\quad+ \underbrace{\int_{|x|<1}x N(t,dx)-t\nu(dx)}_{\text{martingale de sauts purs de carré intégrable}} \nonumber\\ &\quad+ \underbrace{\int_{|x|>1}x N(t,dx)}_{\text{processus de Poisson composé}} \\ + &= X_1(t) + X_2(t) + X_3(t). +\end{align} + +La portion mouvement brownien ($X_1(t)$) décrit le comportement +général du processus, en spécifiant le rendement espéré $a$ et la +volatilité du titre $\sigma^2$. La première intégrale ($X_2(t)$) +décrit le bruit causé par les transactions financières quotidiennes +qui font varier un peu le prix, alors que la seconde ($X_3(t)$) +introduit les sauts occasionnés par des évènements plus rares, comme +les catastrophes naturelles et les crises politiques. Dans les deux +cas, $N(t,dx)$ est une mesure aléatoire de Poisson. + +\subsection{Processus subordonné} +\label{sec:procsubordonne} + +On considère les processus de Lévy $\lbrace X(t) \rbrace$ et $\lbrace +Z(t) \rbrace$. Celui qui suit est défini comme étant un processus +subordonné et aussi un processus de Lévy, comme le démontrent +\cite{sato1999levy} et \cite{schoutens2003levy}: +\begin{equation} + \label{eq:processussubordonne} + \lbrace Y(t) \rbrace = \lbrace X\left(Z(t)\right) \rbrace. +\end{equation} + +Les processus de Lévy les plus couramment utilisés en finance sont des +mouvements browniens subordonnés. Ils sont plus faciles à manipuler et +permettent néanmoins de représenter les phénomènes de queues longues +présents dans les distributions de rendements. Une bonne introduction +à ce sujet est faite par \cite{kyprianou2007introductory}. + +Si $\Lambda(\theta)$ est l'exposant caractéristique du processus +$\lbrace X(t)\rbrace$ et $\Xi(\theta)$, celui du processus subordonné +$\lbrace Z(t)\rbrace$, alors celui du processus $\lbrace Y(t)\rbrace$ +prend la forme suivante: +\begin{align} + \label{eq:exposantcaractYt} + \Psi(\theta) &= \Xi(\theta)\circ i\Lambda(\theta) \nonumber\\ + &= \Xi(i\Lambda(\theta)). +\end{align} + +La fonction caractéristique du processus $\lbrace Y(t) \rbrace$ est +alors +\begin{equation} + \label{eq:fncaractYt} + \phi_{Y(t)}(\xi) = e^{-t\Xi(i\Lambda(\xi))}. +\end{equation} + +La densité de la variable aléatoire $Y(t)$ s'obtient à l'aide de la +formule de l'espérance conditionnelle: +\begin{align} + f_{Y(t)}(y) &= E\left[ f_{X(t)}(y|Z(t)=z) \right] \nonumber\\ + &= \int_{-\infty}^{\infty} f_{X(t)}(y|Z(t)=z) \cdot f_{Z(t)}(z) + dz \label{eq:densiteprocessusyt}. +\end{align} + +\section{Théorèmes d'intégration} +\label{sec:integration} + +Ces deux théorèmes sont présentés sans démonstration afin de +complémenter la démonstration de la méthode d'Epps de la section +\ref{sec:epps2007}. La démonstration de ceux-ci nécessite des +connaissances en théorie de l'intégration. On peut retrouver davantage +d'explications sur ceux-ci dans \cite{teschl2004topics}. + +\subsection{Théorème de convergence dominée de Lebesgue} +\label{sec:theor-de-conv} + +Le théorème de convergence dominée de Lebesgue est un des principaux +éléments de la théorie de l'intégration. Il sert à démontrer qu'une +fonction est intégrable, sachant qu'une autre l'est aussi et qu'elle +répond à certaines conditions. + +Soit $(f_n)_{n\in \mathbb{N}}$, une suite de fonctions réelles ou +complexes intégrables dans un intervalle $\mathit{I}$ qui convergent +vers une fonction $f$: +\begin{align} + \label{eq:sequencetheoremconvdom} + f(x) = \lim_{n \to \infty} f_n(x). +\end{align} + +On suppose qu'une fonction $g$ intégrable dans l'intervalle +$\mathit{I}$ existe telle que pour toute valeur de $n$ et pour tout +point $x\in I$ où elle est définie, la valeur absolue de $f_n(x)$ est +inférieure ou égale à $g(x)$: +\begin{align} + \label{eq:theoremeconvdom} + \forall n \in \mathbb{N}, \forall x \in \mathit{I}, |f_n(x)| \leq + g(x). +\end{align} + +Alors, la fonction $f$ est intégrable dans l'intervalle +$\mathit{I}$. + +\subsection{Théorème de Fubini} +\label{sec:theoreme-de-fubini} + +Le théorème de Fubini permet, entre autres, d'inverser l'ordre +d'intégration lorsque certaines conditions sont remplies. + +Soit une fonction $f$ continue sur le rectangle $\mathit{R}$ suivant: +\begin{align} + \label{eq:rectanglefubini} + \mathit{R} &= \left\{(x,y) | a \leq x \leq b, c \leq y \leq d \right\}. +\end{align} + +on peut inverser l'ordre d'intégration: +\begin{align} + \label{eq:thfubini2} + \int_{X\times Y}f(x,y)~ d(\mu\times\nu)(x,y)=\int_X\left[\int_Y + f(x,y)~ d\nu(y)\right]~ d\mu(x)=\int_Y\left[\int_X f(x,y)~ + d\mu(x)\right]~ d\nu(y). +\end{align} + +Il est possible de généraliser cet énoncé en utilisant la théorie de +la mesure. +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/annexe2.tex b/depotfinal/908144032/annexe2.tex new file mode 100644 index 0000000..7d80845 --- /dev/null +++ b/depotfinal/908144032/annexe2.tex @@ -0,0 +1,145 @@ +\chapter{Éléments de statistique mathématique} + +\section{Loi faible des grands nombres} +\label{sec:loifaible} + +La \textbf{loi faible des grands nombres} est un résultat important en +probabilité, car il permet de définir la notion d'estimateur +convergent. Soit une suite de variables aléatoires indépendantes et +identiquement distribuées $\left\{X_T \right\}_{T=1}^{\infty}$ ayant +une espérance $E\left[ X \right]$ et une variance $V\left[ X \right]$ +finies. Selon la loi faible des grands nombres, pour tout nombre réel +strictement positif $\varepsilon$, la probabilité que la différence +entre la moyenne empirique $Y_T=\frac{X_1+X_2+\cdots+X_T}{T}$ et +l'espérance $E\left[ X \right]$ soit supérieure à la valeur +$\varepsilon$ tend vers 0 lorsque $T$ tend vers l'infini. +\begin{align} + \label{loifaible} + \lim_{T \to +\infty} \mathbb{P}\left(\left|Y_T - E\left[ X + \right]\right| \geq \varepsilon\right) = 0 ,\quad + \forall\varepsilon>0 +\end{align} + +On dit alors que la suite d'estimateurs +$\left\{Y_T\right\}_{T=1}^{\infty}$ converge en probabilité vers +l'espérance $E\left[ X \right]$. L'estimateur de l'espérance $Y_T$ est +alors \textbf{convergent}. + +\section{Théorème central limite} +\label{sec:theor-centr-limite} + +Le \textbf{théorème central limite} est un résultat fondamental en +probabilité qui énonce le rôle de la distribution normale. Il démontre +que toute somme de variables aléatoires indépendantes et identiquement +distribuées suit approximativement une loi normale. Ce résultat permet +entre autres d'identifier la distribution limite d'un estimateur +convergent. + +\subsection{Cas univarié} +\label{sec:cas-univarie} + +Soit une suite d'observations $X_1, \ldots, X_T$ d'un échantillon +aléatoire d'une distribution de moyenne $\mu$ et de variance +$\sigma^2$: +\begin{align} + \label{eq:TCL} + Y_T &= \frac{1}{\sqrt{T}\sigma} \left(\sum_{t=1}^T X_t - T\mu\right) \nonumber\\ + &= \frac{\sqrt{T}}{\sigma}\left(\overline{X}_T-\mu\right). +\end{align} +Alors, cette variable aléatoire converge en distribution vers une +variable aléatoire normale centrée réduite: +\begin{align} + \label{eq:TCL2} + Y_T \stackrel{L}{\rightarrow} \mathcal{N}(0,1). +\end{align} + +\subsection{Cas multivarié} +\label{sec:cas-multivarie} + +On peut aussi généraliser ce théorème pour des observations +multivariées. On considère alors une série d'observations +multivariées $\mathbf{X_1}, \ldots, \mathbf{X_T}$ où +\begin{align} + \label{eq:defmultiX} + \mathbf{X_t}=\begin{bmatrix} X_{t(1)} \\ \vdots \\ + X_{t(k)} \end{bmatrix}, \quad t=1,\ldots,T. +\end{align} + +On définit maintenant la variable aléatoire correspondante: +\begin{align} + \label{eq:TCLmulti1} + \mathbf{Y}_T &= \frac{1}{T}\begin{bmatrix} \sum_{t=1}^{T} \left [ + X_{t(1)} \right ] \\ \vdots \\ \sum_{t=1}^{T} \left [ X_{t(k)} + \right ] \end{bmatrix}. +\end{align} + +Dans cette situation, la variable aléatoire converge en distribution +vers une variable aléatoire de distribution normale multivariée +centrée de matrice de variance-covariance $\mathbf{\Sigma}$: +\begin{align} + \label{eq:TCLmulti2} + \sqrt{T}\left(\mathbf{Y}_T - \boldsymbol{\mu}\right)\ \stackrel{L}{\rightarrow}\ + \mathcal{N}_k(0,\mathbf{\Sigma}) +\end{align} + +où +\begin{align*} + \mathbf{\Sigma} &= \begin{bmatrix} + \omega_{(1,1)} &\cdots& \omega_{(1,k)} \\ + \vdots & \ddots & \vdots \\ + \omega_{(k,1)} &\cdots& \omega_{(k,k)} \\ + \end{bmatrix} ,\quad \mbox{avec } \omega_{(j,k)} = \begin{cases} + Var\left[X_{1(j)}\right] , &(j=k) \\ + Cov\left[X_{1(j)},X_{1(k)}\right] , &(j \neq k). + \end{cases} +\end{align*} + +\section{Méthode delta multivariée} +\label{sec:deltamethod} + +Dans le cas univarié, on utilise la méthode delta pour évaluer la +distribution d'une fonction d'un estimateur, en supposant que la +distribution de cet estimateur est asymptotiquement normale de +variance connue. + +Dans le cas multivarié, on estime la distribution d'une fonction d'un +vecteur d'estimateurs, dont la distribution asymptotique est normale +multivariée, avec une matrice de variance-covariance $\Sigma$. + +On a donc, pour un estimateur convergent $\hat\theta_T$, en appliquant +le théorème central limite, le résultat suivant: +\begin{align} + \sqrt{T} (\hat\theta_T - \theta_0) \stackrel{L}{\longrightarrow} + \mathcal{N}(0,\Sigma). +\end{align} +On cherche la distribution d'une fonction $h(\hat\theta_T)$. On +développe cette fonction sous la forme d'une série de Taylor et en +conservant seulement les deux premiers termes: +\begin{align} + h(\hat\theta_T) \approx h(\theta_0) + \nabla + \left[h(\theta_0)\right]'\cdot(\hat\theta_T - \theta_0). +\end{align} +On a donc, après quelques manipulations, que la variance de la +fonction $h(\hat\theta_T)$ est approximativement +\begin{align} + Var\left[h(\hat\theta_T) \right] \approx \nabla + \left[h(\theta_0)\right]' \left(\frac{\Sigma}{T}\right) \nabla + \left[h(\theta_0)\right]. +\end{align} + +La distribution de la fonction $h(\hat\theta_T)$ est alors +asymptotiquement +\begin{align} + \label{eq:deltamethodmult} + h(\hat\theta_T) \stackrel{L}{\longrightarrow} + N\left(h(\theta_0),\nabla \left[h(\theta_0)\right]' + \left(\frac{\Sigma}{T}\right) \nabla + \left[h(\theta_0)\right]\right). +\end{align} + + + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/annexe3.tex b/depotfinal/908144032/annexe3.tex new file mode 100644 index 0000000..a29e339 --- /dev/null +++ b/depotfinal/908144032/annexe3.tex @@ -0,0 +1,31 @@ +\chapter{Données} +\label{chap:donnees} + +Ce tableau présente l'échantillon de données utilisées au chapitre 9. + +\begin{table}[htbp] + \caption{Prix du titre Abbey National (penny sterling) du 31 juillet au 8 octobre 1991} + \begin{center} + \begin{tabular}{cccccccccccccc} + \hline + \textbf{Date} & \multicolumn{1}{c}{\textbf{Prix}} & & \textbf{Date} & \multicolumn{1}{c}{\textbf{Prix}} & & \textbf{Date} & \multicolumn{1}{c}{\textbf{Prix}} & & \textbf{Date} & \multicolumn{1}{c}{\textbf{Prix}} & & \textbf{Date} & \multicolumn{1}{c}{\textbf{Prix}} \\ \hline + 31/07 & 296 & & 14/08 & 294 & & 28/08 & 303 & & 11/09 & 296 & & 25/09 & 306 \\ + 01/08 & 296 & & 15/08 & 293 & & 29/08 & 304 & & 12/09 & 301 & & 26/09 & 303 \\ + 02/08 & 300 & & 16/08 & 295 & & 30/08 & 304 & & 13/09 & 298 & & 27/09 & 301 \\ + 05/08 & 302 & & 19/08 & 287 & & 02/09 & 309 & & 16/09 & 295 & & 30/09 & 303 \\ + 06/08 & 300 & & 20/08 & 288 & & 03/09 & 309 & & 17/09 & 295 & & 01/10 & 308 \\ + 07/08 & 304 & & 21/08 & 297 & & 04/09 & 309 & & 18/09 & 293 & & 02/10 & 305 \\ + 08/08 & 303 & & 22/08 & 305 & & 05/09 & 307 & & 19/09 & 292 & & 03/10 & 302 \\ + 09/08 & 299 & & 23/08 & 307 & & 06/09 & 306 & & 20/09 & 297 & & 04/10 & 301 \\ + 12/08 & 293 & & 26/08 & 307 & & 09/09 & 304 & & 23/09 & 294 & & 07/10 & 297 \\ + 13/08 & 294 & & 27/08 & 304 & & 10/09 & 300 & & 24/09 & 293 & & 08/10 & 299 \\ + \hline + \end{tabular} + \end{center} + \label{prixabbeyn} +\end{table} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/by-sa.pdf b/depotfinal/908144032/by-sa.pdf new file mode 100644 index 0000000000000000000000000000000000000000..575f606b7821f69f0f88297a8ea6138535cb643c GIT binary patch literal 6761 zcmb_h2{@E(_qV6X)?!OEQcYyciZP6B23Z?R*~^$2OpKXkv6m=I*;1BlQ4%4`TOkyZ 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zXsYz{nTg96I#v6aOp#8kUs(As?p)qN<-B>CooA$TqIEkgx;@6429pw~db8U*wk!qJu(Mmi^!BiV{trV}ON7NoIbY3n)l6 zT&#fpvoNqUG%&O@vNSb{GB8jFB26xR-~1H(N|2miQIwj-WdO953ldl$Qo+pB)Yw!Z zO#v!qXlMY88n8?rOw7Q{5>w359GDW()EO87!wgN#%p4edXkzA;z>q`}voObE2c~(3 z2Bt{nl@ujr=A;&Z3+Ld>s#Kug6!e4g^Gg&!Q3XuXo_T5cz$k^VU4W5lV`O2GY+-1U zVr*iTW@u=XW}KFkWNK+-oRpekY>;AXWXDBV2_#VzmjI&;9HxdwCYD^Ps;>TSTmY?q BHo*V@ literal 0 HcmV?d00001 diff --git a/depotfinal/908144032/chapitre1.tex b/depotfinal/908144032/chapitre1.tex new file mode 100644 index 0000000..a60d6d6 --- /dev/null +++ b/depotfinal/908144032/chapitre1.tex @@ -0,0 +1,834 @@ +\chapter{Les modèles de rendements financiers} + +\section{L'utilisation de modèles en finance} +\label{sec:utilisationmodeles} + +On doit considérer les implications de l'utilisation de modèles en +finance avant d'entreprendre leur étude. On doit aussi prendre +connaissance des différents types ainsi que les risques liés à chacun +d'entre eux. Pour ce faire, on se réfère à la note «Model Risk» +publiée par \cite{derman1996modelrisk}. + +Durant les dernières décennies, plusieurs modèles sont apparus afin de +fournir une approche fondamentale aux concepts de tarification, +d'offre et de demande et d'arbitrage aux intervenants des milieux +financiers. Au cours des années 1970, on se préoccupe particulièrement +des fluctuations des taux d'intérêt, un phénomène qui marque cette +époque. Les notions de duration et de convexité font alors leurs +débuts. Sur les marchés de capitaux propres, on s'intéresse à la +discordance entre le prix négocié des contrats à terme et le prix +raisonnable calculé selon une perspective théorique. + +Puis, la confiance développée envers le modèle de tarification +d'options de \cite{black1973pricing} et ses extensions a favorisé la +croissance du marché des produits dérivés. La puissance de calcul +croissante des ordinateurs a aussi permis l'élaboration et +l'utilisation de modèles de plus en plus sophistiqués. La dépendance +qui peut se développer envers ceux-ci apporte son lot de +considérations. On doit donc se rappeler l'utilisation désirée par les +auteurs de ceux-ci et le risque associé à leur usage à grande échelle. + +\subsection{Différents types de modèles} +\label{sec:differentsmodeles} + +Toujours selon Derman, un modèle financier peut être classé parmi au +moins trois catégories: + +\begin{enumerate} +\item Le \textbf{modèle fondamental}, basé sur un système de postulats + et de données, entre lesquels on peut établir différentes + relations. Le modèle de Black-Scholes en est un exemple. +\item Le \textbf{modèle phénoménologique}, qui présente une + description ou une analogie, afin d'illustrer quelque chose qui ne + peut être directement observé. C'est un modèle moins fondamental, + basé aussi sur des liens de cause à effet. Un modèle qui chercherait + à expliquer l'impact du retrait du porteur de parts majoritaire + d'une entreprise sur la valeur des actions de celle-ci serait + phénoménologique. +\item Le \textbf{modèle statistique}, basé sur une régression ou un + réglage optimal entre différents ensembles de données. On ne cherche + pas ici à expliquer une dynamique, mais à décrire une tendance ou + une corrélation. Le modèle d'évaluation des actifs financiers et + celui des trois facteurs de \cite{fama1993common} en sont des exemples. +\end{enumerate} + +Un modèle financier est en partie basé sur des variables qui +représentent des opinions et des anticipations, et non seulement des +quantités mesurables. Ces variables peuvent être, entre autres, le +rendement et la volatilité future espérés. Cette considération sera +importante notamment lorsque l'on voudra déterminer le prix +raisonnable d'un produit dérivé. En effet, un modèle de tarification +est essentiellement un moyen de refléter l'intuition des acteurs du +marché à propos de ces variables sous la forme d'un prix exprimé dans +une unité monétaire. Un bon modèle doit faciliter l'extrapolation de +ce prix sous certaines conditions de marché. + +Contrairement à la physique classique, un principe fondamental en +finance est l'incertitude. On ne peut anticiper la valeur d'un titre à +un moment donné dans le futur avec la même précision qu'on peut +prévoir la position d'un objet à cet instant. Les outils mathématiques +principalement utilisés seront alors les processus stochastiques, les +statistiques et les distributions de probabilités, en plus du calcul +différentiel et intégral. + +\subsection{Le risque de modélisation} +\label{sec:modelerisque} + +Plusieurs risques inhérents à la modélisation en finance +existent. Quelques-uns d'entre eux seront décrits dans cette section. + +La modélisation peut tout simplement ne pas être applicable à la +situation étudiée. L'exemple le plus probant serait de tenter de +prévoir les mouvements du prix d'un titre financier à court terme. + +Un modèle peut être incorrect pour plusieurs raisons. Entre autres, il +peut ignorer certains facteurs ou poser une hypothèse déterministe +inappropriée sur ceux-ci. Il peut aussi considérer une dynamique +incorrecte pour un des facteurs ou encore une relation inappropriée +entre ceux-ci. Enfin, il peut n'être applicable que sous certaines +conditions bien précises ou encore que son utilisation soit limitée à +court terme, notamment lorsqu'il nécessite un temps de calibration +pour être statistiquement valable. Il peut aussi être inutilisable par +une mauvaise estimation des paramètres. + +Un modèle peut aussi être correct, mais avoir une solution +erronée. Cela se produit notamment lorsqu'on tente de dériver une +solution analytique ou que l'on doit utiliser des méthodes numériques +pour obtenir celle-ci. On se doit, dans ce cas, de connaître l'erreur +maximale possible de la méthode utilisée. Un modèle correct peut aussi +être utilisé dans le mauvais contexte. Par exemple, on pourrait avoir +recours à des paramètres inadéquats de simulation, ou encore réutiliser le +modèle dans une autre situation sans tenir compte des +conditions de validité de celui-ci. + +Son utilisation peut génèrer des prix déraisonnables; on parle alors +d'arbitrage de modèle. Par exemple, si un titre est évalué à l'aide du +modèle d'évaluation des actifs financiers, son prix sera différent de +celui qui serait obtenu avec la régression à trois facteurs de Fama et +French. Un investisseur peut alors faire du profit en achetant le +titre à celui qui demande le prix le plus faible pour le revendre à +celui qui offre le plus élevé. + +L'utilisation de données instables peut produire des résultats +différents selon la période étudiée. La possibilité qu'une estimation +basée sur des données historiques soit erronée doit être considérée. + +Enfin, comme la plupart des modèles financiers sont implémentés sous +forme de logiciels, différents bogues informatiques peuvent se +retrouver dans le code source. On considère entre autres des erreurs +d'arrondissement, de logique et de clarté du code, ainsi que des +particularités du matériel qui n'auraient pas été prises en compte par +le programmeur. Ces erreurs peuvent être difficiles à détecter, c'est +pourquoi un grand nombre de tests devraient être effectués avant de +publier un logiciel de modélisation financière. + +\section{Les rendements financiers} +\label{sec:prixrendements} + +Le \textbf{rendement} est défini comme étant le gain ou la perte de +valeur d'un actif sur une période donnée. Il est constitué des revenus +occasionnés et des gains en capitaux d'un investissement et est +habituellement représenté sous la forme d'un pourcentage. Ces derniers +peuvent prendre la forme de coupons pour les titres à revenus fixes et +de dividendes pour les actions échangées sur les marchés boursiers. On +ne considèrera, dans ce texte, que les titres boursiers sans +dividende, dont le rendement est lié uniquement aux gains en capitaux. + +\subsection{Définitions et notations} +\label{sec:defrendements} + +On définit le prix $S(t)>0$ d'un titre financier observé au temps +$t$. Implicitement, le prix considéré est celui à la fermeture. On +définit aussi le taux de rendement effectif $R(t)$ sur une période +comprise dans l'intervalle de temps $\left[t-1,t\right]$. C'est le +taux composé continument, aussi appelé force d'intérêt, qui aurait +occasionné les mêmes gains ou pertes sur un montant déposé en banque +au cours de la période concernée. Le taux de rendement est la variable +d'intérêt dans le contexte de la modélisation financière. + +On associe le taux de rendement effectif à la différence entre le +logarithme du prix initial et final. Dans la situation où le taux de +rendement est déterministe et non aléatoire, on obtient l'équation +différentielle suivante: +\begin{align*} + \frac{dS(t)}{dt} &= R(t) \cdot S(t). +\end{align*} + +On peut interpréter cette équation en affirmant que la variation du +prix $dS(t)$ sur un intervalle de temps infiniment petit $dt$ est +proportionnelle à la valeur actuelle $S(t)$. Cette équation +différentielle a pour solution générale: +\begin{align} + \label{eq:solutiondiffrendement} + S(t) &= S(0)e^{R(t) \cdot t}. +\end{align} + +Afin de définir les propriétés de l'échantillon sélectionné, on pose +comme hypothèse: +\begin{hypothese} + Le rendement $R(t)$ est constant durant la période définie par + l'intervalle de temps $\left[t-1,t\right]$, mais il est différent + d'une à l'autre: $R(s) \neq R(t), s \neq t$. +\end{hypothese} + +On peut alors représenter le rendement $R(t)$ comme étant la +différence entre les logarithmes des prix observés au temps $t$ et +$t-1$, ou encore le logarithme du quotient de ces mêmes prix: +\begin{align} + R(t) &= \ln{(S(t))} - \ln{(S(t-1))} \nonumber\\ + &= + \ln{\left(\frac{S(t)}{S(t-1)}\right)}. \label{eq:rendementlogprix} +\end{align} + +On définit aussi le \textbf{rendement cumulé} $L(t)$. Il correspond à +la somme des rendements effectifs observés sur l'intervalle +$\left[0,t\right]$: +\begin{align} + L(t) &= \sum_{i=1}^{t} R(i) \nonumber\\ + &= \sum_{i=1}^{t} \left[\ln{(S(i)} - \ln{(S(i-1))}\right] \nonumber\\ + &= \ln{(S(t))} - \ln{(S(0))} \nonumber\\ + &= \ln{\left(\frac{S(t)}{S(0)}\right)}. \label{eq:rendementcumL} +\end{align} + +Cette représentation permet d'exprimer le prix actuel $S(t)$ en +fonction de la valeur initiale $S(0)$ sous une forme similaire à la +solution \eqref{eq:solutiondiffrendement}, mais tenant compte de +l'hypothèse émise précédemment: +\begin{align} + e^{L(t)} &= \frac{S(t)}{S(0)} \nonumber\\ + S(t) &= S(0) \cdot e^{L(t)} \nonumber\\ + &= S(0) \cdot \exp\left(\sum_{i=1}^{t} R(i)\right). +\end{align} + +\subsection{Rendements cumulés} +\label{sec:rendementscum} + +On pose l'hypothèse suivante: +\begin{hypothese} + Les rendements $R(i), i \in 1, \ldots, t$ sont indépendants, mais + pas nécessairement identiquement distribués. +\end{hypothese} + +On peut alors obtenir la distribution du rendement cumulé $L(t)$ en +utilisant le produit de convolution \eqref{eq:convocaract}. +Considérons $\phi_{R(i)}(\xi)$ la fonction caractéristique d'un +rendement $R(i)$ et $\phi_{L(t)}(\xi)$ celle du cumulé $L(t)$. On +obtient alors que cette dernière est égale au produit des fonctions +caractéristiques des rendements effectifs sur chacune des périodes de +l'intervalle $\left[0,t\right]$: +\begin{equation} + \label{eq:convolutionLR} + \phi_{L(t)}(\xi) = \prod_{i=1}^t \phi_{R(i)}(\xi). +\end{equation} + +On considère la situation où l'on posera plutôt l'hypothèse suivante: +\begin{hypothese} + Les rendements $R(i), i \in 1, \ldots, t$ sont à la fois + indépendants et identiquement distribués. +\end{hypothese} + +Alors, la fonction caractéristique des rendements est égale pour +chaque période: +\begin{align} + \label{eq:rendementsIID} + \phi_{R}(\xi) = \phi_{R(1)}(\xi) = \ldots &= \phi_{R(t)}(\xi). +\end{align} + +On peut donc simplifier l'expression \eqref{eq:convolutionLR} pour +obtenir la fonction caractéristique: +\begin{equation} + \label{eq:convolution2} + \phi_{L(t)}(\xi) = \left[\phi_{R}(\xi)\right]^t. +\end{equation} + +Considérer une distribution qui est fermée sous la convolution pour +modéliser les rendements sur une période $R(i)$ peut alors être +intéressant. Le rendement cumulé $L(t)$ pourra aussi être +modélisé à l'aide de la même distribution. Pour ce faire, on modifie +un paramètre d’échelle en fonction de la longueur $t$ de l'intervalle +de temps considéré. + +\subsection{Données disponibles} +\label{sec:donneesdisponibles} + +Les données disponibles auprès des fournisseurs d'informations +financières prennent habituellement la forme de séries chronologiques +discontinues. Celles-ci incluent les prix à l'ouverture, le plus bas +et le plus élevé au courant de la journée ainsi qu'à la fermeture, +pour chaque jour où les marchés financiers sont en activité. Afin de +mesurer le rendement quotidien d'un titre, seuls les prix à la +fermeture seront considérés. + +\section{Les premiers modèles} + +\subsection{Le modèle de Bachelier} +\label{sec:bachelier} + +Un des premiers modèles proposés afin de représenter les rendements +financiers a été celui de \cite{bachelier1900theorie} . + +Le prix d'un titre peut varier, durant une période, de n'importe +quelle valeur comprise dans l'intervalle $\left[ -S(t),\infty +\right]$. Il propose donc que cet intervalle soit remplacé par +l'ensemble du domaine réel $\mathbb{R}$. La probabilité que le titre +atteigne une valeur nulle ou négative ou que celle-ci double devrait +donc être négligeable. Il ajoute aussi que la variation est +indépendante du prix actuel du titre $S(t)$ et que la distribution de +probabilités de celle-ci est symétrique et centrée en ce point. + +Il utilise le principe selon lequel la probabilité que deux évènements +indépendants consécutifs aient lieu est le produit de celles que +chacun d'entre eux se réalise, pour établir la distribution des +variations du prix. Par exemple, la variation du prix sur une première +période prend la valeur $x$ et celle sur une seconde, $z-x$, comme +illustré à la figure \ref{fig:bachelier1}. + +%% ligne du temps +\begin{figure}[!ht] + \centering + \begin{tikzpicture} + \draw [->] (0,0) -- (6,0); \foreach \x in {0,2,4} \draw (\x + cm,3pt) -- (\x cm,-3pt); \draw (0,0) node[below=3pt] {$ 0 $} + node[above=3pt] {$ S(0) $}; \draw (2,0) node[below=3pt] {$ t_1 $} + node[above=3pt] {$ S(0)+x $}; \draw (4,0) node[below=3pt] {$ + t_1+t_2 $} node[above=3pt] {$ S(0)+z $}; + \end{tikzpicture} + \caption{Modèle de Bachelier: probabilité composée} + \label{fig:bachelier1} +\end{figure} + +On définit $f(x,t)$ la fonction de densité de la variation du prix +$S(t)$ par rapport au niveau initial $S(0)$. Alors, selon le principe +précédent, on obtient l'expression +\begin{align} + \label{eq:probcomposeeB} + f(z,t_1+t_2) = \int_{-\infty}^{\infty} f(x,t_1)\cdot f(z-x,t_2) + \cdot dx. +\end{align} + +La solution proposée est que la densité de probabilité soit de la forme +\begin{align*} + \label{eq:formeprobB} + f(x,t) = A \cdot \exp \left\{-B^2x^2 \right\}. +\end{align*} + +Afin que la fonction $f(x,t)$ soit une densité de probabilité, la +condition suivante doit être respectée: +\begin{align} + \int_{-\infty}^{\infty} A \cdot \exp \left\{-B^2x^2 \right\} dx = 1. +\end{align} + +Ceci implique que +\begin{align*} + B&= A\sqrt{\pi}. +\end{align*} + +En posant $x=0$, on a $A=f(0,t)$ et l'on en déduit: +\begin{align} + f(x,t) = f(0,t) \cdot \exp \left\{-\pi \cdot f(0,t)^2 \cdot x^2 + \right\}. +\end{align} + +En reprenant l'intégrale \eqref{eq:probcomposeeB}, on obtient que la +densité de probabilité $f(z,t_1+t_2)$ soit aussi de la forme +\eqref{eq:formeprobB}: +\begin{align} + f(z,t_1+t_2) = \frac{f(x,1)f(z-x,2)}{\sqrt{f(x,1)^2+f(z-x,2)^2}} + \exp \left\{-\pi \frac{f(x,1)f(z-x,2)}{f(x,1)^2+f(z-x,2)^2} z^2 + \right\}. +\end{align} + +On reconnaitra que cette densité est, à un changement de variable +près, une loi normale. La démarche suggère qu'il recherchait une +distribution qui était fermée sous la convolution, une propriété +souhaitable pour un modèle cohérent des rendements financiers. + +Ce modèle implique un processus de Wiener-Bachelier selon lequel les +incréments, ou les changements de prix, suivent une distribution +normale: +\begin{equation} + \label{eq:bachelier00} + S(T)-S(t) \sim N\left(0,\sigma^2 \left(T-t\right)\right). +\end{equation} + +On doit noter que ce modèle implique que la variance des fluctuations +n'est pas proportionnelle au prix initial. Une première correction +sera apportée au modèle afin de considérer le logarithme du prix. Ce +changement permettra d'obtenir un modèle où elle est désormais +proportionnelle au prix initial. Le processus du prix suivra alors un +mouvement brownien géométrique: +\begin{align} + \label{eq:browniengeom} + S(T)-S(t) &\sim LN \left(0, \sigma (T-t) \right). +\end{align} + +Le logarithme du prix suivra alors un processus de Wiener-Bachelier: +\begin{align} + \label{eq:bachelierwiener2} + \ln\left(S(T)\right)-\ln\left(S(t)\right) &\sim N\left(0,\sigma + \left(T-t\right)\right). +\end{align} + +Un des principaux avantages du processus de Bachelier modifié est que +le rendement cumulé $L(t)$ est aussi une variable aléatoire +gaussienne. Cette propriété est appelée L-stabilité ou invariance sous +l'addition. La distribution gaussienne est la seule ayant cette +propriété où le second moment est fini. Le sujet des distributions L +stables sera aussi abordé à la section \ref{sec:mandelbrot}. + +Quelques années après sa publication, ce modèle est l'objet de +critiques de la part d'économistes et de financiers. En se référant à +\cite{mitchell1916critique}, on observe que, sur une base annuelle, +les variations négatives par rapport à la moyenne (149) sont plus +fréquentes que celles qui sont positives (126), pour un ensemble de 40 +titres boursiers, entre 1890 et 1915 (figure \ref{fig:mitchell1}). +Une asymétrie négative des rendements sera alors présente. +\begin{figure}[!ht] + \centering + \includegraphics[scale=0.75]{./graphiques/mitchell1.pdf} + \caption{Distribution des rendements annuels de 40 titres boursiers, + de 1890 à 1915, Table XVIII de \cite{mitchell1916critique}} + \label{fig:mitchell1} +\end{figure} + +De plus, les variations extrêmes sont plus fréquentes que ne pourrait +le prédire un modèle basé sur un mouvement brownien. La distribution +des rendements aurait donc des queues plus épaisses +\footnote{traduction de l'anglais heavy tailed} que la normale. On +doit trouver un modèle qui permet de tenir compte de ces +particularités. + +\subsection{Proposition de Mandelbrot} +\label{sec:mandelbrot} + + +\cite{mandelbrot1963variation} propose un modèle qui vise à combler +les lacunes du processus brownien géométrique +\eqref{eq:browniengeom}. Il explique que les distributions empiriques +des changements de prix sont habituellement trop \emph{pointues} pour +être considérées comme des échantillons d'une population gaussienne. + +Il identifie différentes caractéristiques qu'un bon modèle des +rendements financiers devrait posséder: + +\begin{enumerate} + \label{enum:mandelbrot} +\item Il doit tenir compte de la fréquence des grands changements de + prix. Il doit donc être basé sur une distribution leptocurtique, + plus pointue au centre que la normale. +\item Il doit permettre des changements instantanés et imprévisibles + de toute amplitude. +\item Il doit admettre une probabilité non nulle que plusieurs + changements consécutifs semblent corrélés. +\item Il doit admettre un processus de prix non stationnaire, car la + variance échantillonnale prend différentes valeurs à travers le + temps. +\end{enumerate} + +La famille de distributions L stables semble être celle qui répond le +mieux à l'ensemble de ces conditions \citep{walterlevy}. L'équation +suivante définit la propriété de L-stabilité de la distribution de la +variable aléatoire des rendements sur une période $R$: +\begin{align} + (a_1 R_1 + b_1) + (a_2 R_2 + b_2) &\stackrel{d}{=} aR + b \\ + \forall a_1,a_2 > 0, \forall b_1, b_2. +\end{align} + +La solution générale de cette équation a été découverte par Lévy en +1925. Le logarithme de la fonction caractéristique de celle-ci prend +la forme suivante: +\begin{align} + \ln{(\phi_{R}(\xi))} = i\delta \xi - \gamma |\xi|^{\alpha} + \left[1+\frac{i\beta \xi}{|\xi|} \tan{\frac{\alpha\pi}{2}} \right]. +\end{align} + +Le domaine et le rôle des paramètres de la distribution L stable sont +décrits à la table \ref{tab:roleparam}. La flexibilité apportée par les +quatre paramètres permet de remplir les quatre conditions établies au +début de cette section. De plus, l'absence, dans la majorité des cas, +de moments finis d'ordre supérieur à l'espérance permet de tenir +compte du mouvement erratique des prix et ainsi produire de larges +discontinuités de son processus. Elle permet aussi d'expliquer +l'apparence de corrélation sérielle, en considérant une probabilité +non négligeable que cette caractéristique soit présente. Cependant, ce +modèle est difficile à appliquer à l'évaluation de produits dérivés +pour cette raison, étant donné que l'on devra être en mesure de +quantifier la volatilité. +\begin{table}[!ht] + \centering + \begin{tabular}{|c|p{1.75cm}|p{2.5cm}|p{6.25cm}|} + \hline + \textbf{Paramètre} & \textbf{Domaine} & \textbf{Rôle} & \textbf{Observations} \\ + \hline + $\alpha$ & $\left]0,2\right]$ & Aplatissement & Plus sa valeur est petite, plus la distribution est leptocurtique. $\alpha=2$ correspond à la distribution normale. \\ + $\beta$ & $\left] -1, 1 \right]$ & Asymétrie & Défini seulement lorsque $\alpha \neq 1$. Lorsque $\alpha=1$ et $\beta=0$, on obtient la distribution de Cauchy. \\ + $\gamma = s^{\alpha}$ & $\mathbb{R}\setminus\{0 \}$ & Échelle & On doit prendre la racine $\alpha$ pour obtenir un paramètre d'échelle $s$ tel que défini par Pearson. \\ + $\delta$ & $\mathbb{R}$ & Localisation & \\ + \hline + \end{tabular} + \caption{Domaine et rôle des paramètres de la distribution L stable de Mandelbrot} + \label{tab:roleparam} +\end{table} + +L'approche classique, selon Mandelbrot, pour expliquer les grands +changements de prix a été de considérer un mélange de deux +distributions normales, dont une pour les fluctuations régulières et +une qui a une variance plus importante, pour les discontinuités. Il +remarque que pour expliquer adéquatement le comportement des données +empiriques, on doit introduire un mélange de plusieurs distributions +normales, ce qui rendrait le modèle plus complexe. Par contre, on +retrouve une approche intéressante avec le modèle présenté à la +section suivante. + +\subsection{Le modèle de Press} +\label{sec:press} + +\cite{press1967compound} propose un modèle statistique basé sur un +processus de Poisson composé auquel on ajoute un mouvement brownien +$W(t)$. C'est donc d'un processus ayant des incréments stationnaires +et indépendants. Il présente donc les caractéristiques d'un processus +de Lévy. Press utilise aussi la transformation logarithmique +\eqref{eq:browniengeom} afin que la variation soit proportionnelle au +prix. Il remarque aussi que le modèle logarithmique de Bachelier est +inadéquat, car il ne tient pas compte des queues de la distribution +empirique des rendements qui sont plus épaisses que celles de la +normale. Il ajoute que le modèle proposé par Mandelbrot est +discutable, car il ne trouve aucune évidence, à partir des données +observées, que la distribution de la population aurait une variance +infinie. + +Le processus de Poisson $\left\{N(t)\right\}$ de paramètre $\lambda t$ +est un processus de comptage qui détermine les occurrences des sauts +$Y_k, k = 1, \ldots, N(t)$. Ces sauts surviennent généralement +lorsqu'une information importante est rendue publique par rapport à un +titre. Ceux-ci sont aussi de distribution normale, mais leur espérance +n'est pas nulle et leur variance est différente de celle du processus +$W(t)$. Cette composante que l'on ajoute au modèle de Bachelier +modifié permet d'expliquer les variations plus importantes et moins +fréquentes observées empiriquement. + +Le processus du logarithme du prix $\left\{s(t)\right\} \equiv +\left\{\ln{(S(t))}\right\}$ est donc représenté par l'équation +suivante: +\begin{align} + \label{eq:press67} + s(t) &= s(0) + \sum_{k=1}^{N(t)} Y_k + W(t). +\end{align} + +On définit les différentes variables aléatoires composant le processus +comme suit: +\begin{align*} + Y_k &\sim N(\theta,\sigma_2^2) \\ + W(t) &\sim N(0,\sigma_1^2 t) \\ + N(t) &\sim Poisson(\lambda t) +\end{align*} + +Comme pour la plupart des processus de Lévy, on ne peut obtenir une +forme explicite pour la fonction de densité, car celle-ci se présente +sous la forme d'une série infinie. On représente alors ces processus +par leur fonction caractéristique, formée par le produit de celles de +leurs différentes composantes. + +La distribution du logarithme du prix $s(t)$ est définie par la +fonction caractéristique $\phi_{s(t)}(\xi)$, qui est le produit de +celle de la constante et celles des processus de Wiener et de Poisson +composé: +\begin{align} + \label{eq:fncaractpress} + \phi_{s(t)}\left(\xi\right) &= E\left[e^{i \xi s(t)} \right] \nonumber \\ + &= exp\left\{ i\xi \cdot s(0) \right\} \times exp \left\{ -\frac{t \sigma_1^2 \xi^2}{2} \right\} \times exp \left\{ \lambda t \left[e^{i \theta \xi-(\sigma_2^2 \xi^2/2)}-1 \right] \right\} \nonumber \\ + &= exp\left\{ i\xi \cdot s(0)- \frac{t}{2}\sigma_1^2\xi^2 + \lambda + t \left[e^{i \theta \xi-(\sigma_2^2 \xi^2/2)}-1 \right] \right\}. +\end{align} + +Afin d'estimer le modèle, on s'intéressera plutôt à la distribution +d'un incrément $\Delta s(t) = s(t)-s(t-1)$ de ce processus. La +fonction caractéristique $\phi_{\Delta s(t)}(\xi)$ de cette variable +aléatoire peut être facilement identifiée à partir de celle du +processus \eqref{eq:fncaractpress}. Essentiellement, on pose $s(0)=0 +\mbox{ et } t=1$, pour obtenir: +\begin{align} + \label{eq:fncaractpress2} + \phi_{\Delta s(t)}\left(\xi\right) &= E\left[e^{i\xi\Delta s(t)} \right] \nonumber \\ + &= exp\left\{-\frac{\sigma_1^2 \xi^2}{2} + \lambda \left[e^{i + \xi\theta -(\sigma_2^2 \xi^2/2)}-1 \right] \right\}. +\end{align} + +Pour estimer les paramètres du modèle, on privilégie la méthode des +cumulants, qui est similaire à la méthode des moments. Considérons les +quatre premiers cumulants de la distribution de l'incrément $\Delta +s(t)$: +\begin{subequations}\label{eq:cumulantspress} + \begin{align} + K_1 &= \lambda\theta \\ + K_2 &= \sigma_1^2+\lambda(\theta^2+\sigma_2^2) \\ + K_3 &= \lambda\theta(\theta^2+3\sigma_2^2) \\ + K_4 &= \lambda(\theta^4 + 6 \theta^2 \sigma_2^2 + 3 \sigma_2^4). + \end{align} +\end{subequations} + +En utilisant les quatre premiers cumulants empiriques +\eqref{eq:cumulantsempiriques}, on obtient les équations suivantes: +\begin{subequations}\label{eq:presscum} + \begin{align} + 0 &= \hat\theta^4 - \frac{\overline{K}_3}{\overline{K}_1} \hat\theta^2 + \frac{3\overline{K}_4}{2\overline{K}_1} \hat\theta - \frac{\overline{K}_3^2}{2\overline{K}_1^2} \label{eq:presscumtheta}\\ + \hat\lambda &= \frac{\overline{K}_1}{\hat\theta} \label{eq:presscumlambda}\\ + \hat\sigma_2^2 &= \frac{\overline{K}_3-\hat\theta^2\overline{K}_1}{3\overline{K}_1} \label{eq:presscumsigma2}\\ + \hat\sigma_1^2 &= \overline{K}_2 - + \frac{\overline{K}_1}{\hat\theta}\left(\hat\theta^2 + + \frac{\overline{K}_3 - \overline{K}_3 \theta^2}{3\overline{K}_1} + \right). \label{eq:presscumsigma1} + \end{align} +\end{subequations} + +En résolvant numériquement l'équation \eqref{eq:presscumtheta} pour le +$\hat\theta$, puis par substitutions successives dans les équations +\eqref{eq:presscum}, on obtient des estimateurs convergents pour les +quatre paramètres du modèle. + +Un modèle similaire a aussi été présenté par \cite{merton1976option}, +cependant, il inclut un paramètre de dérive $\alpha$, et considère que +les sauts $Y$, qui sont des facteurs multiplicatifs, peuvent suivre une autre distribution que la normale. Il +présente le modèle sous la forme d'une équation différentielle +stochastique: +\begin{align} + \label{eq:modelemerton} + \frac{dS}{S} = (\alpha - \lambda k)dt + \sigma dW + dq. +\end{align} +La constante $k$ représente l'espérance de la variation relative si un +saut se produit et $q$, le processus de Poisson composé. La solution +de cette équation est, selon le lemme d'Itô: +\begin{align} + S(t) &= \tilde{S}(0) \exp \left\{ + (\alpha-\frac{1}{2}\sigma^2-\lambda k)t + + \sigma W(t) \right\} + \end{align} + où + \begin{align} + \tilde{S}(0) &= \begin{cases} + S(0) & \text{si } N(t) = 0\\ + S(0) \sum_{k=1}^{N(t)} Y_k & \text{si } N(t) \geq 1. \nonumber + \end{cases} +\end{align} + +En spécifiant un paramètre de dérive $\delta = +\alpha-\frac{1}{2}\sigma^2-\lambda k$ et en considérant que les +sauts $Y$ sont de distribution lognormale, on peut réécrire la +fonction caractéristique d'un incrément \eqref{eq:fncaractpress2} du +modèle de Press: +\begin{align} + \label{eq:fncaractmerton} + \phi_{\Delta s(t)}\left(\xi\right) &= E\left[e^{i\xi\Delta s(t)} \right] \nonumber \\ + &= \exp\left\{i\delta \xi -\frac{\sigma^2 \xi^2}{2} + \lambda + \left[e^{i \xi\theta -(\sigma^2 \xi^2/2)}-1 \right] \right\}. +\end{align} + +L'utilisation de ce modèle présente deux désavantages. L'estimation du +modèle est difficile lorsque la moyenne s'approche de 0, car le +quotient \eqref{eq:presscumlambda} tend alors vers une +indétermination. De plus, contrairement à d'autres modèles, il est +difficile d'identifier le rôle des paramètres par rapport à un moment +en particulier (classification de Pearson), contrairement à ce qu'on +pourra observer avec la distribution de Laplace asymétrique +généralisée. + +\subsection{Le modèle de Praetz} + +\cite{praetz1972distribution} propose un modèle inspiré par la +physique des particules. Il pose comme hypothèse que deux intervalles +qui ne se chevauchent pas forment une marche aléatoire, et que les +éléments qui composent la séquence des rendements financiers $\left\{ + R(t) \right\}$ sont mutuellement indépendants. Il considère qu'un +état stable existe où les rendements suivent une loi normale de +paramètres $\mu$ et $\sigma^2$. + +Cependant, cet état stable n'est jamais réellement atteint, et la +fonction de densité empirique généralement observée suppose une +distribution symétrique concave, pointue au centre et ayant des queues +épaisses. Il fait une analogie entre la température d'un gaz et le +niveau d'activité sur les marchés, où la variance du mouvement +brownien est proportionnelle à ces deux quantités. Il propose que le +paramètre de variance de la normale $\sigma^2$ suive une distribution +$g(\sigma^2)$ ayant un support positif. La distribution conditionnelle +est normale lorsque ce paramètre est connu. +\begin{align} + h_{R(t)}(r) &= \int_0^{\infty} f_{R(t)}(r|\sigma^2) g(\sigma^2) d\sigma^2 \\ + f_{R(t)}(r|\sigma^2) &= \frac{1}{\sigma\sqrt{2\pi}}exp\left\{-\frac{(r-\mu)^2}{2\sigma^2} \right\} \label{eq:praetz72} +\end{align} + +Il propose comme solution acceptable pour la densité $g(\sigma^2)$, la +distribution gamma inverse de paramètres $m$ et $s^2$: +\begin{align} + \label{eq:gpraetz} + g(\sigma^2) &= + \frac{s^{2m}(m-1)^me^{-(m-1)\frac{s^2}{\sigma^2}}}{\sigma^{2(m-1)}\Gamma(m)}. +\end{align} + +Cette distribution a pour moyenne $s^2$ et variance $\frac{s^2}{m-2}$. +La distribution non conditionnelle des rendements $h_{R(t)}$ est +approximativement une Student avec $2m$ degrés de liberté à un facteur +d'échelle de $\left(\frac{m}{m-1}\right)^{1/2}$ près: +\begin{align} + \label{eq:hpraetz} + h_{R(t)}(r) &= \frac{\Gamma(m)\left[\ 2(m-1)\pi + \right]^{1/2}s}{\left[1+\frac{(y-\mu)^2}{s^2(2m-2)} + \right]^{m+1/2}}. +\end{align} + +D'autres distributions pourraient être utilisées au lieu de la gamma +inverse. En utilisant la loi gamma, on obtient la distribution de +Laplace asymétrique généralisée, qui sera l'objet d'une étude +approfondie aux chapitres suivants. Il propose enfin d'utiliser aussi +la distribution a priori gamma inverse pour le paramètre $\mu$. Par +contre, il remarque qu'il obtient aussi une distribution similaire à +celle de Student. Cette généralisation n'est donc pas nécessaire. + +\section{Conditions essentielles de Madan et Seneta} +\label{sec:madanseneta90} + +Inspirés par les travaux de Mandelbrot, Press et Praetz, +\cite{madan1990variance} présentent un ensemble de conditions +considérées essentielles dans l'élaboration d'un modèle de rendements +financiers. Ils se baseront sur celles-ci pour proposer le modèle +Variance Gamma: + +\begin{enumerate} +\item La distribution des rendements $R$ doit avoir une queue + épaisse. Ainsi, la probabilité que cette variable aléatoire ait une + valeur supérieure à $r+t$ avec un $t$ petit, sachant qu'elle est + supérieure à $r$, doit tendre vers 1, ce qui signifie que la + fonction de survie converge lorsque cette quantité est grande. + \begin{eqnarray} + \label{eq:condmadan1} + \lim_{r\rightarrow \infty} P\left[R > r+t | R > r \right] &=& 1 \\ + \bar{F}(r+t) &\sim& \bar{F}(r), \qquad r \rightarrow \infty \nonumber + \end{eqnarray} +\item La distribution doit posséder des moments finis pour les $n$ + premières puissances des rendements $R$. Étant donné que l'on + cherche à modéliser la queue de la distribution, on fixe $n=4$. + \begin{equation} + \label{eq:condmadan2} + E\left[R^k\right] < \infty, \qquad k \in \lbrace 1,2,3,4 \rbrace + \end{equation} +\item + \begin{enumerate} + \item Le modèle doit proposer un processus de temps continu ayant + des accroissements stationnaires et indépendants. + \item Les distributions des accroissements doivent appartenir à la + même famille, quelle que soit leur longueur. Cette condition est + essentielle afin de permettre l'échantillonnage et l'analyse des + séries chronologiques. + \end{enumerate} +\item Le modèle doit permettre une extension multivariée avec une + distribution elliptique afin de conserver la validité du modèle + d'évaluation des actifs financiers. +\end{enumerate} + +Chacun des modèles présentés précédemment respecte la majorité ou +toutes ces conditions. Les résultats se retrouvent à la table +\ref{tab:condmadan}. +\begin{table}[!ht] + \centering + \begin{tabular}{ccccc} + & \multicolumn{4}{c}{\textbf{Conditions}} \\ + \hline + \textbf{Modèles} & 1 & 2 & 3 & 4 \\ + \hline + Mouvement brownien de Bachelier & & $\ast$ & $\ast$ & $\ast$ \\ + Distribution stable symétrique de Mandelbrot & $\ast$ & & & $\ast$ \\ + Processus de Poisson composé de Press & $\ast$ & $\ast$ & $\ast$ & $\ast$ \\ + Mélange gaussien/inverse gamma de Praetz & $\ast$ & $\ast$ & & $\ast$ \\ + Modèle Variance Gamma de Madan et Seneta & $\ast$ & $\ast$ & $\ast$ & $\ast$ \\ + \hline + \end{tabular} + \caption{Respect des conditions émises par Madan et Seneta pour les différents modèles présentés} + \label{tab:condmadan} +\end{table} + +On remarque que le modèle de Press remplit toutes les conditions +émises par Madan et Seneta. Cependant, ils remarqueront que ce n'est +pas un processus de sauts, car il contient aussi une composante de +diffusion (Section \ref{sec:levykhintchine}), ce qui va à l'encontre +de l'intuition derrière la continuité de la trajectoire du prix. C'est +cette dernière observation qui les incitera à proposer le modèle +Variance Gamma, qui est un processus de sauts. Ce modèle, aussi étudié +sous le nom de distribution de Laplace asymétrique généralisée par +\cite{kotz2001laplace}, a acquis beaucoup de notoriété dans le domaine +de la finance mathématique. De plus, avec le développement de +l'informatique et des méthodes numériques, on peut maintenant utiliser +de manière efficace la fonction caractéristique dans le cadre de la +calibration, des tests statistiques et de la tarification +d'options. C'est pourquoi un intérêt particulier est apporté à cette +distribution dans ce texte. + +% \section{La volatilité} +% \label{sec:volatilite} + +% La \textbf{volatilité} est une mesure de l'ampleur des variations du +% prix d'un actif. Elle sert à quantifier le risque lié à un +% investissement, le plus souvent sur un horizon à court terme. Elle se +% calcule le plus souvent à partir des prix des options observés sur les +% marchés, on parle alors de volatilité implicite. Comme elle n'est pas +% mesurable, cette volatilité est le reflet de l'anticipation des +% investisseurs quant aux perspectives du marché. + +% Cependant, on peut toujours mesurer la volatilité historique du prix +% d'un titre à travers les rendements passés. La première étude à ce +% sujet a été faite par Black et Scholes, les auteurs du célèbre modèle +% qui porte désormais leur nom. Ils ont conclu que leur modèle +% surestimait le prix des options pour des actifs sous-jacents ayant une +% volatilité historique élevée, le contraire se produisait lorsqu'elle +% l'était peu. Leur modèle est donc utile à condition que les +% investisseurs puissent faire de bonnes prévisions +% \citep{musiela2005martingale}. + +% De plus, comme il a été expliqué précédemment, les données historiques +% démontrent que la volatilité n'est pas constante avec le temps, mais +% plutôt aléatoire. Dans cette perspective, on pourrait identifier la +% distribution de la volatilité à travers le temps. +% \subsection{Mesure de la volatilité historique} +% \label{sec:mesurevolatilite} + +% Afin d'obtenir un ensemble d'observations de la volatilité historique, +% on utilise une approche par fenêtre mobile. \cite{randal2004non} +% considère un estimateur de variance mobile $\hat\sigma^2(t)$, basé sur +% les rendements centrés $R(t) = Y(t)-E\left[Y(t)\right]$ de la forme +% \begin{align} +% \hat\sigma^2(t) = \frac{1}{2r+1} \sum_{j=-r}^r R(t+j)^2, \qquad +% t\in\left[r+1,n-r\right] \label{mobilevariance} +% \end{align} + +% Étant donné la taille limitée $n$ de l'échantillon des rendements, on +% doit faire un compromis entre la précision des observations de la +% volatilité et le nombre $n-2r$ de celles-ci. L'hypothèse de volatilité +% stochastique fera pencher en faveur d'une fenêtre étroite, ce qui +% procurera un grand nombre d'observations la décrivant à court +% terme. On pourra donc ajuster une distribution de probabilités à +% celles-ci. + +% \subsection{Biais de la volatilité implicite} +% \label{sec:impvolsmile} + +% Le biais de volatilité implicite est un concept qui explique pourquoi +% la volatilité des options croît lorsque le prix d'exercice s'éloigne +% de la valeur actuelle du titre sous-jacent. Selon +% \cite{hull1999options}, ce phénomène a été remarqué sur les marchés +% financiers américains à partir du krach boursier du lundi noir +% \footnote{19 octobre 1987}, et n'est toujours pas entièrement +% expliqué. Généralement, on observe que, pour les options sur indices +% boursiers et taux de change, la courbe de volatilité implicite est +% plutôt symétrique, alors qu'elle est asymétrique pour celles sur +% actions (voir figure \ref{fig:volimplicite} pour un exemple). +% \begin{figure}[!ht] +% \centering +% \includegraphics[]{./bbry-echeance-06-2013-20mai2013.png} +% \caption{Courbe de volatilité implicite, titre BBRY, option d'achat +% avec échéance 06-2013, observée le 20-05-2013, prix de 15.83, +% source: \cite{thevolskew}} +% \label{fig:volimplicite} +% \end{figure} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/chapitre2.tex b/depotfinal/908144032/chapitre2.tex new file mode 100644 index 0000000..c9d440e --- /dev/null +++ b/depotfinal/908144032/chapitre2.tex @@ -0,0 +1,979 @@ +\chapter{La distribution de Laplace asymétrique + généralisée} % numéroté + +Dans ce chapitre, on présente, en premier lieu, le processus de +Laplace ainsi que les deux processus sous-jacents à sa construction, +le processus gamma et le processus de Wiener. Ensuite, on présente la +distribution de Laplace asymétrique généralisée et ses principales +propriétés qui seront utilisées pour modéliser les rendements de +titres financiers. Puis, on présente quelques cas particuliers. Dans +le chapitre suivant, on présentera différentes méthodes pour obtenir +une approximation de la fonction de densité et la fonction de +répartition. + +La distribution de Laplace asymétrique généralisée a été +principalement étudiée par \cite{kozubowski1999class}. Cependant, elle +a été introduite près d'une décennie auparavant par +\cite{madan1990variance}, sous le nom de distribution Variance +Gamma. La différence entre les approches des deux auteurs est +majeure. \cite{madan1990variance} développent un modèle financier à +partir du mouvement brownien géométrique, qu'ils généralisent en +proposant que la variance suive une distribution +gamma. \cite{kozubowski1999class} généralisent la distribution de +Laplace asymétrique. Leur approche est plus générale, car ils ne +cherchent pas à développer un modèle financier, mais une nouvelle +classe de distributions utilisable dans divers domaines +scientifiques. Étant donné leur approche plus détaillée et plus +intuitive, c'est leur formulation du modèle qui sera développée. On +rappellera enfin que les deux modèles sont équivalents même si leurs +paramétrisations sont différentes. + +\section{Le processus de Laplace} +\label{sec:processusGAL} + +Le processus de Laplace est défini comme étant un processus de Wiener +subordonné par un processus gamma. En d'autres termes, c'est un +processus de Wiener évalué à des temps aléatoires déterminés par un +processus gamma. Selon \cite{kotz2001laplace}, ce dernier est à la +distribution de Laplace ce que le mouvement brownien est à la loi +normale. Il est aussi un cas particulier des processus de Lévy, et en +conserve donc la principale propriété, celle d'être infiniment +divisible. + +Il a certains points en commun avec le mouvement brownien dont des +moments finis pour tout ordre et des incréments indépendants et +stationnaires. Cependant, la plupart des caractéristiques diffèrent: +\begin{itemize} +\item Discontinuité des trajectoires (processus de sauts); +\item Distribution asymétrique des accroissements; +\item Paramètres d'échelle et de temps entièrement dissociés. +\end{itemize} + +Enfin, il possède une représentation alternative qui n'implique aucun +processus de Wiener. Il peut en fait être représenté comme la +différence de deux processus gamma indépendants. On peut le +représenter en utilisant la forme générale des processus de Lévy. + +\subsection{Le processus gamma} +\label{sec:processusgamma} + +Le processus gamma, noté $\left\{G(t;\tau,\beta)\right\}$, est un +processus de sauts purs (donc aucune composante de dérive ni de +diffusion) dont les incréments $G(t+1;\tau,\beta) - G(t;\tau,\beta)$ +suivent une distribution gamma de paramètres de forme $\tau$ et +d'échelle $\beta$, définie par les fonctions de densité +$f_{\tau,\beta}(x)$ et caractéristique $\phi_{\tau,\beta}(\xi)$: +\begin{align} + f_{\tau,\beta}(x) &= \frac{\beta^\tau}{\Gamma(\tau)} x^{\tau \,-\, 1} e^{- \beta x } 1_{\lbrace x\geq\,0 \rbrace} \label{eq:densitegamma} \\ + \phi_{\tau,\beta}(\xi) &= E\left[e^{i\xi\,X} \right] \nonumber\\ + &= \int_{0}^{\infty} e^{i\xi\,x} f_{\tau,\beta}(x) dx \nonumber\\ + &= + \frac{1}{\left(1-\frac{i\xi}{\beta}\right)^{\tau}} \label{eq:fncaractgamma}. +\end{align} + +On s'intéresse à la situation où le paramètre d'échelle est de valeur +unitaire ($\beta=1$). Le processus gamma agit alors à titre de +compteur et sa valeur $G(t;\tau,\beta=1)$ au temps $t$ correspondra au +nombre de sauts depuis $t=0$. La fonction de densité +$G(t+1;\tau,\beta=1) - G(t;\tau,\beta=1)$ sera alors: +\begin{align} + f_{\tau,\beta=1}(x) &= \frac{1}{\Gamma(\tau)} x^{\tau \,-\, 1} e^{- x} 1_{\lbrace x\geq\,0 \rbrace}. \label{eq:densitegamma1} +\end{align} + +Le paramètre $\tau$, qui définit la forme de la distribution, +déterminera la fréquence moyenne des sauts du processus gamma +$\Gamma(t;\tau,\beta=1)$, étant donné l'espérance $E[G(t)] = \tau\cdot +t$. La fonction caractéristique de ce processus sera donc +$\phi(\xi,t;\tau,\beta=1)$, en utilisant la propriété de convolution +\eqref{eq:convocaractIID} (même si le temps $t$ n'est pas entier, car +la distribution est infiniment divisible): +\begin{align} + \phi(\xi,t;\tau,\beta=1) &= \left[\frac{1}{\left(1-\frac{i\xi}{1}\right)^{\tau}}\right]^t \nonumber\\ + &= \frac{1}{\left(1-i\xi\right)^{\tau\cdot t}}. +\end{align} + +On peut réécrire la fonction caractéristique d'un incrément de ce +processus $\phi(\xi;t=1;\tau,\beta=1)$ sous la représentation de +Lévy-Khintchine \eqref{eq:levykhintchine}, avec l'exposant +caractéristique $\Xi(\zeta)$: +\begin{align} + \label{eq:exposantchargamma} + \Xi(\zeta;t=1;\tau,\beta=1) &= \tau\ln{\left(1-i\zeta\right)} \\ + &= \tau \left(e^{0} - e^{-\infty}\right) \ln{\left(1-i\zeta \right)} \nonumber\\ + &= \tau \int_{0}^{\infty} \frac{e^{-x} - e^{-(1-i\zeta)x}}{x} dx \nonumber\\ + &\qquad\mbox{(intégrale de Frullani \citep{spiegel1999schaum}, p.115)} \nonumber\\ + &= \tau \int_{0}^{\infty} + \left(1-e^{i\zeta\,x}\right)\frac{1}{x}e^{-x}dx. +\end{align} + +On a donc, par cette représentation, la démonstration que le processus +gamma est un processus de sauts purs. Il pourra donc être utilisé +comme subordonnant dans la construction d'un processus subordonné +\eqref{eq:processussubordonne}. + +\subsection{Le processus de Wiener} +\label{sec:mouvementbrownien} + +Le processus de Wiener $\left\{W(t;\mu,\sigma^2)\right\}$ est un +processus de diffusion avec dérive. Il n'a donc pas de composante de +saut. Ses incréments suivent une distribution normale: +\begin{align} + \label{eq:incrwiener} + W(t+1;\mu,\sigma^2) - W(t;\mu,\sigma^2) \sim N(\mu,\sigma^2). +\end{align} +Cette distribution est définie par la fonction de densité +$f_{\mu,\sigma}(x)$ et la fonction caractéristique +$\phi_{\mu,\sigma}(\xi)$: +\begin{align} + f_{\mu,\sigma}(x) &= \frac{1}{\sqrt{2\pi\sigma^2}}\exp{-\left\{\frac{1}{2} \left(\frac{x-\mu}{\sigma} \right)^2\right\}} \label{eq:fndensitenormale} \\ + \phi_{\mu,\sigma}(\xi) &= \exp\left\{ + i\mu\xi-\frac{\sigma^2\xi^2}{2} + \right\} \label{eq:fncaractnormale}. +\end{align} + +Notons que la variance d'un incrément est proportionnelle à la +longueur de celui-ci. Soit deux incréments indépendants d'un même +processus: $I_1 = W(t+q;\mu,\sigma^2) - W(t;\mu,\sigma^2) \sim +N(q\mu,q\sigma^2) \mbox{ et } I_2 = W(t+q+s;\mu,\sigma^2) - +W(t+q;\mu,\sigma^2) \sim N(s\mu,s\sigma^2)$. La somme de ces +incréments suit une distribution normale dont la moyenne et la +variance sont respectivement la somme de celles des deux incréments: +\begin{align} + I_1+I_2 \sim N((q+s)\mu, (q+s)\sigma^2). +\end{align} + +Comme la distribution normale est aussi infiniment divisible, on peut +obtenir la fonction caractéristique du processus +$\phi(\xi;t;\mu,\sigma^2)$ en utilisant la propriété de convolution +\eqref{eq:convocaractIID}: +\begin{align} + \phi(\xi;t;\mu,\sigma^2) = \exp\left\{ i\mu t + \xi-\frac{\sigma^2t\xi^2}{2} \right\}. +\end{align} + +On déduit donc facilement l'exposant caractéristique +$\Lambda(\xi;t=1;\mu,\sigma^2)$ d'un incrément de ce processus, sous +la représentation de Lévy-Khintchine: +\begin{align} + \label{eq:exposantcaractnormale} + \Lambda(\xi;t=1;\mu,\sigma^2) = -(i\mu \xi-\frac{\sigma^2\xi^2}{2}). +\end{align} + +Ceci démontre que le processus de Wiener est un processus avec dérive +et diffusion, mais sans composante de saut. Il pourra donc être +utilisé pour construire un processus subordonné +\eqref{eq:processussubordonne}. + +\subsection{Le processus de Laplace est un processus subordonné} +\label{sec:browniensub} + +On considère un processus gamma $G(t;\tau,\beta=1)$ et un processus de +Wiener $W(t;\mu,\sigma^2)$. On se rappelle que la variance d'un +incrément \eqref{eq:incrwiener} de ce dernier est proportionnelle à la +longueur de l'intervalle de temps. En utilisant une propriété appelée +la subordination, on peut modifier l'échelle de temps du processus de +Wiener de sorte que la variance soit aléatoire pour tout intervalle +. Tout processus de Lévy peut être utilisé comme subordonnant pour +définir cette échelle de temps. Si on utilise le processus gamma, on +obtiendra le processus de Laplace sans dérive +$\left\{Y(t;\sigma,\mu,\tau)\right\}$ défini comme suit: +\begin{align} + \label{eq:VGsubordinne} + \lbrace Y(t;\sigma,\mu,\tau)\rbrace &\equiv \lbrace + W(G(t;\tau,\beta=1);\mu,\sigma^2)\rbrace. +\end{align} + +On obtient l'exposant caractéristique $\Psi(\xi,t=1;\sigma,\mu,\tau)$ +d'un incrément $Y(t+1;\sigma,\mu,\tau)-Y(t;\sigma,\mu,\tau)$ en +utilisant la propriété de subordination définie par l'équation +\eqref{eq:exposantcaractYt}, où $\Xi(\zeta,t=1;\tau,\beta=1)$ est +l'exposant caractéristique du processus gamma et +$\Lambda(\xi,t=1;\mu,\sigma^2)$ celui du processus de Wiener: +\begin{align} + \label{eq:exposantcaractLaplace} + \Psi(\xi,t=1;\sigma,\mu,\tau) &= \Xi(i\Lambda(\xi,t=1;\mu,\sigma^2),t=1;\tau,\beta=1) \nonumber\\ + &= \tau \ln{\left(1-i(i\Lambda(\xi)) \right)} \nonumber\\ + &= \tau \ln{\left(1+(\frac{\sigma^2 \xi^2}{2} - i \mu \xi) \right)}. +\end{align} + +Le processus de Laplace sans dérive est donc, par définition, un +processus de Lévy et par conséquent infiniment divisible. En utilisant +l'exposant caractéristique \eqref{eq:exposantcaractLaplace} et la +définition \eqref{eq:fncaractYt}, on obtient sa fonction +caractéristique: +\begin{align} + \label{eq:fonctioncaractlaplacesansdrift} + \phi_{Y(t;\sigma,\mu,\tau)}(\xi) &= \exp{\left\{-t \cdot \Psi(\xi,t=1;\sigma,\mu,\tau)\right\}} \nonumber\\ + &= \exp{\left\{-t \cdot \left(\tau \ln{\left(1+(\frac{\sigma^2 + \xi^2}{2} - i \mu + \xi) \right)} \right)\right\}} \nonumber\\ + &= \left(1+\frac{\sigma^2 \xi^2}{2} - i \mu \xi\right)^{-\tau \cdot + t}. +\end{align} + +Une manière simple pour expliquer le mécanisme derrière le processus +subordonné est d'en construire une trajectoire à l'aide de la +simulation. + +On simule un temps d'arrivée $T_1$, de distribution gamma, puis une +hauteur de saut $X_1$, de distribution normale. On obtient ainsi le +premier incrément de la trajectoire, tel qu'illustré à la figure +\ref{fig:increment1}. +\begin{figure}[!ht] + \centering \input{"./graphiques/increment1.tex"} + \caption{Premier incrément d'un processus subordonné} + \label{fig:increment1} +\end{figure} + +Une réalisation d'une trajectoire de ce processus par simulation se +trouve à la figure \ref{fig:simgammagauss}. + +\begin{figure}[!ht] + \centering + \includegraphics[scale=.8]{"./graphiques/CH3-SIMGAMMAGAUSS"} + \caption{Simulation d'un processus de Wiener subordonné par un + processus gamma} + \label{fig:simgammagauss} +\end{figure} + +Le processus gamma $G(t;\tau,\beta=1)$, en tant que subordonnant dans +ce cas-ci, définit une échelle de temps économique , selon laquelle +on situe l'arrivée d'évènements pouvant influencer le prix d'un titre +financier. Cette dernière ne peut être mesurée, elle est donc +abstraite. L'échelle de temps où sont effectuées les observations du +processus correspond au temps calendrier. C'est la seule qui +puisse être mesurée. Étant donné que ces deux échelles sont +indépendantes, plusieurs sauts entre deux observations sont possibles. +L'échelle de temps économique est donc soit étirée, soit compressée, +en comparaison au temps calendrier. Autrement dit, si l'on définit +une journée économique comme étant l'intervalle de temps entre deux +sauts, on peut en avoir plusieurs au cours d'une seule journée de +calendrier $(G(t+1;\tau,\beta=1)-G(t;\tau,\beta=1) > 1)$. À l'opposé, +une d'entre elles peut chevaucher plusieurs journées calendrier +$(G(t+1;\tau,\beta=1)-G(t;\tau,\beta=1) \leq 1)$. + +Un processus stochastique qui représente le comportement du prix d'un +titre financier doit inclure une composante de dérive. Celle-ci +exprime le rendement moyen réalisé et est indépendante du processus de +sauts. Pour cette raison, on ajoute un coefficient de dérive $\theta$ +au processus de Laplace sans dérive, pour obtenir sa forme +générale. Comme ce coefficient est constant, on peut multiplier la +fonction caractéristique \eqref{eq:fonctioncaractlaplacesansdrift} par +la transformée de Fourier inverse du produit de celui-ci et de la +longueur de l'intervalle de temps $t$, $\mathcal{F}^{-1}(\theta \cdot +t) = e^{i\xi\theta \cdot t}$, pour obtenir celle du processus de +Laplace: +\begin{align} + \phi_{Y(t;\theta,\sigma,\mu,\tau)}(\xi) &= \frac{e^{i\xi\theta\cdot t}}{\left(1+\frac{\sigma^2\xi^2}{2}- i\mu \xi \right)^{\tau \cdot t}} \nonumber\\ + &= \left(\frac{e^{i\xi\theta}}{\left(1+\frac{\sigma^2\xi^2}{2}- i\mu + \xi + \right)^{\tau}}\right)^{t} \label{eq:fncaractprocessuslaplace}. +\end{align} + +Le processus $\left\{Y(t;\theta,\sigma,\mu,\tau)\right\}$ définit, +dans le contexte financier, l'évolution du logarithme du prix, tel que +présenté par \cite{kotz2001laplace}. Pour des fins de simplification, +on fixe le prix initial à 1: $Y(0;\theta,\sigma,\mu,\tau) = 0$ + +La fonction caractéristique \eqref{eq:fncaractprocessuslaplace} +constituera la principale représentation du processus de Laplace pour +la suite de ce texte. La construction du modèle «Variance Gamma» de +\cite{madan1990variance} est similaire, à l'exception que la +paramétrisation et le processus gamma utilisés sont différents, ce qui +rend leur approche moins intuitive, bien que le résultat soit +équivalent. + +\section{Distribution de Laplace asymétrique généralisée} +\label{sec:distributionGAL} + +La distribution de Laplace asymétrique généralisée caractérise un +intervalle du processus de Laplace avec dérive. Aussi appelée +distribution de Bessel, elle a été introduite par Karl Pearson en +1929, en lien avec la covariance d'un échantillon tiré d'une +population normale à deux variables. C'est aussi une généralisation de +la distribution de Laplace asymétrique qui sera présentée à la section +\ref{sec:distributionAL}. + +\subsection{Fonction caractéristique} +\label{sec:fncaractGAL} + +On définit cette distribution principalement par sa fonction +caractéristique. Celle-ci s'obtient facilement à partir de la fonction +caractéristique du processus de Laplace avec dérive +\eqref{eq:fncaractprocessuslaplace}, en considérant un incrément de +longueur $t=1$. À partir de la définition de ce dernier +\eqref{eq:VGsubordinne}, on déduit qu'elle est en fait un mélange de +la loi normale dont le paramètre de variance suit une distribution +gamma. + +$Y$ est une variable aléatoire définie comme étant la somme: +\begin{itemize} +\item d'un paramètre de translation $\theta$, +\item du produit: + \begin{itemize} + \item d'une variable aléatoire $W$ issue d'une distribution gamma + \eqref{eq:densitegamma1} + \item et d'un paramètre $\mu$ + \end{itemize} +\item et du produit: + \begin{itemize} + \item d'un paramètre $\sigma$, + \item de la racine carrée de la variable aléatoire $W$ + \item et d'une variable aléatoire $Z$ issue d'une distribution + normale centrée réduite: + \begin{align} + \label{eq:defvarY-GAL} + &Y = \theta + \mu W + \sigma \sqrt{W} Z + \end{align} + où + \begin{align} + Z \sim N(0,1) \mbox{ et } W \sim \Gamma(\tau,\beta=1). \nonumber + \end{align} + \end{itemize} +\end{itemize} + +Alors, la variable aléatoire $Y$, sachant que $W=w$, suit une +distribution normale de moyenne $w\mu$ et de variance $w\sigma^2$: +\begin{align} + \label{eq:Ynormalconditionnel} + (Y|W=w) \sim N(w\mu,w\sigma^2). +\end{align} + +La fonction caractéristique de la variable aléatoire $Y$ peut donc +être obtenue en utilisant celle de la loi normale $\phi^{N}_{\mu + w,w\sigma^2}(t)$ et la formule de l'espérance conditionnelle: +\begin{align*} + \phi_Y(t;\theta,\sigma,\mu,\tau) &= E\left[E\left[e^{itY} | W \right] \right] \\ + &= \int_0^{\infty} E \left[ e^{it(\theta + \mu w+\sigma\sqrt{w}Z)} \right] g(w) dw \quad \mbox{(en utilisant \eqref{eq:defvarY-GAL})} \\ + &= e^{i\theta t}\int_0^{\infty} \phi^{N}_{\mu w,w\sigma^2}(t)g(w)dw \\ + &= e^{i\theta t}\int_0^{\infty} e^{ iw\mu t-\frac{w\sigma^2t^2}{2}} \times \frac{1}{\Gamma (\tau)} w^{\tau-1}e^{-w} dw \\ + &= e^{i\theta t}\int_0^{\infty} \frac{1}{\Gamma (\tau)} w^{\tau-1} e^{-w(1+\frac{1}{2} \sigma^2 t^2 - i\mu t)} dw. +\end{align*} + +En complétant l'intérieur de l'intégrale de façon à retrouver la +densité de la loi gamma de paramètres $\alpha=\tau$ et +$\beta=\left(1+\frac{1}{2} \sigma^2 t^2 - i\mu t \right)^{\tau}$, on +obtient la fonction caractéristique de la variable aléatoire $Y$ de +distribution Laplace asymétrique généralisée: +\begin{align} + \label{eq:fncaractGALmu} + \phi_Y(t;\theta,\sigma,\mu,\tau) &= \frac{e^{i\theta t}}{{\left(1+\frac{1}{2} \sigma^2 t^2 - i\mu t \right)^{\tau}}}\int_0^{\infty} \frac{\left(1+\frac{1}{2} \sigma^2 t^2 - i\mu t \right)^{\tau}}{\Gamma (\tau)} w^{\tau-1} e^{-w(1+\frac{1}{2} \sigma^2 t^2 - i\mu t)} dw \nonumber\\ + &= \frac{e^{i\theta t}}{\left(1+\frac{1}{2} \sigma^2 t^2 - i\mu t + \right)^{\tau}}. +\end{align} + +\subsection{Invariance d'échelle} +\label{sec:invariance-dechelle} + +Une seconde paramétrisation pour la famille de distributions de +Laplace introduit la propriété d'invariance d'échelle. Cette propriété +permet d'appliquer un changement d'échelle à une variable aléatoire en +modifiant un seul paramètre sans que la valeur des autres ne soit +affectée. Si l'on revient à la définition de la variable aléatoire +conditionnelle $Y|W$ \eqref{eq:Ynormalconditionnel}, on remarque que +les paramètres $\mu \text{ et } \sigma^2$ sont influencés par la valeur +de $W$. Ils sont donc corrélés. En introduisant un paramètre +d'invariance d'échelle $\kappa$ en remplacement de $\mu$, on élimine +cette corrélation. Le paramètre $\kappa$ est obtenu à l'aide de la +transformation suivante: +\begin{equation} + \label{eq:mukappa} + \kappa = \frac{\sqrt{2\sigma^2+\mu^2}-\mu}{\sqrt{2}\sigma}. +\end{equation} + +À l'inverse, on peut retrouver le paramètre $\mu$ en l'isolant dans +l'équation précédente. On obtient donc: +\begin{equation} + \label{eq:kappamu} + \mu = \frac{\sigma}{\sqrt{2}}\left(\frac{1}{\kappa}-\kappa \right). +\end{equation} + +On définit la fonction vectorielle $T_{\mu\rightarrow\kappa}$ comme +étant la transformation qui permet le passage de la forme en $\mu$ à +celle en $\kappa$: +\begin{align} + T_{\mu\rightarrow\kappa}(\theta, \sigma, \mu, \tau) &= + \left[\begin{array}{c} \theta \\ \sigma \\ + \frac{\sqrt{2\sigma^2+\mu^2}-\mu}{\sqrt{2}\sigma} \\ \tau + \end{array}\right]. +\end{align} + +On définit aussi la transformation inverse $T_{\kappa\rightarrow\mu}$: +\begin{align} + T_{\kappa\rightarrow\mu}(\theta, \sigma, \kappa, \tau) &= + \left[\begin{array}{c} \theta \\ \sigma \\ + \frac{\sigma}{\sqrt{2}}\left(\frac{1}{\kappa}-\kappa \right) \\ + \tau + \end{array}\right]. +\end{align} + +Cette notation permet d'utiliser un vecteur de paramètres. On peut +obtenir la matrice de variance-covariance d'une forme paramétrique en +connaissant celle de l'autre et en utilisant le gradient de ces +transformations. + +Pour la première transformation, on a: +\begin{align} + \nabla T_{\mu\rightarrow\kappa} &= \left[ + \begin{array}[]{cccc} + 1&0&0&0 \\ + 0&1&0&0 \\ + 0&\frac{\mu\sqrt{4\sigma^2+\mu^2}-\mu^2}{2\sigma^2\sqrt{4\sigma^2+mu^2}} & -\frac{\sqrt{4\sigma^2+\mu^2}-\mu}{2\sigma\sqrt{4\sigma^2+\mu^2}} & 0 \\ + 0&0&0&1 + \end{array} + \right]. +\end{align} + +Pour la seconde transformation, on a: +\begin{align} + \nabla T_{\kappa\rightarrow\mu} &= \left[ + \begin{array}[]{cccc} + 1&0&0&0 \\ + 0&1&0&0 \\ + 0&-\frac{\kappa^2-1}{\sqrt{2}\kappa} & -\frac{\left(\kappa^2+1\right)\sigma}{\sqrt{2}\kappa^2} & 0 \\ + 0&0&0&1 + \end{array} + \right]. +\end{align} + +La forme en $\mu$ sera privilégiée pour l'estimation, car elle est +plus compacte. Cependant, certaines propriétés de la distribution font +appel à la forme utilisant le paramètre $\kappa$. + +\subsection{Fonctions génératrices} + +En utilisant la relation \eqref{eq:fncaractfgm}, on obtient la +fonction génératrice des moments à partir de la fonction +caractéristique: +\begin{align} + M_{Y}(\xi) &= \phi_{Y}(-i\xi) \nonumber\\ + &=\frac{e^{\theta \xi}}{\left(1-\frac{1}{2} \sigma^2 \xi^2 - \mu \xi + \right)^{\tau}}, \label{eq:fgmGAL}\\ + &\quad \mbox{où } 1-\frac{1}{2} \sigma^2 \xi^2 - \mu \xi > 0. + \label{eq:fgmGALcond} +\end{align} + +La condition \ref{eq:fgmGALcond} permet de s'assurer que le +dénominateur prend une valeur réelle strictement positive. Cette +condition impose aussi une restriction à l'espace des paramètres +$\Omega$. La fonction génératrice des moments permet d'obtenir tous +les moments $E[Y^r]$ d'une variable aléatoire en la dérivant +successivement par rapport à la variable de transformation et en +égalant cette dernière à 0: +\begin{align} + \label{eq:fgmmomentsGAL} + E[Y^r] = \left[ \frac{d^r M_Y(\xi)}{d\xi^r} \right]_{\xi=0}. +\end{align} + +La fonction génératrice des cumulants $K_{Y}(\xi)$ est aussi +intéressante à utiliser. On l'obtient à partir du logarithme de la +fonction génératrice des moments \eqref{eq:fgmmoments}: +\begin{align} + \label{eq:fgcGAL} + K_Y(\xi) &= \ln(M_Y(\xi)) \nonumber\\ + &= \ln\left(\frac{e^{\theta \xi}}{\left(1-\frac{1}{2} \sigma^2 + \xi^2 - \mu \xi \right)^{\tau}}\right),\qquad 1-\frac{1}{2} + \sigma^2 \xi^2 - \mu \xi > 0. +\end{align} + +Elle a une utilité similaire à la fonction génératrice des moments, +sauf qu'elle permet d'obtenir les cumulants $K_r$ de la distribution +en la dérivant successivement par rapport à la variable de +transformation et en égalant celle-ci à 0: +\begin{align} + \label{eq:fgccumGAL} + K_r = \left[ \frac{d^r \ln{(M_Y(\xi))}}{d\xi^r} \right]_{\xi=0}. +\end{align} + +\subsection{Moments et rôle des paramètres} +\label{sec:momentsGAL} + +En mathématiques, les moments sont des quantités décrivant la forme +d'un ensemble de points. En statistique, les moments décrivent +certaines caractéristiques d'une population ou d'un échantillon. Ces +caractéristiques sont utilisées dans la sélection d'une distribution +de probabilité appropriée pour représenter la population à partir de +l'échantillon, ce qu'on appelle l'inférence statistique. Les moments +bruts et centraux sont évalués par rapport à 0 et à la moyenne +respectivement. + +On obtient les premiers moments bruts de cette distribution à l'aide +de la relation décrite précédemment \eqref{eq:fgmmomentsGAL}. +\begin{align*} + E[Y] &= \theta+\tau\,\mu \\ + E[Y^2] &= {\theta}^{2}+2\,\mu\,\tau\,\theta+\tau\,{\sigma}^{2}+{\mu}^{2}\,{\tau}^{2}+{\mu}^{2}\,\tau \\ + E[Y^3] &= {\theta}^{3}+3\,\mu\,\tau\,{\theta}^{2}+\left( 3\,\tau\,{\sigma}^{2}+3\,{\mu}^{2}\,{\tau}^{2} +3\,{\mu}^{2}\,\tau\right) \,\theta \\ + &\quad + \left( 3\,\mu\,{\tau}^{2}+3\,\mu\,\tau\right) \,{\sigma}^{2}+{\mu}^{3}\,{\tau}^{3}+3\,{\mu}^{3}\,{\tau}^{2}+2\,{\mu}^{3}\,\tau \\ + E[Y^4] &= {\theta}^{4}+4\,\mu\,\tau\,{\theta}^{3}+\left( 6\,\tau\,{\sigma}^{2}+6\,{\mu}^{2}\,{\tau}^{2}+6\,{\mu}^{2}\,\tau\right) \,{\theta}^{2}\\ + &\quad+\left( \left( 12\,\mu\,{\tau}^{2}+12\,\mu\,\tau\right) \,{\sigma}^{2}+4\,{\mu}^{3}\,{\tau}^{3}+12\,{\mu}^{3}\,{\tau}^{2}+8\,{\mu}^{3}\,\tau\right) \,\theta \\ + &\quad+\left( 3\,{\tau}^{2}+3\,\tau\right) \,{\sigma}^{4}+\left( + 6\,{\mu}^{2}\,{\tau}^{3}+18\,{\mu}^{2}\,{\tau}^{2}+12\,{\mu}^{2}\,\tau\right) + \,{\sigma}^{2} \\ + &\quad+{\mu}^{4}\,{\tau}^{4}+6\,{\mu}^{4}\,{\tau}^{3}+11\,{\mu}^{4}\,{\tau}^{2}+6\,{\mu}^{4}\,\tau. +\end{align*} + +À partir de ceux-ci, on obtient aussi les quatre premiers moments +centraux: +\begin{subequations}\label{eq:momentsGAL} + \begin{align} + m_1 &= E[Y] = \theta+\tau\,\mu \label{eq:moments1GAL}\\ + m_2 &= E[(Y-m_1)^2] = \tau\,\sigma^2+\tau\,\mu^2\label{eq:moments2GAL}\\ + m_3 &= E[(Y-m_1)^3] = 3\,\tau\,{\sigma}^{2}\mu+2\,\tau\,{\mu}^{3}\label{eq:moments3GAL}\\ + m_4 &= E[(Y-m_1)^4] = + 3\,{\tau}^{2}{\sigma}^{4}+3\,{\tau}^{2}{\mu}^{4}+3\,\tau\,{\sigma}^{4}+6\,\tau\,{\mu}^{4}+6\,{\tau}^{2}{\mu}^{2}{\sigma}^{2}+12\,\tau\,{\sigma}^{2}{\mu}^{2}.\label{eq:moments4GAL} + \end{align} +\end{subequations} + +On résume le domaine et le rôle des paramètres à la table +\ref{tab:roleparamGAL}. +\begin{table}[!ht] + \centering + \begin{tabular}{cp{1.75cm}p{2.5cm}p{6.25cm}} + \hline + \textbf{Paramètre} & \textbf{Domaine} & \textbf{Rôle} & \textbf{Observations} \\ + \hline + $\theta$ & $\mathbb{R}$ & Localisation & N'influence que la moyenne. Équivaut au mode lorsque $\mu=0$. \\ + $\sigma$ & $\mathbb{R}^{+} \setminus \lbrace 0 \rbrace$ & Échelle & Vrai paramètre d'échelle lorsque $\kappa$ est utilisé. \\ + $\mu$ & $\mathbb{R}$ & Asymétrie & Distribution asymétrique à gauche lorsque négatif et à droite lorsque positif. Déplace la moyenne dans la même direction. Corrélation positive avec la variance et le coefficient d'aplatissement.\\ + $\kappa$ & $\mathbb{R}^{+} \setminus \lbrace 0 \rbrace$ & Asymétrie & Valeur dans l'intervalle $\left[0,1\right[$ lorsque $\mu<0$, dans $\left[1,\infty \right]$ lorsque $\mu \geq0$. \\ + $\tau$ & $\mathbb{R}^{+} \setminus \lbrace 0 \rbrace$ & Aplatissement & Négativement corrélé avec le coefficient d'aplatissement. Une petite valeur donne une distribution pointue. \\ + \hline + \end{tabular} + \caption{Domaine et rôle des paramètres de la distribution de Laplace asymétrique généralisée} + \label{tab:roleparamGAL} +\end{table} + +On retrouve quelques exemples de courbes de la fonction de densité +avec différents paramètres d'asymétrie et d'aplatissement à la figure +\ref{fig:densiteGAL}. +\begin{figure}[!ht] + \centering + \includegraphics[scale=0.8]{./graphiques/dGAL-exemples.pdf} + \caption{Fonction de densité de la distribution Laplace asymétrique + généralisée avec différents paramètres: + $GAL(y;\theta,\sigma,\kappa,\tau)$} + \label{fig:densiteGAL} +\end{figure} + +Afin de comparer la distribution de Laplace asymétrique généralisée +avec la normale, que l'on cherche à remplacer dans le contexte des +rendements financiers, le comportement des coefficients d'asymétrie +$\gamma_1$ et d'aplatissement $\gamma_2$ de celle-ci peut être +intéressant à observer. On obtient ces derniers à partir des moments +centraux \eqref{eq:momentsGAL} : +\begin{subequations}\label{eq:moments56GAL} + \begin{align} + \gamma_1(Y) &= \frac{m_3}{(m_2)^{3/2}} = + \frac{2\,{\mu}^{3}+3\,{\sigma}^{2}\,\mu}{{\left( {\mu}^{2}+{\sigma}^{2}\right) }^{3/2}\,\sqrt{\tau}}\label{eq:moments5GAL}\\ + \gamma_2(Y) &= \frac{m_4}{(m_2)^{2}} - 3 = \frac{\left( + 3\,{\mu}^{4}+6\,{\sigma}^{2}\,{\mu}^{2}+3\,{\sigma}^{4}\right) + \,{\tau}^{2}+\left( + 6\,{\mu}^{4}+12\,{\sigma}^{2}\,{\mu}^{2}+3\,{\sigma}^{4}\right) + \,\tau}{{\left( {\mu}^{2}+{\sigma}^{2}\right) }^{2}\,{\tau}^{2}} + - 3.\label{eq:moments6GAL} + \end{align} +\end{subequations} + +Le coefficient d'asymétrie de la normale vaut $\gamma_1^N(Y)=0$ et +celui d'excès d'aplatissement, $\gamma_2^N(Y)=0$, puisqu'il a été +défini à partir de cette distribution. Si l'on fixe tous les +paramètres sauf $\mu$, la valeur minimale du coefficient d'excès +d'aplatissement $\gamma_2(Y)$ est atteinte lorsque $\mu$ est de 0: +\begin{align*} + \min_{\mu} \gamma_2(Y) &= \left[\gamma_2(Y)\right]_{\mu=0} \\ + &= \frac{3}{\tau}. +\end{align*} +Étant donné que le paramètre $\tau$ est strictement positif, la valeur +minimale que peut prendre le coefficient d'excès d'aplatissement +$\gamma_2(Y)$ est plus grande que 0. Par conséquent, l'aplatissement de la +distribution de Laplace asymétrique généralisée sera toujours +supérieur à celui de la normale; c'est donc une distribution +leptocurtique, ce qui est une propriété désirable selon les +observations de \cite{madan1990variance}. + +\subsection{Changement d'échelle et de localisation} +\label{sec:transGAL} + +Parfois, on doit modifier un ensemble de données afin d'effectuer des +comparaisons ou pour bénéficier des avantages d'une méthode +numérique. Les principales transformations utilisées sont: +\begin{enumerate} +\item un changement de localisation, où l'on additionne une constante + à l'ensemble des données +\item un changement d'échelle, où l'on multiplie l'ensemble des + données par un facteur +\item une combinaison des deux transformations précédentes. +\end{enumerate} + +Soit deux constantes $a$, $b\neq0$ et une variable aléatoire $X$ qui +suit une distribution Laplace asymétrique généralisée: $X \sim +GAL(\theta,\sigma,\kappa,\tau)$. $Y$ correspond à la somme: +\begin{itemize} +\item de la constante $b$ et +\item du produit: + \begin{itemize} + \item de la constante $a$ et + \item de la variable aléatoire $X$. + \end{itemize} +\end{itemize} + +Selon \cite{kotz2001laplace}, en utilisant le paramètre $\kappa$, on +peut effectuer un changement d'échelle et de localisation et obtenir +une variable aléatoire qui suit encore cette distribution: +\begin{itemize} +\item le paramètre de localisation $\theta$ subit la même + transformation que la variable aléatoire $X$; +\item le paramètre d'échelle $\sigma$ est multiplié par la constante + $a$; +\item le paramètre $\kappa$ est inversé si cette constante est + négative. Ce dernier cas est en fait une réflexion de la + distribution autour du mode. +\end{itemize} + +On a donc: +\begin{align} + \label{eq:GALscaletrans} + Y = aX + b \sim GAL(a \theta + b,a\sigma,\kappa^{\sgn{a}\cdot + 1},\tau). +\end{align} + +On estime les paramètres $\hat\theta, \hat\sigma, \hat\mu$ et +$\hat\tau$ sur un ensemble de données centrées et réduites $X_t$ à +partir d'un échantillon original $Y_t$. $\hat{m} \text{ et } +\hat{s}>0$ sont respectivement la moyenne et l'écart-type de +l'échantillon $Y_t$. En utilisant l'équation de transformation +\eqref{eq:GALscaletrans} avec les paramètres estimés précédemment, on +peut alors obtenir ceux correspondants pour l'échantillon: +\begin{align} + \label{eq:transparamGALNS} + Y_t = \hat{s} X_t + \hat{m} \sim GAL(\hat{s} \theta + + \hat{m},\hat{s}\sigma,\kappa,\tau) . +\end{align} + +Le paramètre $\kappa$ n'est pas modifié puisque l'écart-type $\hat{s}$ +est strictement positif. Cette propriété permettra de pratiquer +l'estimation sur des données centrées et réduites, ce qui diminue le +risque d'erreurs numériques sans nuire à sa précision, puisqu'aucune +information contenue dans l'échantillon n'est perdue. + +\subsection{Représentation alternative et simulation} +\label{sec:simulationGAL} + +Le processus de Laplace $Y(t;\theta,\sigma,\kappa,\tau)$ peut être +représenté sous la forme d'une différence de deux processus gamma +$G_1(t;\tau) - G_2(t;\tau)$ à laquelle on additionne une composante de +dérive. Le premier processus compte les gains, alors que le second, +les pertes. Les deux ont des incréments qui suivent une distribution +gamma $\Gamma(\alpha=\tau, \beta=1)$, c'est-à-dire la même +distribution que le processus +subordonnant utilisé à la section \ref{sec:browniensub}: +\begin{align} + \label{eq:distributiongammaformealt} + G_i(\tau) \sim \Gamma\left(\tau,\beta=1 \right), \qquad i\in\left\{ + 1,2 \right\}. +\end{align} + +Il peut être représenté sous la forme du processus composé suivant: +\begin{align} + \label{eq:processuslaplace2gamma} + Y(t) \stackrel{d}{=} \theta + + \frac{\sigma\sqrt{2}}{2}\left(\frac{1}{\kappa} G_1(t) - \kappa + G_2(t)\right). +\end{align} + +La distribution de Laplace asymétrique généralisée peut alors être +représentée sous la forme d'un incrément de ce processus. Les +variables aléatoires $G_1$ et $G_2$ sont respectivement des +réalisations indépendantes de distribution gamma avec densité +\eqref{eq:densitegamma1}: +\begin{align} + \label{eq:differencegamma} + Y \stackrel{d}{=} \theta + \frac{\sigma\sqrt{2}}{2} \left( + \frac{1}{\kappa} G_1 - \kappa G_2 \right). +\end{align} + +Simuler une réalisation de chacune d'entre elles suffira pour obtenir +une réalisation de la distribution gamma. On peut aussi réécrire la +définition précédente de la variable aléatoire $Y$ +\eqref{eq:differencegamma} sous la forme simplifiée suivante: +\begin{align} + \label{eq:differencegamma2} + Y \stackrel{d}{=} \theta + \left(G_3-G_4 \right). +\end{align} + +Les deux variables aléatoires sont alors définies comme suit: +\begin{align*} + G_3 \sim \Gamma\left(\tau,\beta=\frac{\sqrt{2}}{\kappa\sigma} \right)\\ + G_4 \sim \Gamma\left(\tau,\beta=\frac{\kappa\sigma}{\sqrt{2}} + \right). +\end{align*} + +On peut facilement démontrer l'équivalence en distribution +\eqref{eq:differencegamma2}. En effectuant le produit des fonctions +caractéristiques respectives $\phi_{Y}(\xi)$, $\phi_{G_3}(\xi)$ et +$\phi_{G_4}(\xi)$ des variables aléatoires $Y$, $G_3$ et $G_4$, on +retrouve celle de la forme en $\kappa$: +\begin{align} + \label{eq:equivalence2gammaFC} + \phi_{Y}(\xi) &= e^{i\xi\theta} \cdot \phi_{G_3}(\xi) \cdot \phi_{G_4}(\xi) \nonumber\\ + &= e^{i\xi\theta} \cdot \left(\frac{1}{1+i\xi\cdot\frac{\kappa\sigma}{\sqrt{2}}}\right)^{\tau} \cdot \left(\frac{1}{1-i\xi\cdot\frac{\kappa\sigma}{\sqrt{2}}}\right)^{\tau} \nonumber\\ + &= + \frac{e^{i\xi\theta}}{\left(1-i\xi\frac{\sigma}{\sqrt{2}}\left(\frac{1}{\kappa}-\kappa + \right) + \frac{\sigma^2\xi^2}{2}\right)^{\tau}}. +\end{align} + +La figure \ref{fig:simulGAL} présente un histogramme et un estimateur +de densité par noyau d'un échantillon de 2500 réalisations de la +variable aléatoire $Y$, suivant une distribution de Laplace +asymétrique généralisée de paramètres $\theta=0$, $\sigma=1$, +$\kappa=2$ et $\tau=1$. On remarquera que le paramètre $\kappa$ +produit une distribution asymétrique à gauche puisqu'il prend une +valeur supérieure à 1. + +\begin{figure}[!ht] + \centering + \includegraphics[scale=0.8]{"./graphiques/CH3-SIMULGAL0121"} + \caption{Histogramme et estimateur de densité par noyau de 2500 + réalisations de la variable aléatoire $Y\sim + GAL(\theta=0,\sigma=1,\kappa=2,\tau=1)$} + \label{fig:simulGAL} +\end{figure} + +\subsection{Fonction de Bessel et densité} +\label{sec:besseldensiteGAL} + +La densité de la distribution Laplace asymétrique généralisée peut +être exprimée à l'aide de la fonction de Bessel modifiée du troisième +type \citep{abramowitz1965handbook}: +\begin{align} + \label{eq:BesselK} + K_{\lambda}(u) = + \frac{(u/2)^{\lambda}\Gamma(1/2)}{\Gamma(\lambda+1/2)} + \int_1^{\infty} e^{-ut} (t^2-1)^{\lambda-1/2}dt,\qquad \lambda \geq + -1/2. +\end{align} + +On définit les paramètres $\delta = \mu\sigma^{-1}$ et $\eta = +\sqrt{1+\delta^2}$ afin de simplifier la notation. La démonstration +qui permet d'obtenir la fonction de densité est fondée sur la +définition de la distribution utilisant la différence de deux +processus gamma \eqref{eq:differencegamma}. Elle est développée par +\cite{kozubowski1999class}: +\begin{align} + \label{eq:densitekotz} + f(x) &= \frac{e^{-\kappa^{\sgn(x)}|x|}}{\Gamma(\tau)} \int_{0}^{\infty} y^{\tau-1}(|x|+y)^{\tau-1}e^{-\eta y} dy \nonumber\\ + &= \frac{1}{\Gamma(\tau)\sqrt{\pi}} \left(\frac{|x|}{\eta} + \right)^{\tau-1/2} e^{\delta x/2} K_{\tau-1/2}\left(\frac{\eta + |x|}{2}\right). +\end{align} + +Afin d'incorporer le paramètre de localisation $\theta$, on remplace +$x$ par la valeur centrée $x-\theta$, et les paramètres $\eta$ et +$\delta$ par ceux d'origine, tels que définis précédemment, pour +obtenir la forme suivante \citep{kotz2001laplace}: +\begin{align} + \label{eq:densitekotz2001} + f(x) = + \frac{\sqrt{2}e^{\frac{\sqrt{2}}{2\sigma}(1/\kappa-\kappa)(x-\theta)}}{\sqrt{\pi}\sigma^{\tau+1/2}\Gamma(\tau)} + \bigg(\frac{\sqrt{2}|x-\theta|}{\kappa+1/\kappa} \bigg)^{\tau-1/2} + K_{\tau-1/2}\left(\frac{\sqrt{2}}{2\sigma}\left(\frac{1}{k}+\kappa + \right)|x-\theta| \right). +\end{align} + +Cette fonction a été implémentée dans un algorithme de maximum de +vraisemblance pour le logiciel GNU R par +\cite{RpackageVarianceGamma}. Elle reste cependant sensible +numériquement en plus de demander un temps de calcul important pour de +grands échantillons. De plus, comme la fonction de densité n'est pas +différentiable, on ne peut pas obtenir les estimateurs du maximum de +vraisemblance et leurs matrices de variance-covariance. Donc, +il est impossible d'effectuer de tests d'adéquation ou d'hypothèse avec ces +résultats. C'est une des principales raisons qui ont motivé la +recherche d'autres méthodes d'estimation plus efficaces pour cette +distribution. + +\section{Cas particuliers} +\label{sec:cas-particuliers} + +La distribution de Laplace asymétrique généralisée peut être +considérée comme une extension de plusieurs cas particuliers. Ces +distributions ainsi que leur notation et leur fonction de densité sont +présentées dans la table \ref{tab:casspeciauxGAL}. +\begin{table}[!ht] + \centering + \begin{tabular}{cccc} + \hline + \textbf{Cas} & \textbf{Distribution} & \textbf{Notation} & \textbf{Densité} \\ + \hline + $\theta=0$ & \multirow{4}{*}{Exponentielle de moyenne $\mu>0$} & \multirow{4}{*}{$GAL(0,0,\mu,1)$} & \multirow{4}{*}{$\frac{1}{\mu}e^{-x/\mu}\,(x > 0)$} \\ + $\sigma=0$ & & & \\ + $\mu>0$ & & & \\ + $\tau=1$ & & & \\ \hline + $\theta=0$ & \multirow{3}{*}{Gamma de paramètres $\alpha=\tau$, $\beta=\mu>0$} & + \multirow{3}{*}{$GAL(0,0,\mu,\tau)$} & \multirow{3}{*}{$\frac{x^{\tau-1}e^{-x/\mu}}{\mu^\tau\Gamma(\tau)},(x > 0)$} \\ + $\sigma=0$ & & & \\ + $\mu>0$ & & & \\ \hline + $\sigma>0$ & \multirow{3}{*}{Laplace symétrique $(\sigma>0)$} & + \multirow{3}{*}{$GAL(\theta,\sigma,0,1)$} & \multirow{3}{*}{$\frac{1}{\sqrt{2}\sigma}e^{-\sqrt{2}|x-\theta|/\sigma},(x \in \mathbb{R})$} \\ + $\mu=0$ & & & \\ + $\tau=1$ & & & \\ \hline + $\sigma>0$ & \multirow{3}{*}{Laplace asymétrique $(\sigma>0)$} & + \multirow{3}{*}{$GAL(\theta,\sigma,\mu,1)$} & \multirow{3}{*}{$\frac{\sqrt{2}\kappa}{\sigma(1+\kappa^2)}\begin{cases}e^{\frac{\sqrt{2}\kappa}{\sigma}(\theta-x)},\quad & x\geq\theta \\ e^{\frac{\sqrt{2}}{\sigma\kappa}(x-\theta)},\quad & x < \theta \end{cases}$} \\ + $\mu \neq 0$ & & & \\ + $\tau=1$ & & & \\ \hline + $\sigma=0$ & \multirow{3}{*}{Dégénérée à $\theta$} & \multirow{3}{*}{$GAL(\theta,0,0,0)$} & \multirow{3}{*}{$1_{x=\theta}$} \\ + $\mu=0$ & & & \\ + $\tau=0$ & & & \\ \hline + \end{tabular} + \caption{Cas spéciaux de la distribution de Laplace asymétrique généralisée} + \label{tab:casspeciauxGAL} +\end{table} + +Les deux premiers cas spéciaux sont des distributions de probabilité +classiques. L’attention sera portée sur la distribution de Laplace +asymétrique et son cas spécial, celle de Laplace symétrique. La +dégénérée n'a aucune application pratique, mais en mentionner +l'existence est pertinent, puisque ce ne sont pas toutes les +distributions qui admettent ce cas. + +\subsection{Distribution de Laplace asymétrique} +\label{sec:distributionAL} + +La distribution de Laplace asymétrique a été introduite +par \cite{hinkley1977estimation} et étudiée en profondeur par +\cite{kozubowski1999class}. Cette distribution a elle aussi plusieurs +caractéristiques intéressantes pour remplacer la normale, dans le +contexte de la modélisation financière, comme le présentent +\cite{kozubowski2001asymmetric}. Elle est notée +$AL(\theta,\sigma,\kappa)$. + +Cette distribution, en tant que cas particulier de la distribution de +Laplace asymétrique généralisée, conserve la majorité de ses +propriétés. Entre autres, elle est infiniment divisible et possède des moments +finis. Les paramètres $\theta$, $\sigma$ et $\kappa$ conservent leurs +rôles et propriétés définis à la section \ref{sec:momentsGAL}. On peut +donc contrôler la localisation, l'échelle et l'asymétrie de la +distribution. Cependant, aucun paramètre ne permet de contrôler le +coefficient d'aplatissement. + +La densité de probabilité de cette distribution +$f(x;\theta,\sigma,\kappa)$ présente une forme analytique simple qui +ne requiert pas le recours à la fonction de Bessel modifiée de +troisième type \eqref{eq:BesselK}. +\begin{align} + \label{eq:densiteAL} + f(x;\theta,\sigma,\kappa) = + \frac{\sqrt{2}}{\sigma}\frac{\kappa}{1+\kappa^2} \begin{cases} + \exp\left(\frac{-\sqrt{2}\kappa}{\sigma}\left( x-\theta \right)\right),\quad & x\geq\theta \\ + \exp\left( \frac{\sqrt{2}}{\theta\kappa}\left(\theta - x \right) + \right),\quad & x<\theta.\end{cases} +\end{align} + +La fonction caractéristique de cette distribution prend la forme +suivante: +\begin{align} + \label{eq:caractALkappa} + \phi(t;\theta,\sigma,\kappa) = \frac{e^{i \theta + t}}{1+\frac{\sigma^2 t^2}{2}- + i\frac{\sigma}{\sqrt{2}}\left(\frac{1}{\kappa}-\kappa \right) + t}\,. +\end{align} + +Cette dernière est utilisée principalement lorsque l'application d'une +propriété nécessite une distribution avec un seul paramètre d'échelle. +La seconde paramétrisation de cette distribution utilise le paramètre +$\mu$ qui n'est pas invariant d'échelle. Cette paramétrisation donne +une forme beaucoup plus simple de la fonction caractéristique +\eqref{eq:caractALkappa}: +\begin{align} + \label{eq:caractALmu} + \phi(t\theta,\sigma,\mu) = \frac{e^{i \theta t}}{1+\frac{\sigma^2 + t^2}{2}- i\mu t}. +\end{align} + +On reconnait la fonction caractéristique d'un incrément du processus +de Laplace \eqref{eq:fncaractprocessuslaplace} avec les paramètres +($\tau=1$, $t=1$). Une forme standardisée de cette distribution existe +aussi avec le paramètre d'échelle $\sigma=1$ et de dérive $\theta=0$. +Enfin, on peut avoir une représentation alternative de cette +distribution, utilisée principalement pour la simulation, basée sur +l'exponentielle: +\begin{align} + \label{eq:ALrepsimul} + Y &= \theta + \frac{\sigma}{\sqrt{2}} \left(\frac{1}{\kappa}W_1 - k W_2 \right) +\end{align} +où +\begin{align*} + W_1 \sim \mbox{Exponentielle}\left(\frac{\sigma}{\sqrt{2}\kappa}\right) \nonumber \mbox{ et } W_2 \sim \mbox{Exponentielle}\left(\frac{\sigma\kappa}{\sqrt{2}}\right) +\end{align*} + +Cette méthode n'est cependant pas la plus efficace pour simuler une +réalisation $Y$ de la distribution de Laplace asymétrique, car on peut +la faire à partir de deux réalisations indépendantes $U_1$ et $U_2 $ +d'une variable aléatoire uniforme définie sur l'intervalle +$\left[0,1\right]$: +\begin{align} + \label{eq:simulALunif} + Y = \theta + \frac{\sigma}{\sqrt{2}} + \ln{\frac{U_1^\kappa}{U_2^{1/\kappa}}}. +\end{align} + +Lorsque le paramètre $\tau$ de la distribution de Laplace asymétrique +généralisée est un entier, on peut générer une réalisation $X$ de +celle-ci en sommant un total de $\tau$ réalisations indépendantes de +la variable aléatoire $Y$: +\begin{align*} + X &= \sum_{i=1}^{\tau} Y_{i} \\ + Y_{i} &\sim AL(\theta,\sigma,\kappa). +\end{align*} + +\section{Relation avec le modèle de \cite{madan1990variance}} +\label{sec:relation-avec-le} + +Le modèle Variance Gamma de \cite{madan1990variance} utilise une +paramétrisation différente de celle de \cite{kotz2001laplace}. On +doit, afin de conserver une compatibilité des développements des +prochains chapitres, détailler les changements de paramètres qui +permettent de passer d'un modèle à l'autre. Le modèle Variance Gamma +utilise les paramètres $C,\sigma,\theta \mbox{ et } \nu$. Afin +d'éviter la confusion, les paramètres seront indicés de 1 à 3 selon le +modèle d'où ils proviennent. Le modèle 1 sera celui de +\cite{madan1990variance}, les modèles 2 et 3 seront respectivement les +paramétrisations en $\mu$ et en $\kappa$ de +\cite{kotz2001laplace}. Les différentes équivalences se trouvent à la +table \ref{tab:chparam}. +\begin{table}[!ht] + \centering + \begin{tabular}{cccccc} + \hline + \textbf{Vers le modèle} & & & \textbf{À partir du modèle} & & \textbf{À partir du modèle} \\ \hline + \textbf{1} & & & \textbf{2} & & \textbf{3} \\ \hline + + & $C$ & $=$ & $\theta_2$ & $=$ & $\theta_3$ \\ + \multicolumn{ 1}{l}{} & $\sigma_1$ & $=$ & $\sqrt{\tau_2}\sigma_2$ & $=$ & $\sqrt{\tau_3}\sigma_3$ \\ + \multicolumn{ 1}{l}{} & $\theta_1$ & $=$ & $\mu\tau_2$ & $=$ & $\tau_3\sigma_3(1/\kappa - \kappa)/\sqrt{2}$ \\ + \multicolumn{ 1}{l}{} & $\nu$ & $=$ & $1/\tau_2$ & $=$ & $1/\tau_3$ \\ + \hline + \textbf{2} & & & \textbf{1} & & \textbf{3} \\ \hline + & $\theta_2$ & $=$ & $C$ & $=$ & $\theta_3$ \\ + \multicolumn{ 1}{l}{} & $\sigma_2$ & $=$ & $\sigma_1\sqrt{\nu}$ & $=$ & $\sigma_3$ \\ + \multicolumn{ 1}{l}{} & $\mu$ & $=$ & $\theta_1*\nu$ & $=$ & $\sigma_3(1/\kappa - \kappa)/\sqrt{2}$ \\ + \multicolumn{ 1}{l}{} & $\tau_2$ & $=$ & $1/\nu$ & $=$ & $\tau_3$ \\ + \hline + \textbf{3} & & & \textbf{1} & & \textbf{2} \\ \hline + & $\theta_3$ & $=$ & $C$ & $=$ & $\theta_2$ \\ + \multicolumn{ 1}{l}{} & $\sigma_3$ & $=$ & $\sigma_1\sqrt{\nu}$ & $=$ & $\sigma_2$ \\ + \multicolumn{ 1}{l}{} & $\kappa$ & $=$ & $\frac{\sqrt{2(\sigma_1\sqrt{\nu})^2 + (\theta_1*\nu)^2} – \theta_1*\nu}{\sigma_1\sqrt{\nu}\sqrt{2}}$ & $=$ & $\frac{\sqrt{2(\sigma_2)^2 + \mu^2} - \mu}{\sigma_2\sqrt{2}}$ \\ + \multicolumn{ 1}{l}{} & $\tau_3$ & $=$ & $1/\nu$ & $=$ & $\tau_2$ \\ \hline + \end{tabular} + \caption{Changements de paramétrisation} + \label{tab:chparam} +\end{table} + + + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: + diff --git a/depotfinal/908144032/chapitre3.tex b/depotfinal/908144032/chapitre3.tex new file mode 100644 index 0000000..d132663 --- /dev/null +++ b/depotfinal/908144032/chapitre3.tex @@ -0,0 +1,631 @@ +\chapter{Approximation de la densité et de la fonction de + répartition} % numéroté + +Il existe plusieurs situations où il n'est pas possible d'obtenir une +forme analytique des fonctions de densité et de répartition de la +distribution d'une variable aléatoire. C'est particulièrement le cas +pour la majorité des distributions issues d'un processus de Lévy, des +sommes de longueur finie ou aléatoire de variables aléatoires et des +mélanges de distributions. Cependant, dans la majorité de ces cas, il +est facile d'obtenir la fonction caractéristique ou encore la fonction +génératrice des moments. D'ailleurs, il pourrait être intéressant +d'utiliser l'intégration numérique afin d'obtenir les transformées de +Fourier ou de Laplace de ces fonctions. Par contre, la convergence de +ces méthodes d'intégration est souvent lente, c'est-à-dire qu'il faut +un pas de discrétisation très fin afin d'obtenir une approximation +satisfaisante \citep{lugannani1980saddle}. Il est aussi possible +d'utiliser l'algorithme de la transformée de Fourier rapide, mais +celui-ci a comme désavantage d'être très peu précis dans l'évaluation +des probabilités aux extrémités du support de la densité (figure +\ref{fig:probdroite}). +\begin{figure}[!ht] + \centering \input{./graphiques/probdroite.tex} + \caption{Probabilité à l'extrémité du support} + \label{fig:probdroite} +\end{figure} + +C'est dans cette optique qu'ont été développées les approximations par +la méthode du point de selle. Puisque celle-ci requiert l'utilisation +de l'approximation de Laplace pour la fonction de densité et de +l'approximation de Temme pour la fonction de répartition, on les +introduit avant de développer la méthode à proprement parler. Il est à +noter que la majorité de ces démonstrations sont tirées de +\cite{butler2007saddlepoint} qui présente une monographie assez +complète sur la méthode du point de selle. + +\section{L'approximation de Laplace} +\label{sec:appr-de-lapl} + +L'approximation de Laplace sera utilisée afin de développer la méthode +du point de selle pour la fonction de densité. Soit $g(x)$, une +fonction régulière sur l'intervalle $\left[c,d\right]$ ayant un +minimum global au point $\hat{x} \in +\left]c,d\right]$. L'approximation de Laplace de premier ordre prend +la forme suivante: +\begin{align} + \label{eq:approxlaplace1} + \int_{c}^{d} e^{-g(x)}dx &\approx + \frac{\sqrt{2\pi}e^{-g(\hat{x})}}{\sqrt{g^{\prime\prime}(\hat{x})}}. +\end{align} + +La démonstration se fait facilement en utilisant le développement de +Taylor de $g(x)$ autour du point $\hat{x}$: +\begin{align} + \label{eq:taylorgx} + g(x) &= g(\hat{x}) + g^{\prime}(\hat{x})(x-\hat{x}) + \frac{1}{2} + g^{\prime\prime}(\hat{x})(x-\hat{x})^2 + R. +\end{align} + +La valeur $R$ représente le reste du développement et est une quantité +relativement petite. Comme $\hat{x}$ est un minimum local, alors la +dérivée première $g^{\prime}(\hat{x})$ vaut $0$ et la dérivée seconde +$g^{\prime\prime}(\hat{x})$ est supérieure à 0. On obtient donc +l'intégrale suivante, qui ressemble à celle de la fonction de +répartition de la distribution normale: +\begin{align} + \label{eq:integraleapproxlaplace1} + e^{-g(\hat{x})}\int_{c}^{d} + \exp{\left\{-\frac{1}{2}g^{\prime\prime}(\hat{x})(x-\hat{x})^2 + \right\}}dx &\approx e^{-g(\hat{x})} + \sqrt{\frac{2\pi}{g^{\prime\prime}(\hat{x})}}. +\end{align} + +En utilisant un terme de plus du développement de Taylor +\eqref{eq:taylorgx}, on obtient l'approximation de Laplace de second +ordre, qui a une plus grande précision: +\begin{align} + \label{eq:approxlaplace2} + \int_{c}^{d} e^{-g(x)}dx &\approx + \frac{\sqrt{2\pi}e^{-g(\hat{x})}}{\sqrt{g^{\prime\prime}(\hat{x})}}\left\{\frac{5}{24}\hat{\kappa}_3^2 + - \frac{1}{8} \hat{\kappa}_4 \right\} +\end{align} + +où +\begin{align} + \hat{\kappa}_i &= + \frac{g^{(i)}(\hat{x})}{\left\{g^{\prime\prime}(\hat{x})\right\}^{i/2}} + ,\quad i\geq 3 \label{eq:kappailaplace}. +\end{align} + +\section{L'approximation de Temme} +\label{sec:lappr-de-temme} + +L'approximation de Temme sera utilisée afin de dériver la méthode du +point de selle pour la fonction de répartition. On considère une +variable aléatoire $Z_{w_0}$ de distribution normale standard tronquée +de telle sorte qu'elle ne puisse prendre des valeurs supérieures au +point $w_0$. +\begin{figure}[!ht] + \centering \input{./graphiques/normaletronque.tex} + \caption{Distribution normale tronquée à $w_0$} + \label{fig:normaletronque} +\end{figure} + +On définit l'espérance d'une fonction $h$ de cette variable aléatoire, +à partir des fonctions de densité $\phi(w)$ et de répartition +$\Phi(w)$ de la loi normale, comme suit: +\begin{align} + E[h(Z_{w_0})] &= \frac{\int_{-\infty}^{w_0} h(w)\phi(w) + dw}{\Phi(w_0)}. +\end{align} + +Une approximation du numérateur est donnée par le développement +suivant: +\begin{align*} + \int_{-\infty}^{w_0} h(w)\phi(w) dw &= h(0)\Phi(w_0)+\int_{-\infty}^{w_0} \frac{h(w)-h(0)}{w} w\phi(w)dw \nonumber\\ + &= h(0)\Phi(w_0)-\int_{-\infty}^{w_0} \frac{h(w)-h(0)}{w} d\phi(w)\quad \mbox{(car }w\phi(w)dw = -d\phi(w) \mbox{)} \nonumber\\ + &\approx h(0)\Phi(w_0) - \left[ \frac{h(w)-h(0)}{w} \phi(w) \right]_{w=-\infty}^{w=w_0}. \nonumber\\ +\end{align*} + +On ignore l'intégrale de l'intégration par parties, puisqu'elle prend +une petite valeur, pour obtenir l'approximation de Temme : +\begin{align} + \label{eq:temmeintegrale} + \int_{-\infty}^{w_0} h(w)\phi(w) dw &\approx h(0)\Phi(w_0) + + \phi(w_0) \left[ \frac{h(w_0)-h(0)}{w_0} \right]. +\end{align} + +\section{La méthode du point de selle} +\label{sec:pointcolGAL} + +L'approximation de la distribution d'une variable aléatoire par la +méthode du point de selle a été introduite par +\cite{daniels1954saddlepoint}. Il cherchait une façon d'estimer la +distribution de la moyenne de variables aléatoires. + +\subsection{Approximation de la densité} +\label{sec:appr-de-prem} + +Soit une variable aléatoire $Y$ dont on cherche à évaluer la fonction +de densité $f_Y(y)$. Les fonctions génératrices des moments $M_Y( s)$ +et des cumulants $K_Y( s)$ sont définies de manière équivalente par +les intégrales suivantes, où l'on cherche une forme qui permettra +d'utiliser l'approximation de Laplace: +\begin{align} + M_Y( s) = e^{K_Y( s)} &= \int_{-\infty}^{\infty} e^{ s y} f_Y(y) dy \label{eq:FGMdefinitionsaddle}\\ + &= \int_{-\infty}^{\infty} e^{ s y + \ln{f_Y(y)}} dy \label{eq:FGMdaniels}\\ + &= \int_{-\infty}^{\infty} e^{-g( s,y)} dy. +\end{align} + +Cette intégrale doit converger pour toute valeur $s$ comprise dans un +intervalle $\left]-c_1,c_2\right[$ contenant l'origine et où la somme +des bornes est positive: $c_1+c_2 > 0$. Cet intervalle doit être +choisi de sorte qu'il soit le plus grand possible. On considère pour +l'instant que la fonction $g( s,y) = - s y - \ln{f_Y(y)}$ répond aux +conditions de l'approximation de Laplace énoncées au début de la +section \ref{sec:appr-de-lapl}. On obtient alors l'approximation +suivante pour une valeur de $s$ fixée: +\begin{align} + \label{eq:laplacesaddle1} + e^{K(s)} &\approx \sqrt{\frac{2\pi}{g^{\prime\prime}( s,\hat{y}_{ + s})}}e^{ s \hat{y}_{ s}} f_Y(\hat{y}_{ s}). +\end{align} + +La valeur $\hat{y}_{s}$ est celle qui minimise la fonction $g(s,y)$ +pour un point $s$ fixé. C'est donc la solution de la condition de +premier ordre: +\begin{align} + 0 &= -g^{\prime}( s,\hat{y}_{ s}) \nonumber\\ + &= s + \frac{\partial \ln{(f_Y(\hat{y}_{ s})}}{\partial \hat{y}_{ s}} \nonumber\\ + s &= - \frac{\partial \ln{(f_Y(\hat{y}_{ s})}}{\partial \hat{y}_{ + s}}. \label{eq:premierordresaddle} +\end{align} + +L'hypothèse posée afin de faire l'approximation est que la fonction +$g(s)$ est convexe, ce qui implique que la dérivée seconde de celle-ci +est positive $(\frac{\partial^2g( s,y)}{\partial y^2} > 0)$ par +définition. On examine la relation entre le point de selle $s$ et le +point $\hat{y}_{ s}$ déterminé par l'équation +\eqref{eq:premierordresaddle}. En dérivant celle-ci par rapport au +point $\hat{y}_{ s}$, on constate qu'il existe une relation croissante +et monotone entre les deux: +\begin{align} + \label{eq:secondordresaddle} + \frac{d s}{d\hat{y}_{ s}} &= - \frac{\partial^2 \ln{(f_Y(\hat{y}_{ + s})}}{\partial \hat{y}_{ s}^2} > 0. +\end{align} + +On doit maintenant trouver une solution de l'équation de premier ordre +\eqref{eq:premierordresaddle}. Pour ce faire, on isole le terme +$\ln{f_Y(y)}$ dans l'équation \eqref{eq:laplacesaddle1}: +\begin{align} + \label{eq:lnfysaddlelaplace} + \ln{f_Y(y)} & \approx K( s) - s \hat{y}_{ s} - \frac{1}{2} + \ln{\left(\frac{2\pi}{-\left(\frac{\partial^2 \ln{(f_Y(\hat{y}_{ + s}))}}{\partial \hat{y}_{ s}^2}\right)}\right)}. +\end{align} + +Si l'on considère que le dernier terme de la soustraction est +relativement constant par rapport au point $\hat{y}_{ s}$, alors sa +dérivée sera pratiquement nulle. Ce terme peut donc être négligé dans +la prochaine étape, où on dérive \eqref{eq:lnfysaddlelaplace} par +rapport à $y_s$: +\begin{align} + \frac{\partial \ln{f_Y(\hat{y}_{s})}}{\partial \hat{y}_{s}} + &\approx \left(K^{\prime}(s) - \hat{y}_{s}\right)\frac{\partial + s}{\partial + \hat{y}_{s} } - s.\label{eq:dlnfysaddlelaplace} +\end{align} + +En combinant l'approximation \eqref{eq:dlnfysaddlelaplace} et la +condition de premier ordre \eqref{eq:premierordresaddle}, il en +résulte que le terme de gauche de la partie de droite est +approximativement égal à 0: +\begin{align} + \label{eq:premierodreapprox} + \left(K^{\prime}( s) - \hat{y}_{ s}\right)\frac{\partial s}{\partial + \hat{y}_{s} } &\approx 0. +\end{align} + +Cela conduit donc à l'équation qui met en relation le point de +selle $s$ et le point $\hat{y}_{ s}$: +\begin{align} + \label{eq:saddlepoint} + \hat{y}_{ s} &= K^{\prime}(s). +\end{align} + +Enfin, la dernière relation consiste à exprimer $g^{\prime\prime}( +s,\hat{y}_{ s})$ seulement en fonction du point de selle $s$ en +utilisant la condition de second ordre \eqref{eq:secondordresaddle} et +en dérivant l'équation précédente \eqref{eq:saddlepoint} en $s$: +\begin{align} + \label{eq:deriveesecondesaddlepoint} + g^{\prime\prime}( s,\hat{y}_{ s}) &= - \frac{\partial^2 + \ln{(f_Y(\hat{y_{s}}))}}{\partial \hat{y}_{ s}^2} \nonumber\\ + &= \frac{d s}{d\hat{y}_{ s}} \nonumber\\ + &= \left(\frac{d\hat{y}_{ s}}{d s}\right)^{-1} \nonumber\\ + &= \left(K^{\prime\prime}( s)\right)^{-1}. +\end{align} + +On peut donc réécrire l'approximation de Laplace +\eqref{eq:laplacesaddle1} en utilisant les relations +\eqref{eq:saddlepoint} et \eqref{eq:deriveesecondesaddlepoint}: +\begin{align} + e^{K( s)} &\approx \sqrt{2\pi K^{\prime\prime}( s)}\exp{\left\{ s + \hat{y}_{ s}\right\}} + f_Y(\hat{y_{s}}) \nonumber\\ + \hat{f_Y}(y_{s}) &\approx \frac{1}{\sqrt{2\pi K^{\prime\prime}( + \hat{s})}} \exp{\left\{K(\hat{s}) - \hat{s} y\right\}} = + \hat{f}_Y(y).\label{eq:approximationsaddlepointordre1} +\end{align} + +L'équation \eqref{eq:approximationsaddlepointordre1} est une +approximation par la méthode du point de selle de la densité de la +variable aléatoire $Y$ au point $y_{s}$. + +Par convention, on considère que l'approximation $\hat{f}_Y(y)$ est +une fonction de densité, mais, elle n'en est pas exactement une +puisqu'elle n'intègre pas à 1. Cependant, il est possible de la +normaliser en calculant la valeur de l'intégrale de celle-ci sur le +domaine $\chi$ de la distribution, puis en divisant l'approximation +$\hat{f}_Y(y)$ par celle-ci: +\begin{subequations}\label{eq:normalisationsaddle} + \begin{align} + c &= \int_{\chi} \hat{f}_Y(y) dy \neq 1 \label{eq:normalisationsaddle1}\\ + \overline{f}_Y(y) &= c^{-1} \hat{f}_Y(y),\quad \forall y \in + \chi. \label{eq:normalisationsaddle2} + \end{align} +\end{subequations} + +\subsection{Unicité du point de selle} +\label{sec:unicite-du-point} + +\cite{daniels1954saddlepoint} démontre que l'équation du point de +selle \eqref{eq:saddlepoint} a une solution unique dans l'intervalle +$\left[-c_1,c_2 \right]$ sélectionné précédemment lorsque $y_0$ est +situé dans un intervalle $\left[a,b\right]$. + +Pour ce faire, il présente une condition essentielle. Les limites +inférieure et supérieure de la dérivée première de la fonction +génératrice des cumulants doivent correspondre aux points $a$ et $b$ +respectivement \eqref{eq:limitesderiveeK}: +\begin{align} + \label{eq:limitesderiveeK} + \lim_{s\to c_2} K'(s) = b \\ + \lim_{s\to -c_1} K'(s) = a. +\end{align} + +On définit la fonction génératrice des moments de la différence entre +la variable aléatoire $Y$ et le point $y_0$ par $M(s,y)$. + +Quand $a < y_0 < b$, la dérivée première prend la forme suivante: +\begin{align} + \label{eq:6} + M'(s,y_0) = \int_{a}^{b} (y-y_0) e^{s(y-y_0)} dF(y). +\end{align} + +Les limites inférieure et supérieure de la fonction $M'(s,y_0)$ sont +respectivement $M'(-\infty,y_0) = -\infty$ et $M'(\infty,y_0) = +\infty$. De plus, la dérivée de celle-ci est toujours positive +$M''(s,y_0)>0$. Donc, pour chaque valeur $a < y_0 < b$, une racine +unique $\hat{s}$ de l'équation $M'(s,y_0)=0$ existe +nécessairement. Par conséquent, l'équation du point de selle +\eqref{eq:saddlepoint}, qui lui est équivalente, a aussi une solution +réelle unique. + +% \subsection{Approximation de second ordre de la densité} +% \label{sec:appr-de-second} + +% On reprend la définition de la fonction génératrice des moments +% \eqref{eq:FGMdefinitionsaddle}. En réorganisant les termes, on définit +% une fonction de densité décalée pour chaque valeur $ s \in +% \left]a,b\right[$: +% \begin{align} +% \label{eq:nouvelledensiteordre2} +% f_Y(y; s) = \exp{\left\{ s y - K( s)\right\}} f_Y(y) +% \end{align} + +% Cet ensemble est défini comme étant la famille de densités décalées +% par rapport au point de selle $s$, et a été développé par +% Esscher. Soit $Y_{ s}$ la variable aléatoire ayant cette densité, dont +% la moyenne et la variance sont définies comme suit: +% \begin{align} +% E[Y_{ s}] &= K^{\prime}( s) \label{eq:moyennetilted}\\ +% Var[Y_{ s}] &= K^{\prime\prime}( s) \label{eq:vartilted} +% \end{align} + +% On remarque que lorsque l'on situe le point de selle à l'origine +% $(s=0)$, on retrouve la définition usuelle des deux premiers +% cumulants. On définit les cumulants standardisés de la variable +% aléatoire $Y_{ s}$, qui sont basés sur les coefficients de +% l'approximation de Laplace du second ordre \eqref{eq:kappailaplace}. +% \begin{align} +% \hat{\kappa}_i &= +% \frac{K^{(i)}(\hat{x})}{\left\{K^{\prime\prime}(\hat{x})\right\}^{i/2}} +% ,\quad i\geq 3 \label{eq:kappaitilting} +% \end{align} + +% On définit l'expansion d'Edgeworth d'ordre 2 +% \nocite{abramowitz1965handbook} autour de la moyenne comme étant +% l'approximation suivante, basée sur la distribution normale +% $\mathcal{N}(\mu,\sigma^2)$: +% \begin{align} +% \label{eq:edgeworthmoyenne} +% f_Y(\mu) &\approx +% \frac{1}{\sigma\sqrt{2\pi}}\left(1+\frac{1}{8}\kappa_4 - +% \frac{5}{24} \kappa_3^2 \right) +% \end{align} + +% La technique développée par Esscher utilise indirectement cette +% expansion. Elle consiste en deux étapes: +% \begin{enumerate} +% \item Représenter la fonction de densité de la variable aléatoire $Y$ +% en l'isolant dans l'expression \eqref{eq:nouvelledensiteordre2}: +% \begin{align} +% \label{eq:etape1esscher} +% f_Y(y) = \exp{\left\{K( s)- s y \right\}} f_Y(y; s) +% \end{align} +% \item Utiliser l'expansion \eqref{eq:edgeworthmoyenne} de la densité +% $f_Y(y; s)$ autour d'un point $ s $ optimal, qui se trouve à être la +% moyenne de la variable $Y_{ s}$ définie par +% \eqref{eq:moyennetilted}, ce qui revient à résoudre l'équation du +% point de selle \eqref{eq:saddlepoint} en $s$: +% \begin{align*} +% E[Y_{\hat{s}}] &= K^{\prime}(\hat{s}) = y +% \end{align*} +% \end{enumerate} + +% L'expansion prend donc la forme suivante en remplaçant l'écart-type +% $\sigma$ par la variance de la variable aléatoire $Y_{ s}$, exprimée à +% l'aide de la fonction génératrice des cumulants \eqref{eq:vartilted}. +% \begin{align} +% \label{eq:edgeworthordre2} +% f_Y(y;\hat{s}) &\approx \frac{1}{\sqrt{2\pi +% K^{\prime\prime}(\hat{s})}} \left(1+\frac{1}{8}\kappa_4(\hat{s}) +% - \frac{5}{24} \kappa_3^2(\hat{s}) \right) +% \end{align} +% En substituant l'expansion \eqref{eq:edgeworthordre2} dans l'équation +% \eqref{eq:etape1esscher}, on obtient l'approximation par la méthode du +% point de selle d'ordre 2 de la densité de la variable aléatoire $Y$. +% \begin{align} +% \label{eq:approximationsaddlepointordre2} +% f_Y(y) &\approx \frac{1}{\sqrt{2\pi K^{\prime\prime}(\hat{s})}} +% \exp{\left\{K(\hat{s})-\hat{s} y \right\}} +% \left(1+\frac{1}{8}\kappa_4(\hat{s}) - \frac{5}{24} +% \kappa_3^2(\hat{s}) \right) +% \end{align} + +\subsection{Approximation de la fonction de + répartition} +\label{sec:appr-de-prem-1} + +La fonction de répartition de la variable aléatoire $Y$ au point $y$ +est définie comme étant la probabilité $P(y \leq Y)$ et s'obtient en +intégrant la densité sur l'intervalle $\left[ -\infty,y \right]$. On +considère le changement de variable $y \mapsto \hat{w}$ défini comme +suit: +\begin{align} + \label{eq:chengementytow} + \frac{\hat{w}^2}{2} = \hat{s} y - K(\hat{s}). +\end{align} + +La valeur $\hat{w}_y$ est obtenue en résolvant l'équation du point de +selle pour la racine $\hat{s}$ et en remplaçant celle-ci dans +l'équation \eqref{eq:chengementytow}. Ce changement de variable est +une transformation monotone, continue et croissante dont la dérivée +est donnée par l'équation suivante: +\begin{align} + \label{eq:derivchengementytow} + \frac{dy}{d\hat{w}} = \begin{cases} + \hat{w} / \hat{s}, & \hat{s} \neq 0 \\ + \sqrt{K^{\prime\prime}(0)}, & \hat{s} = 0. + \end{cases} +\end{align} + +L'approximation de premier ordre de la fonction de répartition +s'obtient en intégrant celle de premier ordre de la densité +\eqref{eq:approximationsaddlepointordre1} sur ce même intervalle à +l'aide du changement de variable \eqref{eq:chengementytow}: +\begin{align} + \label{eq:approxintegraleREPordre1} + F_Y(y) &\approx \int_{-\infty}^{y} \frac{1}{\sqrt{2\pi + K^{\prime\prime}( s)}} + \exp{\left\{K( s) - s x\right\}} dx \nonumber\\ + &= \int_{-\infty}^{y} \frac{1}{\sqrt{K^{\prime\prime}( s)}} + \phi(\hat{w}) dx \nonumber\\ + &= \int_{-\infty}^{\hat{w}_y} + \frac{\hat{w}}{\hat{s}\sqrt{K^{\prime\prime}( s)}}\phi(\hat{w}) + d\hat{w}. +\end{align} + +On applique l'approximation de Temme à l'intégrale précédente avec +l'équation: +\begin{align*} + h(\hat{w})=\frac{\hat{w}}{\hat{s}\sqrt{K^{\prime\prime}( s)}}. +\end{align*} + +On obtient ainsi l'approximation de premier ordre de la fonction de +répartition: +\begin{align} + \label{eq:approximationsaddlepointREPordre1} + F_Y(y) &\approx \Phi(\hat{w}_y) + + \phi(\hat{w}_y)\frac{1-h(\hat{w}_y)}{\hat{w}_y} \nonumber\\ + & \approx \Phi(\hat{w}_y) + \phi(\hat{w}_y) + \left(\frac{1}{\hat{w}_y}-\frac{1}{\hat{u}_y} \right). +\end{align} + +Les deux constantes $\hat{w}_y$ et $\hat{u}_y$ sont respectivement +définies comme suit: +\begin{subequations}\label{eq:wuysaddlerepart2} + \begin{align} + \hat{w}_y &= \sgn(\hat{s})\sqrt{2\left\{\hat{s} y - K(\hat{s})\right\}} \label{eq:wysaddlerepart2}\\ + \hat{u}_y &= + \hat{s}\sqrt{K^{\prime\prime}(\hat{s})} \label{eq:uysaddlerepart2}. + \end{align} +\end{subequations} + +% \subsection{Approximation de second ordre de la fonction de +% répartition} +% \label{sec:appr-de-second-1} + +% On reprend le développement précédent +% \eqref{eq:approxintegraleREPordre1}, mais avec l'approximation de +% second ordre de la densité \eqref{eq:approximationsaddlepointordre2} +% \citep{daniels1987tail}. +% \begin{align} +% \label{eq:approxintegraleREPordre2} +% F_Y(y) &\approx \int_{-\infty}^{y} \frac{\exp{\left\{K( s) - s +% x\right\}}}{\sqrt{2\pi K^{\prime\prime}( s)}} +% \left(1+\frac{1}{8}\kappa_4(\hat{s}) - \frac{5}{24} +% \kappa_3^2(\hat{s}) \right) dx \nonumber\\ +% &= \int_{-\infty}^{\hat{w}_y} +% \frac{\hat{w}}{\hat{s}\sqrt{K^{\prime\prime}( +% s)}}\left(1+\frac{1}{8}\kappa_4(\hat{s}) - \frac{5}{24} +% \kappa_3^2(\hat{s}) \right)\phi(\hat{w}) d\hat{w} +% \end{align} + +% La fonction à utiliser pour l'approximation de Temme est alors +% \begin{align*} +% h(\hat{w})=\frac{\hat{w}}{\hat{s}\sqrt{K^{\prime\prime}( +% s)}}\left(1+\frac{1}{8}\kappa_4(\hat{s}) - \frac{5}{24} +% \kappa_3^2(\hat{s}) \right)\phi(\hat{w}) +% \end{align*} + +% Ainsi, on obtient l'approximation de second ordre de la fonction de +% répartition. +% \begin{align} +% \label{eq:approximationsaddlepointREPordre2} +% F_Y(y) & \approx \Phi(\hat{w}_y) + \phi(\hat{w}_y) +% \left(\frac{1}{\hat{w}_y}-\frac{1}{\hat{u}_y}(1-\frac{1}{8} +% \hat{\kappa}_4+\frac{5}{24}\hat{\kappa}_3^2) \right) +% \end{align} + +\subsection{Quelques propriétés des approximations} +\label{sec:proprietesaddle} + +Les approximations par la méthode du point de selle ont pour +principale propriété de respecter le principe d'invariance. Cette +propriété permet notamment d'obtenir une approximation de la densité +ainsi que la fonction de répartition en utilisant des paramètres +estimés à partir d'un échantillon de données centrées et +réduites. Puis, elle permet d'appliquer la transformation +linéaire $Y = \sigma{X}+\mu$, telle qu'utilisée à la section +\ref{sec:transGAL}, tout en conservant la même approximation. + +Une autre propriété de ces approximations est qu'elle respecte la +symétrie de la distribution. Ainsi, lorsque la distribution est +asymétrique, on doit s'attendre à ce que l'approximation le soit +aussi. + +\section{Application de la méthode du point de selle} + +\subsection{Approximation de la densité} +\label{sec:applicationsaddleGAL} + +On applique la méthode du point de selle à la distribution de Laplace +asymétrique généralisée, afin d'obtenir une approximation de la +densité \eqref{eq:densitekotz} et aussi pouvoir représenter la +fonction de répartition, car la première n'est pas intégrable +analytiquement. Cependant, pour des fins de simplification, la +paramétrisation $\mu$ sera utilisée. On rappelle qu'on peut passer +d'une forme à l'autre à l'aide des équations \eqref{eq:mukappa} et +\eqref{eq:kappamu}. + +On évalue d'abord la dérivée première de la fonction génératrice des +cumulants \eqref{eq:fgcGAL} par rapport à la variable de +transformation $s$: +\begin{align} + \label{eq:derivcumulantGAL1} + K^{\prime}_Y({s}) &= + \frac{\partial}{\partial{s}}\ln\left(\frac{e^{\theta + {s}}}{\left(1-\frac{1}{2} \sigma^2 {s}^2 - \mu {s} + \right)^{\tau}}\right) \nonumber\\ + &= \frac{{\sigma}^{2}\theta{{s}}^{2}+\left( + 2\mu\theta-2{\sigma}^{2}\tau\right) + {s}-2\theta-2\mu\tau}{{\sigma}^{2}{{s}}^{2}+2\mu{s}-2}, \\ & \quad + \mbox{où } 1-\frac{1}{2} \sigma^2 {s}^2 - \mu {s} > 0. +\end{align} + +On détermine ensuite le point de selle $\hat{s}_y$ correspondant à la +valeur $y$ pour laquelle on veut estimer la fonction de densité ou de +répartition, à l'aide de l'équation \eqref{eq:saddlepoint} et de la +première dérivée de la fonction génératrice des cumulants +\eqref{eq:derivcumulantGAL1}: +\begin{align*} + y &= K^{\prime}_Y({\hat{s}_y}) \\ + &= \frac{{\sigma}^{2}\theta{\hat{s}_y}^{2}+\left( + 2\mu\theta-2{\sigma}^{2}\tau\right) + {\hat{s}_y}-2\theta-2\mu\tau}{{\sigma}^{2}{{\hat{s}_y}}^{2}+2\mu{\hat{s}_y}-2}. +\end{align*} + +L'équation précédente a deux solutions dont une seule correspond à un +minimum. Pour déterminer quelle solution est appropriée, on doit +évaluer la dérivée seconde en ces deux points: +\begin{align} + \label{eq:saddlepointGAL} + \hat{s}_y &= + \frac{\pm\sqrt{2{\sigma}^{2}{{y}}^{2}+{\mu}^{2}{{y}}^{2}-4{\sigma}^{2}\theta{y}-2{\mu}^{2}\theta{y}+2{\sigma}^{2}{\theta}^{2}+{\mu}^{2}{\theta}^{2}+{\sigma}^{4}{\tau}^{2}}}{{\sigma}^{2}(y-\theta)} + -\frac{\mu}{\sigma^{2}} -\frac{\tau}{y-\theta}. +\end{align} + +On évalue la dérivée seconde de la fonction génératrice des cumulants: +\begin{align} + \label{eq:derivcumulantGAL2} + K^{\prime\prime}_Y({s}) &= \frac{2\left( + {s}^{2}{\sigma}^{4}+2\mu{s}{\sigma}^{2}+2{\sigma}^{2}+2{\mu}^{2}\right) + \tau}{{\left( {s}^{2}{\sigma}^{2}+2\mu{s}-2\right) }^{2}} > 0. +\end{align} + +On peut maintenant déterminer quelle valeur $\hat{s}_y$ utiliser, car +la fonction $K^{\prime\prime}_Y({s})$ sera positive pour celle-ci. De manière équivalente, on utilise la condition +\ref{eq:fgmGALcond}, qui permet l'existence de la fonction génératrice +des moments, afin d'identifier plus aisément la solution unique. + +On peut maintenant évaluer \eqref{eq:approximationsaddlepointordre1}, +pour obtenir l'approximation de premier ordre de la fonction de +densité de la distribution de Laplace asymétrique généralisée. + +% \subsection{Approximation de second ordre de la densité} + +% L'évaluation de l'approximation de second ordre nécessite les +% dérivées troisième et quatrième de la fonction génératrice des +% cumulants: +% \begin{align} +% \label{eq:K3K4GAL} +% K^{\prime\prime\prime}({s}) &= -\frac{4\tau\left( {\sigma}^{2}{s}+\mu\right) \left( {\sigma}^{4}{{s}}^{2}+2\mu{\sigma}^{2}{s}+6{\sigma}^{2}+4{\mu}^{2}\right) }{{\left( {\sigma}^{2}{{s}}^{2}+2\mu{s}-2\right) }^{3}} \\ +% K^{iv}({s}) &= \frac{12\tau\left( +% {\sigma}^{8}{s}^{4}+4\mu{\sigma}^{6}{s}^{3}+12{\sigma}^{6}{s}^{2}+12{\mu}^{2}{\sigma}^{4}{s}^{2}+24\mu{\sigma}^{4}{s}+16{\mu}^{3}{\sigma}^{2}{s}+4{\sigma}^{4}+16{\mu}^{2}{\sigma}^{2}+8{\mu}^{4}\right) +% }{{\left( {\sigma}^{2}{s}^{2}+2\mu{s}-2\right) }^{4}} +% \end{align} + +% On peut ainsi évaluer les cumulants normalisés $\kappa_3(s)$ et +% $\kappa_4(s)$ à l'aide de la définition \eqref{eq:kappaitilting}. +% \begin{align} +% \kappa_3(s) &= -\frac{\sqrt{2}\left( s{\sigma}^{2}+\mu\right) \left( {s}^{2}{\sigma}^{4}+2\mu{s}{\sigma}^{2}+6{\sigma}^{2}+4{\mu}^{2}\right) \left| {s}^{2}{\sigma}^{2}+2\mu{s}-2\right| \tau}{\left( {s}^{2}{\sigma}^{2}+2\mu{s}-2\right) {\left( \left( {s}^{2}{\sigma}^{4}+2\mu{s}{\sigma}^{2}+2{\sigma}^{2}+2{\mu}^{2}\right) \tau\right) }^{\frac{3}{2}}} \label{eq:kappa3gal}\\ +% \kappa_4(s) &= \frac{3\left( +% {s}^{4}{\sigma}^{8}+4\mu{s}^{3}{\sigma}^{6}+12{s}^{2}{\sigma}^{6}+12{\mu}^{2}{s}^{2}{\sigma}^{4}+24\mu{s}{\sigma}^{4}+4{\sigma}^{4}+16{\mu}^{3}{s}{\sigma}^{2}+16{\mu}^{2}{\sigma}^{2}+8{\mu}^{4}\right) +% }{{\left( +% {s}^{2}{\sigma}^{4}+2\mu{s}{\sigma}^{2}+2{\sigma}^{2}+2{\mu}^{2}\right) +% }^{2}\tau} \label{eq:kappa4gal} +% \end{align} + +% En remplaçant ces valeurs dans l'équation +% \eqref{eq:approximationsaddlepointordre2}, on obtient +% l'approximation de second ordre de la densité de la distribution de +% Laplace asymétrique généralisée. + +\subsection{Approximation de la fonction de répartition} + +En évaluant les valeurs $\hat{w}_y \mbox{ et } \hat{u}_y$ +\eqref{eq:wuysaddlerepart2} à l'aide de celles du point de selle +\eqref{eq:saddlepointGAL}, de la fonction génératrice des cumulants +\eqref{eq:fgcGAL} et de la dérivée seconde de cette dernière +\eqref{eq:derivcumulantGAL2}, on obtient l'approximation de premier +ordre de la fonction de répartition. + +% Pour évaluer l'approximation de second ordre de la fonction de +% répartition \eqref{eq:approximationsaddlepointREPordre2}, on reprend +% les cumulants normalisés $\kappa_3(s) \text{ et } \kappa_4(s)$ +% \eqref{eq:kappa3gal} \eqref{eq:kappa4gal} et on les remplace dans +% l'expression \eqref{eq:approximationsaddlepointREPordre2}. + +Tout au long de ce chapitre, on a supposé que les paramètres de la +distribution étaient connus. Cependant, ceux-ci doivent être estimés à +partir des données d'un échantillon représentatif de la population +étudiée. C'est d'ailleurs ce qui sera fait dans les chapitres +suivants. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/chapitre4.tex b/depotfinal/908144032/chapitre4.tex new file mode 100644 index 0000000..8ca051d --- /dev/null +++ b/depotfinal/908144032/chapitre4.tex @@ -0,0 +1,1033 @@ +\chapter{Méthode des moments généralisée} % numéroté + +Si l'on pose l'hypothèse selon laquelle un échantillon issu d'une +population a une certaine distribution de probabilité, on doit +ensuite déterminer quels sont les paramètres de celle-ci. Ces +paramètres peuvent être estimés à l'aide de différentes méthodes +statistiques. La méthode des moments généralisée est une de celles-ci. + +\section{Introduction} +\label{sec:intromethodeGMM} + +On considère un échantillon de taille $T$ formé de plusieurs +réalisations $\mathbf{y} = y(1),\ldots,y(T)$ indépendantes entre elles +et identiquement distribuées. Selon le modèle paramétrique, cet +échantillon est formé de réalisations d'une variable aléatoire $Y$ +suivant une distribution particulière. Le vecteur de paramètres de +celle-ci, $\theta$, de longueur $a$, appartenant à l'espace $\Omega +\subset \mathbb{R}^a$, doit être estimé à partir de l'échantillon. Une +première approche consiste à maximiser la fonction de vraisemblance +$L(\theta;\mathbf{y})$, qui équivaut au produit de la densité +$f(y;\theta)$ évaluée à chacune des réalisations $y(t)$: +\begin{align} + \label{eq:vraisemblance} + L(\theta;\mathbf{y}) = \prod_{t=1}^T + f(y(t);\theta),\quad\theta\in\Omega. +\end{align} + +L'estimateur $\hat\theta$ est, dans ce cas, la valeur pour +laquelle l'échantillon a la plus grande probabilité d'être +observé. Cependant, puisque cela apporte plusieurs simplifications +intéressantes, on cherchera à maximiser la fonction de +log-vraisemblance $l(\theta;\mathbf{y}) = +\ln{(L(\theta;\mathbf{y}))}$: +\begin{align} + \label{eq:thetavraisemblance} + \hat\theta = \underset{\theta}{\operatorname{arg\,max}} \, + l(\theta;\mathbf{y}). +\end{align} + +Le maximum est obtenu en résolvant la condition de premier ordre, +c'est-à-dire en égalant le gradient de la fonction de +log-vraisemblance à 0: +\begin{align} + \label{eq:EEvraisemblance} + \frac{\partial{l(\theta;\mathbf{y})}}{\partial{\theta}} &= 0. +\end{align} + +Cette méthode permet d'obtenir un estimateur convergent de variance +minimale, car elle utilise l'ensemble de l'information contenue dans +l'échantillon. Cependant, la fonction de vraisemblance doit pouvoir +être spécifiée sous une forme analytique, et de plus, celle-ci doit +être différentiable afin de pouvoir utiliser cette méthode, ce qui +n'est pas le cas avec la distribution de Laplace asymétrique +généralisée. Lorsque cette situation se présente, une méthode +alternative est préférée, celle des moments généralisée, comme +proposée par \cite{hansen1982large}. Celle-ci est décrite par +\cite{hamilton1994time} dans le contexte de l'étude des séries +chronologiques. Elle a pour avantage de nécessiter seulement la +spécification de certaines conditions de moment. Par contre, elle +n'utilise pas toute l'information fournie par l'échantillon, ce qui ne +permettra pas d'obtenir un estimateur de variance minimale. + +\subsection{Méthode classique des moments} +\label{sec:methodemoments} + +Pour certaines distributions, on ne peut directement estimer les +paramètres. On cherchera alors des fonctions des paramètres qui sont +facilement estimables de manière convergente. L'ensemble de fonctions +le plus commun qui répond à cette condition est celui des moments, +d'où le nom de la méthode. + +La méthode classique des moments a été introduite par Pearson en +1894. On considère un échantillon de taille $T$ dont les observations +seront notées $y(t)$. On veut estimer le vecteur de paramètres +$\theta$, de longueur $a$ de la distribution. On définit les moments +$\mathbf{m} = m_1, \ldots, m_a$ de la population totale, +représentée par la variable aléatoire $Y$, comme étant l'espérance des +puissances de celle-ci, et donc une fonction des paramètres de la +distribution. On considèrera le même nombre de moments que de +paramètres à estimer. +\begin{align} + \label{eq:momentspopulation} + m_i\left( \theta \right) = E \left[ Y^i \right] ,\qquad i=1, + \ldots, a +\end{align} + +Cette méthode consiste à résoudre un système d'équations +\eqref{eq:momentsechantillon} où l'on égale les moments empiriques +$m_i(\theta)$ à ceux de la distribution $\hat{m}(\mathbf{y})$. +\begin{align} + \label{eq:momentsechantillon} + \left\{\begin{array}{rcl} + m_1(\theta) &=& \frac{1}{T}\sum_{t=1}^T y_t\\ + m_2(\theta) &=& \frac{1}{T}\sum_{t=1}^T y_t^2\\ + &\vdots& \\ + m_a(\theta) &=& \frac{1}{T}\sum_{t=1}^T y_t^a + \end{array}\right\} +\end{align} + +L'estimateur des moments $\hat\theta_T$ est celui qui résout ce +système. + +\section{Méthode des moments généralisée} +\label{sec:methodeGMM} + +La méthode classique des moments utilise le même nombre d'équations +d'estimation que de paramètres $(r=a)$. De plus, le système +d'équations formé par celles-ci doit admettre une solution réelle +appartenant à l'espace des paramètres $\Omega$, ce qui n'est pas +toujours le cas. Lorsque ces deux conditions ne sont pas réunies, on +doit choisir un vecteur de paramètres $\theta$ pour lequel les moments +de la population $m_i$ ont une valeur la plus près possible de ceux +de l'échantillon $\hat{m}_i$ correspondants. Cette distance est notée +par le vecteur $g(\theta;\mathbf{y})$ et correspond au cas le plus +simple de la méthode des moments généralisée. +\begin{align} + \label{eq:1} + g(\theta;\mathbf{y}) &= \begin{bmatrix} + m_1 - \hat{m}_1\\ + \vdots\\ + m_r - \hat{m}_r\\ + \end{bmatrix}. +\end{align} + +Pour obtenir ces estimateurs, on cherchera plutôt à minimiser une +fonction objectif notée $Q\left(\theta;\mathbf{y} \right)$, qui +correspond à une norme quadratique pondérée par une matrice définie +positive $W$: +\begin{align} + Q\left(\theta;\mathbf{y} \right) \equiv g(\theta;\mathbf{y})' W + g(\theta;\mathbf{y}). +\end{align} + +\cite{hansen1982large} nomme cette procédure «méthode des moments +généralisée». Elle est aussi nommée «méthode du $\chi^2$ minimum» par +\cite{berkson1980minimum}, bien qu'elle n'en soit qu'un cas +particulier. On retrouve aussi le nom d'estimateur de distance +minimale, par \cite{wolfowitz1957minimum}. + +\subsection{Définition} +\label{sec:definitionGMM} + +On considère un vecteur $\mathbf{y}_{T}$ de longueur $T$ contenant les +données $y(t)$ de l'échantillon tirées d'une population représentée +par la variable aléatoire $Y$. On considère de plus un vecteur de +paramètres $\theta \in \Omega$ de longueur $a$ dont la vraie valeur, +qui est inconnue, est représentée par la constante $\theta_0$. Soit +une fonction vectorielle de longueur $r$ de la variable aléatoire $Y$ +appelée condition de moment ou d'orthogonalité: +\begin{align} + \label{eq:def1condmoment} + h\left(\theta,Y\right):\left(\mathbb{R}^a \times \mathbb{R}\right) + \longrightarrow \mathbb{R}^r. +\end{align} + +L'espérance de cette fonction, sous l'hypothèse $\theta = +\mathbf{\theta_0}$, est un vecteur nul noté $\mathbf{0}_r$. +\begin{align} + E \left[h\left(\theta_0,Y \right) \right] = \mathbf{0}_r. +\end{align} + +On définit aussi la fonction $g(\theta,\mathbf{y}_{T}):\mathbb{R}^a +\longrightarrow \mathbb{R}^r$ comme étant la moyenne empirique des +conditions de moment $h\left(\theta,y \right)$: +\begin{align} + \label{eq:estimateurfonctiongh} + g(\theta,\mathbf{y}_{T}) \equiv \frac{1}{T} \sum_{t=1}^T + h\left(\theta,y(t) \right). +\end{align} + +L’idée derrière la méthode des moments généralisée est de choisir un +ensemble de paramètres $\theta$ de sorte que la valeur de la fonction +$g(\theta,\mathbf{y}_{T})$ soit aussi près que possible du vecteur nul +$\mathbf{0}_r$. Selon la norme utilisée pour mesurer cette distance, +les propriétés de l'estimateur $\hat{\theta}_T$ vont varier. Étant +donné les nombreuses propriétés bien établies dans le domaine des +statistiques, on utilise la norme quadratique avec pondération, +appelée aussi moindres carrés généralisés, qui prend la forme +suivante: +\begin{align} + \label{eq:objectifGMM1} + Q(\theta,\mathbf{y}_{T}) = \left[g(\theta,\mathbf{y}_{T}) \right]' + W_T \left[g(\theta,\mathbf{y}_{T}) \right]. +\end{align} + +Cette fonction de minimisation permettra d'utiliser des tests +statistiques basés sur la distribution $\chi^2$ de Pearson. La matrice +carrée $W_T$ de dimension $r\times r$ est définie positive et est +habituellement une fonction des données de l'échantillon +$\mathbf{y}_{T}$ et des paramètres $\theta$. Une matrice de +pondération optimale sera déterminée à la section +\ref{sec:matriceWoptimaleGMM}. + +\subsection{Convergence} +\label{sec:convergenceGMM} + +Si le nombre de paramètres $a$ est égal à celui de conditions de +moments $r$, alors la fonction objectif atteindra un minimum de 0 au +point $\mathbf{\hat\theta}$. On obtiendra ce dernier en résolvant +l'équation suivante pour le paramètre $\theta$: +\begin{align} + g(\hat\theta_T,\mathbf{y}_{T})=0. \label{eq:paramegalecondmomentsGMM} +\end{align} + +Lorsque le nombre de conditions de moments est plus grand que celui +des paramètres, on ne pourra pas obtenir une solution pour l'équation +précédente \eqref{eq:paramegalecondmomentsGMM}. La proximité entre la +valeur de chaque condition de moment et $0$ sera déterminée par la +matrice de pondération $W_T$. Étant donné que la fonction +$g(\hat\theta_T,\mathbf{y}_{T})$ est la moyenne échantillonnale de la +fonction aléatoire $h\left(\theta_0,Y \right)$, on a, par la loi des +grands nombres, la relation suivante entre ces deux quantités: +\begin{align} + g(\theta,\mathbf{y}_{T}) \stackrel{P}{\longrightarrow} E + \left[h\left(\theta,Y \right) \right]. \label{eq:normequadratique} +\end{align} + +On considère la suite d'observations $\left\{y(t) \right\}_{t=1}^T$ +comme un processus stochastique. L'ensemble des conditions de +régularité suivantes permet d'obtenir un estimateur convergent +\citep{hansen1982large}. +\begin{enumerate} +\item Le processus stochastique $\left\{y(t) \right\}_{t=1}^T$ est + \textbf{stationnaire}, dont la distribution conjointe de plusieurs + observations ne change pas dans le temps: + \begin{align} + \label{eq:conditionGMM1.1} + F_{Y}(y({t(1)+\tau}) ,\ldots, y({t(k)+\tau})) = + F_{Y}(y({t(1)}),\ldots, y({t(k)})), \quad \forall \tau \in + \mathbb{R}. + \end{align} + + Il est aussi \textbf{ergodique}, c'est-à-dire que l'on peut déduire + les propriétés du processus à partir d'un échantillon (ou + réalisation) suffisamment long de celui-ci. Entre autres, la moyenne + et la variance d'un échantillon recueilli sur une période en + particulier sont représentatives de celles de n'importe quel autre + intervalle de temps de ce processus. Dans ces deux dernières + situations, on parle aussi de convergence en moyenne + ($\mathbb{L}^1$) et en norme quadratique ($\mathbb{L}^2$): + \begin{align} + \label{eq:conditionGMM1.2} + \lim_{T\longrightarrow\infty}\mathrm{E}\left(\left|Y_T-Y\right|^r\right)=0, + \quad r\in\left\{1,2\right\}. + \end{align} + +\item L'\textbf{espace métrique} $(\Omega,\sigma)$, défini par + l'espace des paramètres $\Omega$ et la norme valeur absolue est + \textbf{séparable}: + \begin{align} + \label{eq:conditionGMM2.1} + \sigma=\left|Y_T-Y\right|^r. + \end{align} + + Cette condition définit l'unicité du vecteur de paramètres, car si + la distance est nulle, les estimateurs correspondent aux vrais paramètres: + \begin{align} + \label{eq:conditionGMM2.2} + \sigma = 0 &\Longleftrightarrow \mathbf{\hat\theta} = \theta_0, + \quad \theta_0 \in \Omega. + \end{align} + +\item La fonction $h\left(\theta,\mathbf{y}\right)$ est + \textbf{mesurable au sens de Borel} pour chaque vecteur $\theta$, + c'est-à-dire qu'un sous-ensemble de l'espace des paramètres $\Omega$ + existe pour chaque valeur qu'elle peut prendre. De plus, la fonction + est continue sur l'ensemble $\Omega$ pour chaque échantillon + $\mathbf{y}$. + +\item L'espérance de la fonction $h\left(\theta, Y \right)$ existe et + est définie pour toute valeur $\theta \in \Omega$. De plus, par + définition, l'espérance de la fonction pour les vrais paramètres est + de 0: + \begin{align*} + E\left[h\left(\theta_0, Y \right)\right] &= 0. + \end{align*} + +\item La séquence de matrices de pondération $\left\{ W_T + \right\}_{T=1}^{\infty}$ converge presque sûrement, élément par + élément, vers une constante $W_0$, en utilisant la norme valeur + absolue définie précédemment \eqref{eq:conditionGMM2.1}. +\end{enumerate} + +$\hat\theta_T \in \Omega$ est un estimateur convergent de $\theta$ +lorsque les conditions précédentes sont respectées. +\begin{align} + \hat{\theta}_T &= \operatorname{arg}\min_{\theta\in\Omega} Q(\theta,\mathbf{y}_{T}) \nonumber\\ + &= \operatorname{arg}\min_{\theta\in\Omega} + \left[g(\theta,\mathbf{y}_{T}) \right]' W_T + \left[g(\theta,\mathbf{y}_{T}) \right]. \label{eq:estimateurGMM} +\end{align} + +\subsection{Matrice de pondération optimale} +\label{sec:matriceWoptimaleGMM} + +On définit la variance-covariance $\mathbf{S}(\theta;\mathbf{y})$ de +la moyenne échantillonnale de la fonction +$h(\theta,\mathbf{y})$. Cette matrice est formée par l'espérance, +élément par élément, du produit extérieur de l'estimateur par sa +transposée, multiplié par la taille de l'échantillon $T$ +\begin{align} + \label{eq:matricevcov1} + \mathbf{S}(\theta;\mathbf{y}) = T \cdot E\left\{ \left[ + h(\theta,\mathbf{y})\right] \left[ h(\theta,\mathbf{y}) \right]' + \right\} . +\end{align} + +La variance-covariance asymptotique de la moyenne échantillonnale de +la fonction $h(\theta_0,y(t))$ est obtenue en évaluant la matrice +\eqref{eq:matricevcov1} au point $\mathbf{\theta_0}$. +\begin{align} + \mathbf{S}(\theta_0;\mathbf{y}) = T \left[\cdot \lim_{T + \longrightarrow \infty} E \left\{ \left[ + h(\theta,\mathbf{y})\right] \left[ h(\theta,\mathbf{y}) + \right]' \right\} \right]_{\theta=\theta_0} +\end{align} + +La valeur optimale de la matrice de pondération $W_T$ de l'équation +\eqref{eq:estimateurGMM} est obtenue en inversant la +variance-covariance asymptotique: +\begin{align} + \label{eq:matriceWinvercevcov} + W_T = \mathbf{S}^{-1}(\theta_0;\mathbf{y}_T). +\end{align} + +Cependant, comme on ne connaît pas la valeur de $\theta_0$, on +utilisera l'estimateur convergent ${\mathbf{\hat\theta}}$, qui +minimise la condition suivante: +\begin{align} + \label{eq:objectifGMM2} + Q_T(\theta,\mathbf{y}_{T}) = \left[g(\theta,\mathbf{y}_{T}) \right]' + \mathbf{S}_T^{-1}(\theta;\mathbf{y}_T) + \left[g(\theta,\mathbf{y}_{T}) \right]. +\end{align} + +Le problème d'optimisation se note alors comme suit: +\begin{align} + \label{eq:estimateurGMM2} + \hat{\theta}_T &= \operatorname{arg}\min_{\theta\in\Omega} + Q_T(\theta,\mathbf{y}_{T}). +\end{align} + +Comme la séquence $\left\{ h(\theta_0,y(t)) +\right\}_{t=-\infty}^{\infty}$ ne présente pas de corrélation +sérielle, on pourrait estimer la variance-covariance $S_T$ de manière +convergente en évaluant la moyenne empirique du produit extérieur de +la condition de moment \citep[p.413]{hamilton1994time}: +\begin{align} + \label{eq:matponderationproduith} + \mathbf{S}_T^{*}(\theta;\mathbf{y}_T) = \frac{1}{T} \sum_{t=1}^T + \left[g\left(\theta_0,y(t) \right) \right] + \left[g\left(\theta_0,y(t) \right) \right]^{\prime}. +\end{align} + +L'estimateur $\mathbf{\hat{S}}_T({\mathbf{\hat\theta}};\mathbf{y}_T)$ +converge en probabilité vers la vraie valeur de la matrice +$\mathbf{S}(\theta_0;\mathbf{y}_T)$. Étant donné que l'on estime la +fonction $h(\theta,\mathbf{y}_{T})$ à l'aide de la fonction +$g(\mathbf{\hat\theta};\mathbf{y}_T)$, on a aussi la convergence en +probabilité: +\begin{align} + \mathbf{\hat{S}}_T({\mathbf{\hat\theta}};\mathbf{y}_T) = \frac{1}{T} + \sum_{t=1}^T \left[g\left(\hat{\theta}_T,w(t) \right) \right] + \left[g\left(\hat{\theta}_T,w(t) \right) \right]^{\prime} + \stackrel{P}{\longrightarrow} \mathbf{S}(\theta_0;\mathbf{y}_T). +\end{align} + +L'estimateur de la matrice de pondération optimale est alors défini +comme étant l'inverse de la variance-covariance estimée: +\begin{align} + \hat{W}_T &= + \mathbf{\hat{S}}_T^{-1}(\mathbf{\hat\theta};\mathbf{y}_T). \label{eq:matvcovGMM} +\end{align} + +Comme cette matrice dépend de l'estimateur $\hat\theta$, qui est pour +l'instant inconnu, elle ne pourra pas être utilisée pour une première +optimisation de l'équation de minimisation +\eqref{eq:estimateurGMM}. Cependant, elle pourra être utilisée dans +une procédure itérative à la section \ref{sec:GMMtwostep}. On devra +considérer l'utilisation d'un point de départ alternatif pour le +vecteur de paramètres dans l'algorithme de minimisation. + +\subsection{Méthode des moments généralisée itérative} +\label{sec:GMMtwostep} +\nocite{wooldridge2001econometric} + +La méthode des moments généralisée itérative de +\cite{hall2005generalized} permet de contourner le problème de +l'estimation de la matrice +$\mathbf{\hat{S}}_T(\mathbf{\hat\theta};\mathbf{y}_T)$. Elle consiste, +en premier lieu, à calculer un estimateur préliminaire +$\mathbf{\hat{\theta}}^{(0)}$, en utilisant la matrice identité $W_T = +I_r$ dans l'équation \eqref{eq:estimateurGMM}. On suggère d'utiliser +un vecteur de valeurs initiales $\hat{\theta}^{I}$ obtenues par une +autre méthode d'estimation, lorsque possible, comme point de départ de +l'optimisation numérique. + +À l'aide de l'estimateur initial $\mathbf{\hat{\theta}}^{(0)}$, on +obtient une première évaluation de la matrice de pondération: +\begin{align} + W_T = \left[\hat{S}_T(\mathbf{\hat{\theta}}^{(0)};\mathbf{y}_T) + \right]^{-1}. +\end{align} + +En utilisant la matrice $W_T$ comme pour pondérer la fonction objectif +\eqref{eq:estimateurGMM}, on obtiendra un nouvel estimateur +$\hat{\theta}_T^{(1)}$. + +Par la suite, on répète cette procédure jusqu'à ce qu'on obtienne deux +estimateurs consécutifs ($\hat{\theta}_T^{(j)}$ et +$\hat{\theta}_T^{(j+1)}$) qui ne sont pas significativement +différents, selon un critère d'arrêt $\epsilon_i < \epsilon$. Ce +critère d'arrêt prendra ici la forme suivante, qui correspond à la +norme euclidienne de la différence entre les deux derniers estimateurs +obtenus: +\begin{align} + \label{eq:criterearret} + \epsilon_i = + \sqrt{\left[\hat\theta^{(i)}_T-\hat\theta^{(i-1)}_T\right]^{\prime}\left[\hat\theta^{(i)}_T-\hat\theta^{(i-1)}_T\right]}. +\end{align} + +On peut aussi fixer un nombre maximal d'itérations $i_{max}$ si l'on +ne parvient pas à obtenir le niveau de précision voulu. Par contre, +dans cette situation, on préfère utiliser un autre point de départ. + +Chaque estimateur $\hat{\theta}_T^{(i)}$ a la même distribution +asymptotique. La méthode itérative a pour avantage, en pratique, que +les estimateurs produits sont invariants d'échelle et ne dépendent pas +de la matrice $W_T$ initiale. + +\subsection{Distribution asymptotique des estimateurs} +\label{sec:matvcovGMM} + +On considère $\hat{\theta}_T$, la valeur qui minimise la fonction +objectif $Q(\theta,\mathbf{y}_{T})$ \eqref{eq:objectifGMM2}. Cette +minimisation équivaut à résoudre le système non linéaire où l'on égale +la dérivée de l'équation d'optimisation à 0: +\begin{align} \label{eq:premierordreGMM} + \frac{d}{d\theta^{\prime}}Q(\theta,\mathbf{y}_{T}) &= + \left[\left.\frac{d}{d\theta^{\prime}}g(\theta,\mathbf{y}_{T})\right|_{\theta=\hat{\theta}} + \right]^{\prime} \cdot \hat{W}_T \cdot g(\theta,\mathbf{y}_{T}) \\ + &= 0. \nonumber +\end{align} + +où le gradient $D'(\theta,\mathbf{y}_{T})$ est +\begin{align} \label{eq:gradientGMM} D'(\theta,\mathbf{y}_{T}) &= + \left[\frac{d}{d\theta^{\prime}}g(\theta,\mathbf{y}_{T}) + \right]^{\prime} \nonumber\\ + &= \begin{bmatrix} + \frac{d}{d\theta_1}g_1(\theta,\mathbf{y}_{T})& \cdots & \frac{d}{d\theta_1}g_a(\theta,\mathbf{y}_{T}) \\ + \vdots & \ddots & \vdots \\ + \frac{d}{d\theta_k}g_1(\theta,\mathbf{y}_{T})& \cdots & + \frac{d}{d\theta_k}g_a(\theta,\mathbf{y}_{T}) + \end{bmatrix}. +\end{align} + +En appliquant le théorème central limite multivarié +\eqref{eq:TCLmulti2} à l'estimateur des conditions de moments +$g(\theta,\mathbf{y}_{T})$, on obtient que, lorsque la taille $T$ est +suffisamment grande, celui-ci converge en loi vers un vecteur +aléatoire de distribution normale multivariée de moyenne +$\mathbf{0}_a$ et de variance-covariance +$T^{-1}\mathbf{S}(\mathbf{\theta};\mathbf{y}_T)$: +\begin{align} + \label{eq:TCL-GMM} + \sqrt{T} g(\theta,\mathbf{y}_{T}) \stackrel{L}{\longrightarrow} + N\left(\mathbf{0}_a,\mathbf{S}(\mathbf{\theta};\mathbf{y}_T)\right). +\end{align} + +Cette relation permet de conclure que l'estimateur $\hat\theta$ est +gaussien. On n'a donc qu'à calculer sa variance-covariance +asymptotique. + +Soit $\left\{\mathbf{\hat{S}}_T \right\}_{T=1}^{\infty}$, une séquence +de matrices $(r \times r)$ définies positives qui convergent en +probabilité vers la variance-covariance asymptotique: +\begin{align*} + \mathbf{\hat{S}}_T \stackrel{P}{\longrightarrow} \mathbf{S}. +\end{align*} + +On ajoute que la fonction $g(\theta,\mathbf{y}_{T})$ doit être +différentiable par rapport au vecteur $\theta$ pour tout échantillon +$\mathbf{y}_{T}$. + +On doit préalablement poser un ensemble de conditions supplémentaires +de régularité: +\begin{enumerate} +\item L'estimateur $\hat\theta_T$ converge en probabilité vers la + vraie valeur des paramètres $\theta_0$: + \begin{align} + \label{eq:varasympt.1} + \hat\theta_T \stackrel{P}{\longrightarrow} \theta_0. + \end{align} +\item Le théorème central limite s'applique pour la fonction + $g(\theta,\mathbf{y}_{T})$ \eqref{eq:TCL-GMM}. +\item Pour toute séquence d'estimateurs $\left\{ \theta_T^{*} + \right\}_{T=1}^{\infty}$ convergents en probabilité $\theta_T^{*} + \stackrel{P}{\longrightarrow} \theta_0$, on peut évaluer le gradient + $D^{\prime}(\theta,\mathbf{y}_{T})$ de l'équation + $g(\theta,\mathbf{y}_{T})$ \eqref{eq:gradientGMM} à l'aide de + limites: + \begin{align} + \label{eq:varasympt.3} + D^{\prime}_T(\theta,\mathbf{y}_{T}) &\equiv plim \left\{\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}|_{\theta=\theta_T^{*}} \right\} \nonumber\\ + &= plim + \left\{\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}|_{\theta=\theta_0} + \right\}. + \end{align} + On note que les colonnes de la matrice $D$ sont linéairement + indépendantes. +\end{enumerate} + +Pour obtenir la distribution asymptotique de l'estimateur +$\hat\theta$, on utilise le premier ordre du développement de Taylor +de la fonction $g(\hat\theta,\mathbf{y}_{T})$ autour de la valeur du +vrai paramètre $\theta_0$, tel qu'avancé par +\cite{gourieroux1989statistique}: +\begin{align} + \label{eq:taylorfonction.g} + g(\hat\theta,\mathbf{y}_{T}) &= g(\theta_0,\mathbf{y}_{T}) + + D^{\prime}_T(\theta,\mathbf{y}_{T}) \left(\hat\theta-\theta_0 + \right). +\end{align} + +On multiplie de part et d'autre par la matrice $ +\left(\left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T \right)$ de dimension $(a \times r)$: +\begin{align} + \label{eq:taylorfonction.gprod} + \left(\left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T\right) \times g(\hat\theta,\mathbf{y}_{T}) &= + \left(\left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T\right) \times g(\theta_0,\mathbf{y}_{T})\nonumber\\ + &+ + \left(\left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T\right) \times D^{\prime}_T(\theta,\mathbf{y}_{T}) + \left(\hat\theta-\theta_0 \right). +\end{align} + +L'équation de premier ordre \eqref{eq:premierordreGMM} nous indique +que le côté gauche de l'égalité précédente +\eqref{eq:taylorfonction.gprod} vaut 0. On retrouve alors une +expression de la distance entre l'estimateur et la vraie valeur des +paramètres, qui dépend de la matrice de pondération, de la fonction +$g(\theta_0,\mathbf{y}_{T})$ et du gradient +$D^{\prime}_T(\theta,\mathbf{y}_{T})$: +\begin{align} + \label{eq:taylorfonction.gprod2} + \left(\hat\theta-\theta_0 \right) &= - + \left[\left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T \times D^{\prime}_T(\theta,\mathbf{y}_{T}) \right]^{-1} \nonumber\\ + &\quad\times + \left.\frac{dg(\theta,\mathbf{y}_{T})}{d\theta^{\prime}}\right|_{\theta=\theta_T} + \times W_T \times g(\theta_0,\mathbf{y}_{T}). +\end{align} + +La condition de régularité \eqref{eq:varasympt.3} permet la +convergence de chaque rangée de l'estimateur +$D^{\prime}_T(\theta,\mathbf{y}_{T})$ vers celles du gradient +$D^{\prime}(\theta_0,\mathbf{y}_{T})$. De plus, l'équation +\eqref{eq:taylorfonction.gprod2} implique la relation de convergence +suivante: +\begin{align} + \label{eq:taylorfonction.gprod3} + \sqrt{T} \left(\hat\theta-\theta_0 \right) + &\stackrel{P}{\longrightarrow} + -\left\{D(\theta,\mathbf{y}_{T})W_TD^{\prime}(\theta,\mathbf{y}_{T}) + \right\}^{-1} \nonumber\\ + &\quad\times \left\{D(\theta,\mathbf{y}_{T})W_T\sqrt{T} \cdot + g(\theta_0,\mathbf{y}_{T}) \right\}. +\end{align} + +Afin de simplifier la notation, on définit la constante +$C(\theta,\mathbf{y}_{T})$: +\begin{align*} + C(\theta,\mathbf{y}_{T})=-\left\{D(\theta,\mathbf{y}_{T})W_TD^{\prime} + (\theta,\mathbf{y}_{T}) \right\}^{-1} \times + D(\theta,\mathbf{y}_{T})W_T. +\end{align*} + +L'équation \eqref{eq:taylorfonction.gprod3} devient +\begin{align} + \label{eq:taylorfonction.gprod4} + \sqrt{T} \left(\hat\theta-\theta_0 \right) + &\stackrel{P}{\longrightarrow} C(\theta,\mathbf{y}_{T})\sqrt{T} + \cdot g(\theta_0,\mathbf{y}_{T}). +\end{align} + +En combinant la relation \eqref{eq:TCL-GMM}, où l'on applique le +théorème central limite à la fonction $g(\theta_0,\mathbf{y}_{T})$, +avec la méthode delta multivariée \eqref{eq:deltamethodmult}, on +retrouve la forme suivante, avec une convergence en loi cependant, +puisque celle-ci est moins forte que celle en probabilité: +\begin{align} + \sqrt{T} (\hat{\theta}-\theta_0) \stackrel{L}{\longrightarrow} + N(0,\mathcal{J}_0^{-1}) +\end{align} +où +\begin{align} + \mathcal{J}_0^{-1} &= + C(\theta,\mathbf{y}_{T})\left\{W_T\right\}^{-1}C(\theta,\mathbf{y}_{T}) + \nonumber\\ + &= \left\{D(\theta,\mathbf{y}_{T})W_TD^{\prime} + (\theta,\mathbf{y}_{T}) \right\}^{-1} D(\theta,\mathbf{y}_{T})W_T + \left\{W_T\right\}^{-1}\nonumber\\ + &\quad\times W_T D^{\prime}(\theta,\mathbf{y}_{T}) + \left\{D(\theta,\mathbf{y}_{T})W_TD^{\prime} (\theta,\mathbf{y}_{T}) + \right\}^{-1}\nonumber\\ + &= + \left\{D(\theta,\mathbf{y}_{T})W_TD^{\prime}(\theta,\mathbf{y}_{T})\right\}^{-1}. \label{matricevcovparamGMMnc} +\end{align} + +\section{Estimation sous contraintes} +\label{sec:estimGMMcontraint} + +La méthode des moments généralisée suppose que le vrai vecteur de +paramètres $\theta_0$ appartient à l'ensemble $\Omega$. En pratique, +les paramètres sont souvent soumis à certaines contraintes à +l'égalité. + +On définit un ensemble de $q$ contraintes linéaires implicites +appliquées au vecteur de paramètres $\theta$ de longueur $a$: +\begin{align} + \label{eq:contraintelin0} + \left\{ + \begin{array}{rcl} + r_{(1,1)}\theta_1 + \ldots + r_{(1,a)}\theta_a &=& r_{(1,0)}\\ + \ldots \\ + r_{(q,1)}\theta_1 + \ldots + r_{(q,a)}\theta_a &=& r_{(q,0)} + \end{array} + \right\}. +\end{align} + +On peut les présenter sous la forme d'un système matriciel: +\begin{align} + \label{eq:contraintelin} + \underbrace{ + \begin{bmatrix} + r_{(1,1)}&\ldots&r_{(1,a)}\\ + \vdots&\ddots&\vdots\\ + r_{(q,1)}&\ldots&r_{(q,a)}\\ + \end{bmatrix}}_{\mathbf{R}} \times \underbrace{\begin{bmatrix} + \theta_1\\ + \vdots\\ + \theta_a + \end{bmatrix}}_{\mathbf{\theta}} &= \underbrace{\begin{bmatrix} + r_{(1,0)}\\ + \vdots\\ + r_{(q,0)} + \end{bmatrix}}_{\mathbf{r}}. +\end{align} + +Afin de les inclure dans un problème de minimisation, on préfèrera +utiliser la notation $a(\theta) = R\theta-r$. On notera au passage +que le gradient du vecteur de contraintes équivaut à la matrice de +coefficients: +\begin{align} + \label{eq:gradientcontrainte} + \frac{\partial}{\partial\theta^{\prime}} a(\theta) = R. +\end{align} + +Ainsi, on peut estimer les paramètres de la distribution contrainte à +l'aide de la méthode des moments généralisée, de manière analogue à la +distribution non contrainte, comme il a été présenté à la section +précédente. On utilisera la technique du multiplicateur de Lagrange +afin d'inclure la contrainte $a(\theta)$ dans l'équation de +minimisation \eqref{eq:estimateurGMM2}. Le vecteur $\gamma$ associe un +multiplicateur à chaque contrainte linéaire. On définit le lagrangien +$\mathcal{L}(\tilde{\theta})$ mettant en relation la fonction objectif +$Q_T(\theta)$ \eqref{eq:objectifGMM2} et les contraintes $a(\theta)$: +\begin{equation} + \label{eq:estimateurGMMlagrange} + \mathcal{L}(\theta) = - Q_T(\theta) - a(\theta)^{\prime} \gamma. +\end{equation} + +L'estimateur contraint $\tilde{\theta}$ est obtenu en maximisant ce +lagrangien: +\begin{align} + \label{eq:lagrangienGMMcontraint} + \tilde{\theta} = \operatorname{arg}\max_{\theta\in\Omega} + \mathcal{L}(\theta). +\end{align} + +La solution optimale s'obtient en résolvant les conditions de premier +ordre par rapport au vecteur de paramètres $\theta$ et celui des +multiplicateurs de Lagrange $\gamma$: + +\begin{align} + \frac{\partial}{\partial\theta}Q_T(\tilde\theta) - + \frac{\partial}{\partial\theta^{\prime}}a(\tilde\theta)^{\prime}\gamma_{\scriptscriptstyle + T} &= 0\label{eq:premierordreGMMlagrange1} \\ a(\tilde\theta) &= + 0. \label{eq:premierordreGMMlagrange2} +\end{align} + +On s'intéresse aussi à la distribution asymptotique de cet estimateur +contraint. Pour ce faire, on doit développer les conditions de premier +ordre comme il a été fait à la section \ref{sec:matvcovGMM} pour +l'estimateur non contraint. + +\subsection{Distribution asymptotique des estimateurs contraints} +\label{sec:matvcovGMMconst} + +Supposons que les conditions de premier ordre sont deux fois +continûment dérivables par rapport au vecteur $\theta$. On développe +les équations \eqref{eq:premierordreGMMlagrange1} et +\eqref{eq:premierordreGMMlagrange2} autour de la vraie valeur du +paramètre contraint $\theta_0$. Puis, on les multiplie par les facteurs +$\frac{1}{\sqrt{T}}$ et $\sqrt{T}$ respectivement. Notons que la +fonction $a(\theta)$ vaut 0 au point $\theta_0$, ce qui permettra de +simplifier la seconde équation: +\begin{subequations} \label{eq:premierordreGMMlagrange1.1-2} + \begin{align} + \frac{1}{\sqrt{T}} \frac{\partial}{\partial\theta}Q_T(\theta_0) + + \frac{1}{T} + \frac{\partial^2}{\partial\theta\partial\theta^{\prime}}Q_T(\theta_0) + \sqrt{T} (\tilde\theta - \theta_0) - + \frac{\partial}{\partial\theta^{\prime}}a(\tilde\theta)^{\prime}\frac{\gamma_{\scriptscriptstyle + T}}{\sqrt{T}} &\approx 0 \label{eq:premierordreGMMlagrange1.1}\\ + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) \sqrt{T} + (\tilde\theta - \theta_0) &\approx + 0. \label{eq:premierordreGMMlagrange1.2} + \end{align} +\end{subequations} + +On définit la matrice d'information de Fisher comme étant la limite de +l'espérance de la valeur de la dérivée seconde de la fonction +objectif. Au point $\theta_0$, on identifie l'estimateur de cette +matrice par $\mathcal{J}_0$, la variance-covariance de l'estimateur +non contraint \eqref{matricevcovparamGMMnc}: +\begin{align} + \label{eq:fisherJGMMlagrange} + \mathcal{J}_0 &= \lim_{T\to\infty} -\frac{1}{T} + \frac{\partial^2}{\partial\theta\partial\theta^{\prime}}Q_T(\theta_0). +\end{align} + +On reprend l'équivalent asymptotique de l'équation +\eqref{eq:premierordreGMMlagrange1.1} pour l'estimateur non contraint: +\begin{align} + \frac{1}{\sqrt{T}} \frac{\partial}{\partial\theta}Q_T(\theta_0) - + \mathcal{J}_0 \sqrt{T} (\hat\theta - \theta_0) &\approx + 0\label{eq:premierordreGMMnc1.1}. +\end{align} + +En combinant ces deux dernières expressions, on peut formuler les +conditions de premier ordre \eqref{eq:premierordreGMMlagrange1.1-2} +comme étant asymptotiquement des fonctions linéaires de l'estimateur +non contraint $\sqrt{T} (\hat\theta - \theta_0)$: +\begin{align} + \mathcal{J}_0 \sqrt{T} (\hat\theta - \theta_0) - \mathcal{J}_0 + \sqrt{T} (\tilde\theta - \theta_0) - + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\frac{\gamma_{\scriptscriptstyle + T}}{\sqrt{T}} &\approx 0 \label{eq:premierordreGMMlagrange2.1}\\ + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) \sqrt{T} + (\tilde\theta - \theta_0) &\approx + 0. \label{eq:premierordreGMMlagrange2.2} +\end{align} + +En réorganisant la première équation +\eqref{eq:premierordreGMMlagrange2.1}, on obtient: +\begin{align} + \label{eq:premierordreGMMlagrange3.1} + \sqrt{T} (\tilde\theta - \theta_0) \approx \sqrt{T} (\hat\theta - + \theta_0) - \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\frac{\gamma_{\scriptscriptstyle + T}}{\sqrt{T}}. +\end{align} + +En la reportant dans la seconde équation +\eqref{eq:premierordreGMMlagrange2.2}, on obtient: +\begin{align} + \label{eq:premierordreGMMlagrange3.2} + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) \sqrt{T} + (\hat\theta - \theta_0) - \frac{\partial}{\partial\theta^{\prime}} + a(\theta_0) \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\frac{\gamma_{\scriptscriptstyle + T}}{\sqrt{T}} + \frac{\partial}{\partial\theta^{\prime}} + a(\theta_0) \sqrt{T} (\tilde\theta - \theta_0) &\approx 0. +\end{align} + +Comme le rang de la matrice $\frac{\partial}{\partial\theta^{\prime}} +a(\theta_0)$ est égal au nombre de contraintes $r$, alors +$\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) +\mathcal{J}_0^{-1} +\frac{\partial}{\partial\theta}a(\theta_0)^{\prime}$ est inversible, +et l'on peut donc isoler le multiplicateur de Lagrange +$\frac{\gamma_{\scriptscriptstyle T}}{\sqrt{T}}$ en fonction des +estimateurs contraints $\sqrt{T} (\tilde\theta - \theta_0)$ et non +contraints $\sqrt{T} (\hat\theta - \theta_0)$: +\begin{align} + \label{eq:LagrangienJ.GMM} + \frac{\gamma_{\scriptscriptstyle T}}{\sqrt{T}} & \approx + \left(\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime} \right)^{-1} + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) \sqrt{T} + (\hat\theta - \theta_0). +\end{align} + +On définit l'estimateur contraint en fonction de l'estimateur non +contraint en utilisant le lagrangien \eqref{eq:LagrangienJ.GMM} dans +la condition \eqref{eq:premierordreGMMlagrange3.1}: +\begin{align} + \label{eq:contraintvsncGMM} + \sqrt{T} (\tilde\theta - \theta_0) &\approx + \left(I-\underbrace{\mathcal{J}_0^{-1}\frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\left(\frac{\partial}{\partial\theta^{\prime}} + a(\theta_0) \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime} + \right)^{-1}\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + }_{P}\right)\mathcal{J}_0^{-1}\sqrt{T} (\hat\theta - \theta_0) \\ + & \approx \left(I-P\right)\mathcal{J}_0^{-1}\sqrt{T} (\hat\theta - + \theta_0). \nonumber +\end{align} + +La variance asymptotique de l'estimateur contraint est donc, à partir +du résultat précédent \eqref{eq:contraintvsncGMM} et de la définition +\eqref{eq:premierordreGMMnc1.1}: +\begin{align} + \label{eq:VcontraintGMM} + V\left[\sqrt{T} (\tilde\theta - \theta_0) \right] = + \left(I-P\right)\mathcal{J}_0^{-1}. +\end{align} + +L'estimateur $\tilde\theta$ suit donc asymptotiquement une +distribution normale multivariée de moyenne $\theta_0$ et de variance +$T\left(I-P\right)\mathcal{J}_0^{-1}$: +\begin{align} + \label{eq:distcontraintGMM} + \tilde\theta &\sim + \mathcal{N}\left(\theta_0,T\left(I-P\right)\mathcal{J}_0^{-1}\right). +\end{align} + +\section{Tests d'hypothèses paramétriques} +\label{sec:testparam} + +Les tests d'hypothèses paramétriques sont utilisés afin d'évaluer une +hypothèse concernant les paramètres d'une distribution, en fonction +d'un échantillon de données. Afin d'effectuer ces tests, on présume +que la différence entre l'estimateur $\hat\theta$ et la vraie valeur +des paramètres $\theta_0$ suit une distribution normale +multivariée. Les hypothèses sont habituellement formulées sous la +forme de contraintes linéaires, ainsi, les statistiques de test sont +obtenues à partir du calcul matriciel. Les trois tests les plus +couramment utilisés dans le cadre de l'estimation par maximum de +vraisemblance peuvent être adaptés à la méthode des moments +généralisée \citep{newey1994large}. Pour l'ensemble de ces tests, +l'hypothèse nulle correspond à la contrainte linéaire suivante: +\begin{equation} + \label{eq:hypcontraintelin} + H_0: a(\theta) = R\theta - r = 0. +\end{equation} + +\subsection{Test de Wald} +\label{sec:testwald} + +Le test de Wald permet de vérifier si la différence entre l'estimateur +non contraint $\hat\theta$ et l'estimateur contraint $\tilde\theta$ +est significative. Les contraintes linéaires posées ne seront pas +applicables lorsque le résultat est positif. Pour ce faire, on doit +connaître la distribution asymptotique de celle-ci. Comme la +distribution asymptotique des deux estimateurs est normale, alors +celle de cette différence l'est aussi: +\begin{align} + \label{eq:7} + (\hat\theta - \tilde\theta) \sim \mathcal{N}(0,TP\mathcal{J}_0^{-1}). +\end{align} + +On obtient l'espérance et la variance de la statistique +$\sqrt{T}\left(\hat\theta - \tilde\theta\right)$ en utilisant le fait +que la somme de deux variables aléatoires normales l'est aussi: +\begin{align} + \label{eq:moyennevariancesomme} + E\left[\sqrt{T}\left(\hat\theta - \tilde\theta\right) \right] &= + E\left[\sqrt{T}\left(\hat\theta - \theta_0\right) \right] + + E\left[\sqrt{T}\left(\theta_0 - \tilde\theta\right) \right]\nonumber\\ + &= 0 - 0 \nonumber\\ + &= 0 \\ + V\left[\sqrt{T}\left(\hat\theta - \tilde\theta\right) \right] &= + V\left[\sqrt{T}\left(\hat\theta - \theta_0\right) \right] + + V\left[\sqrt{T}\left(\theta_0 - \tilde\theta\right) \right]\nonumber\\ + &= T\left(I+(P-I)\right)\mathcal{J}_0^{-1}\nonumber\\ + &= TP\mathcal{J}_0^{-1}. +\end{align} + +On définit la statistique $\chi^{WALD,1}$, qui a asymptotiquement une +distribution $\chi^2$ avec $q$ degrés de liberté: +\begin{align} + \label{eq:statistiqueWald} + \chi^{WALD,1} &= T \left(\hat\theta - \tilde\theta\right)^{\prime} P + \left(\hat\theta - \tilde\theta\right) \\ + &=T \left(\hat\theta - \tilde\theta\right)^{\prime} + \mathcal{J}_0^{-1}\frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\left(\frac{\partial}{\partial\theta^{\prime}} + a(\theta_0) \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime} + \right)^{-1}\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \left(\hat\theta - \tilde\theta\right). \nonumber +\end{align} + + +On rejettera l'hypothèse nulle \eqref{eq:hypcontraintelin} lorsque la +valeur de la statistique $\chi^{WALD,1}$ sera supérieure à un seuil +critique $\chi_{q,1-\alpha}^2$ au niveau de confiance $1-\alpha$. + +Une version asymptotiquement équivalente de ce test qui ne requiert +pas de calculer la valeur de l'estimateur contraint existe. Ce test +est équivalent lorsque les contraintes définissent certains paramètres +comme des constantes. On vérifie si un cas particulier d'une +distribution s'applique, par exemple avec celle de Laplace asymétrique +généralisée. + +On définit alors la statistique $\chi^{WALD,2}$, qui a aussi +asymptotiquement une distribution $\chi^2$ avec $q$ degrés de +liberté. Par contre, ici, on teste si la valeur de la contrainte +linéaire est significativement différente de 0: +\begin{align} + \label{eq:statistiqueWald2} + \chi^{WALD,2} &= T a^{\prime}(\hat\theta) + \left(\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a^{\prime}(\theta_0) \right)^{-1} + a(\hat\theta). +\end{align} + + +\subsection{Test du multiplicateur de Lagrange} +\label{sec:testscore} + +Le test du multiplicateur de Lagrange, ou du score, introduit par +\cite{newey1987hypothesis}, est basé uniquement sur l'estimateur +contraint et est équivalent asymptotiquement au test de Wald présenté +à la section précédente. Il vérifie l'application de la contrainte +\eqref{eq:contraintelin} à l'estimateur $\tilde\theta$. Selon la +définition du lagrangien \eqref{eq:estimateurGMMlagrange}, si la +contrainte est vérifiée, alors la restriction $a(\tilde\theta)$ vaudra +0. Selon la condition de premier ordre +\eqref{eq:premierordreGMMlagrange2}, la dérivée de la fonction +objectif de l'estimateur non contraint +$\frac{\partial}{\partial\theta}Q_T(\theta)$, le score, $\tilde\theta$ +devrait aussi être égale à 0. On cherchera donc à tester si cette +valeur est significativement différente de 0. + +On définit la statistique du multiplicateur de Lagrange $\chi^{LM,1}$, +qui a asymptotiquement une distribution $\chi^2_q$ avec $q$ degrés de +liberté: +\begin{align} + \label{statistiqueLM} + \chi^{LM,1} &= T \frac{\partial}{\partial\theta}Q_T(\tilde\theta) P + \frac{\partial}{\partial\theta}Q_T(\tilde\theta). +\end{align} + +On rejettera l'hypothèse nulle \eqref{eq:hypcontraintelin} lorsque la +valeur de la statistique $\chi^{LM,1}$ sera supérieure à un seuil +critique $\chi_{q,1-\alpha}^2$ au niveau de confiance $1-\alpha$. + +On peut construire un test équivalent, basé sur la valeur du +multiplicateur, dont la matrice de variance-covariance est l'inverse +de celle de la contrainte. On définit alors la statistique +$\chi^{LM,2}$ suivante: +\begin{align} + \label{eq:statistiqueLM2} + \chi^{LM,2} &= T \gamma^{\prime} + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a^{\prime}(\theta_0) \gamma. +\end{align} + +\subsection{Test basé sur la statistique de métrique de distance} + +La statistique de métrique de distance est basée sur la différence +entre les valeurs minimales de la fonction objectif $Q_T(\theta)$ +obtenues lors de l'optimisation avec contraintes +\eqref{eq:estimateurGMMlagrange} et sans contraintes +\eqref{eq:estimateurGMM}. + +On définit la statistique $\chi^{DM}$: +\begin{align} + \label{statistiqueD} + \chi^{DM} &= -T \left[Q_T(\tilde\theta) - Q_T(\hat\theta)\right]. +\end{align} + +Cette statistique a asymptotiquement une distribution $\chi^2_q$ avec +$q$ degrés de liberté. Elle est l'analogue de la statistique du ratio +de vraisemblance dans le cadre de l'estimation par la méthode du +maximum de vraisemblance. + +Le test basé sur la métrique de distance vérifie que la contrainte +\eqref{eq:contraintelin} posée lors de l'estimation du vecteur +$\tilde\theta$ est valide. Un des désavantages de ce test est qu'il +requiert deux optimisations. Par contre, on peut facilement récupérer +les valeurs de $Q_T(\theta)$ lors de l'estimation. + +On rejettera l'hypothèse nulle \eqref{eq:hypcontraintelin} lorsque la +valeur de la statistique $\chi^{DM}$ sera supérieure à un seuil +critique $\chi_{q,1-\alpha}^2$ au niveau de confiance $1-\alpha$. + +\subsection{En résumé} +\label{sec:resumetests} + +On rassemble les différentes statistiques permettant d'effectuer un +test d'hypothèse paramétrique à la table \ref{tab:testsparamGMM}. +\begin{table}[!ht] + \centering + \begin{tabular}{cc} + \hline + \textbf{Statistique} & \textbf{Valeur} \\ + \hline + $\chi^{WALD,1}$ & $T \left(\hat\theta - \tilde\theta\right)^{\prime} + \mathcal{J}_0^{-1}\frac{\partial}{\partial\theta}a(\theta_0)^{\prime}\left(\frac{\partial}{\partial\theta^{\prime}} + a(\theta_0) \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a(\theta_0)^{\prime} + \right)^{-1}\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \left(\hat\theta - \tilde\theta\right)$ \\ + $\chi^{WALD,2}$ & $T a^{\prime}(\hat\theta) + \left(\frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a^{\prime}(\theta_0) \right)^{-1} + a(\hat\theta)$ \\ + $\chi^{LM,1}$ & $T \frac{\partial}{\partial\theta}Q_T(\tilde\theta) P + \frac{\partial}{\partial\theta}Q_T(\tilde\theta)$ \\ + $\chi^{LM,2}$ & $T \gamma^{\prime} + \frac{\partial}{\partial\theta^{\prime}} a(\theta_0) + \mathcal{J}_0^{-1} + \frac{\partial}{\partial\theta}a^{\prime}(\theta_0) \gamma$ \\ + $\chi^{DM}$ & $-T \left[Q_T(\tilde\theta) - Q_T(\hat\theta)\right]$ \\ + \hline + \end{tabular} + \caption{Tests d'hypothèse paramétriques pour la méthode des moments généralisée} + \label{tab:testsparamGMM} +\end{table} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/chapitre5.tex b/depotfinal/908144032/chapitre5.tex new file mode 100644 index 0000000..6833381 --- /dev/null +++ b/depotfinal/908144032/chapitre5.tex @@ -0,0 +1,391 @@ +\chapter{Méthode de l'équation d'estimation + optimale} % numéroté + +Une équation d'estimation est une fonction des données de +l'échantillon et des paramètres d'un modèle qui spécifie de quelle +manière on doit procéder pour estimer ces derniers, lorsque la +distribution de la population est inconnue. Cette approche a un +avantage sur les méthodes de vraisemblance, car elle ne requiert pas +l'utilisation de la fonction de densité ou de répartition, mais +seulement les moments de la distribution. \citep{everitt2006cambridge} + +\cite{crowder1986consistency} définit l'équation d'estimation sous une +forme générale, composée d'une matrice d'échelle $S(\theta;Y)$, un +vecteur de fonctions aléatoires $u(\theta;Y)$ et un vecteur de +pondération $\mathbf{w}(\theta;Y)$: +\begin{align} + \label{eq:generalEE86} + g(\theta;Y) = S^{-1}(\theta;Y) \sum_{t=1}^T u(\theta;Y) + \mathbf{w}(\theta;Y). +\end{align} + +En faisant abstraction du vecteur de pondération +$\mathbf{w}(\theta;Y)$, on retrouve une forme qui rappelle la méthode +des moments généralisée. La fonction d'estimation $u(\theta;Y)$ +définit, dans ce cas, des conditions de moment. Toutefois, en faisant +abstraction de la matrice $S(\theta;Y)$, on obtient une classe +d'équations d'estimation $g(\theta;Y)$ qui généralise plusieurs +méthodes connues, comme développé par \cite{crowder1987linear}: +\begin{align} + \label{eq:generalEE87} + g(\theta;Y) = \sum_{t=1}^T u(\theta;Y) \mathbf{w}(\theta;Y). +\end{align} + +La propriété fondamentale de toute équation d'estimation est que son +espérance est nulle. L'équation d'estimation \eqref{eq:generalEE86}, +et donc \eqref{eq:generalEE87}, produit des estimateurs sans biais: +\begin{align} + \label{eq:EEsansbiais} + E\left[ g(\theta;Y) \right] = 0. +\end{align} + +Par analogie avec la méthode du maximum de vraisemblance, on pourra +alors la considérer comme une équation de quasi-score. On peut en +effet représenter l'équation précédente par celle du score +\eqref{eq:scoreEMV} donnée par l'espérance de la dérivée de la +fonction de log-vraisemblance par rapport au vecteur de paramètres +$\theta$: +\begin{align} + \label{eq:scoreEMV} + E\left[g\left(\theta;Y \right)^{EMV}\right] &= E\left[\frac{d \ln + L\left(\theta;Y\right)}{d\theta}\right] = 0. +\end{align} + +On désigne la moyenne et la variance de la distribution par les +notations $\mu\left(\theta\right)$ et $\sigma^2(\theta)$ +respectivement. On définit aussi les dérivées premières de la moyenne +et de l'écart-type par rapport au vecteur de paramètres: +\begin{subequations}\label{eq:derivmom} + \begin{align} + \mu^{\prime}\left( \theta \right) &= \frac{d \mu\left(\theta\right)}{d\theta} \label{eq:derivmom1}\\ + \sigma^{\prime}\left( \theta \right) &= \frac{d + \sqrt{\sigma^2(\theta)}}{d\theta}.\label{eq:derivmom2} + \end{align} +\end{subequations} + +Dans ce chapitre, on considère une équation d'estimation quadratique +de la forme suivante: +\begin{equation} + \label{eq:generalquad} + g\left(\theta;Y \right) = \sum_{t=1}^n \left[ \mathbf{a}(\theta;y_t)(y_t-\mu\left(\theta\right)) + \mathbf{b}(\theta;y_t)\left((y_t-\mu\left(\theta\right))^2-\sigma^2\left(\theta\right) \right)\right]. +\end{equation} + +Le vecteur de pondération $\mathbf{w}(\theta;Y)$ de l'équation +\eqref{eq:generalEE87} est composé de deux fonctions déterministes: +$\mathbf{a}(\theta;Y)$ et $\mathbf{b}(\theta;Y)$: +\begin{align} + \label{eq:vecteuroptimal.ab} + \mathbf{w}(\theta;Y) &= \begin{bmatrix} \mathbf{a}(\theta;Y) \\ + \mathbf{b}(\theta;Y) \end{bmatrix}. +\end{align} + +On démontre facilement que c'est une équation d'estimation en +calculant son espérance: +\begin{align} + \label{eq:demogeneralquad-convergent} + E\left[g\left(\theta;Y \right)\right] &= E\left[ \sum_{t=1}^n \left[ \mathbf{a}(\theta;y_t)(y_t-\mu\left(\theta\right)) + \mathbf{b}(\theta;y_t)\left((y_t-\mu\left(\theta\right))^2-\sigma^2\left(\theta\right) \right)\right]\right] \nonumber\\ + &= \sum_{t=1}^n \left( \mathbf{a}(\theta;y_t)\cdot E\left[y_t-\mu\left(\theta\right)\right] + \mathbf{b}(\theta;y_t)\cdot E\left[(y_t-\mu\left(\theta\right))^2-\sigma^2\left(\theta\right) \right]\right) \nonumber\\ + &= 0. +\end{align} + +La forme quadratique regroupe certaines méthodes d'estimation bien +établies, recensées à la table \ref{tab:methodesquad}. +\begin{table}[!ht] + + \centering + \begin{tabular}{ccc} + \hline + \textbf{Méthode} & $\mathbf{a}(\theta;y_t)$ & $\mathbf{b}(\theta;y_t)$ \\ + \hline + Moindres carrés non pondérés & $\mu^{\prime}\left( \theta \right)$ & $0$ \\ + Maximum de quasi-vraisemblance & $\frac{\mu^{\prime}\left( \theta \right)}{ \sigma^2\left( \theta \right)}$ & $0$ \\ + Estimation gaussienne de Whittle + \citep{fox1986large} & $\frac{\mu^{\prime}\left( \theta \right)}{ \sigma^2\left( \theta \right)}$ & $\frac{\sigma^{\prime}\left( \theta \right)}{ \sigma^2\left( \theta \right)^{3/2}}$ \\ + \hline + \end{tabular} + \caption{Méthodes d'estimation représentables par la forme quadratique \eqref{eq:generalquad} + }\label{tab:methodesquad} +\end{table} + +La méthode d'estimation gaussienne de Whittle présente un avantage par +rapport aux modèles linéaires conventionnels (moindres carrés pondérés +et maximum de quasi-vraisemblance) du fait qu'elle prend en +considération la variance comme critère d'optimisation et non +seulement comme pondération. + +\cite{crowder1987linear} développe, à partir de cette dernière, +une équation d'estimation optimale qui vise à +remplacer la méthode de quasi-vraisemblance basée sur la moyenne et la +variance jusqu'alors utilisée. + +\section{Équation d'estimation optimale} +\label{sec:equationsquadopt} + +Soit une équation d'estimation de la forme générale +\eqref{eq:generalEE87}. On a, par définition, que l'espérance de la +composante stochastique $u(\theta;Y_t)$ de celles-ci sous les vrais +paramètres $\theta_0$ est nulle. De plus, si l'on considère que les +données ne sont pas corrélées, l'espérance du produit de deux de ces +fonctions est nulle, sauf lorsqu'elle est évaluée au même point $Y_t$: +\begin{align} + \label{eq:Crowder86-th4.1-def} + E \left[u(\theta_0;Y_t) \right] &= 0 \\ + E \left[u(\theta_0;Y_s)u(\theta_0;Y_t) \right] &= + \delta_{s,t}Var\left[u(\theta_0;Y_t)\right]. +\end{align} + +Les matrices non singulières $M(\theta_0) \mbox{ et } V(\theta_0)$ +représentent respectivement, pour l'équation d'estimation, l'espérance +de sa dérivée par rapport à $\theta$ et l'inverse de sa +variance-covariance, évaluée à la vraie valeur des paramètres +$\theta_0$: +\begin{subequations}\label{eq:Crowder86-th3.3-def} + \begin{align} + M(\theta_0;Y) &= \sum_{t=1}^{T} \mathbf{w}(\theta_0) E\left[u^{\prime}(\theta_0;Y_t) \right] \label{eq:Crowder86-th3.3-def-1}\\ + V(\theta_0;Y) &= \sum_{t=1}^{T} \mathbf{w}(\theta_0) + Var\left[u(\theta_0;Y_t)\right] + \mathbf{w}^{T}(\theta_0)\label{eq:Crowder86-th3.3-def-2}. + \end{align} +\end{subequations} + +La variance asymptotique de l'estimateur $\hat\theta$ prend alors la +forme suivante, en utilisant la méthode delta multivariée: +\begin{align} + \label{eq:VarAsymptEstEE} + Var\left[\hat\theta\right] &= + M^{-1}(\theta_0)V(\theta_0)\left[M^{-1}(\theta_0)\right]^{T}. +\end{align} + +On désire obtenir des équations d'estimation optimales de telle sorte +que la variance soit minimale. Le \emph{théorème 4.1} de +\cite{crowder1986consistency} permet d'établir quel vecteur de +pondération $\mathbf{w}^{*}(\theta)$ utiliser. + +Selon ce théorème, lorsqu'on utilise le vecteur de pondération optimal +\eqref{eq:Crowdervecteuroptimal}, on obtient que la dérivée de la +fonction d'estimation $M^{*}(\theta)$ soit égale à l'inverse de la +matrice de variance-covariance $V^{*}(\theta)$: +\begin{align} + \label{eq:Crowdervecteuroptimal} + \mathbf{w}^{*}(\theta) &= \left\{E \left[u^{\prime}(\theta_0;Y_t) + \right]\right\}^{T}\left\{Var\left[u(\theta;Y_t)\right] + \right\}^{-1}. +\end{align} + +La variance asymptotique \eqref{eq:VarAsymptEstEE} du vecteur de +paramètres optimal $\hat\theta^{*}$ qui résout l'équation d'estimation +$g^{*}(\theta)$ se simplifie alors sous la forme: +$\left[V^{*}(\theta_0;y_t)\right]^{-1}$ +\begin{align} + \label{eq:13} + M^{*}(\theta_0) &= \sum_{t=1}^{T} \left\{E + \left[u^{\prime}(\theta_0;Y_t) + \right]\right\}^{T}\left\{Var\left[u(\theta_0;Y_t)\right] + \right\}^{-1} E\left[u^{\prime}(\theta_0;Y_t) \right] \\ + V^{*}(\theta_0) &= \sum_{t=1}^{T} \left\{E + \left[u^{\prime}(\theta_0;Y_t) + \right]\right\}^{T}\left\{Var\left[u(\theta_0;Y_t)\right] + \right\}^{-1} Var\left[u(\theta_0;Y_t)\right] \nonumber\\ + &\quad \times \left[\left\{E \left[u^{\prime}(\theta_0;Y_t) + \right]\right\}^{T}\left\{Var\left[u(\theta_0;Y_t)\right] + \right\}^{-1}\right]^{T} \nonumber\\ + &= \sum_{t=1}^{T} \left\{E \left[u^{\prime}(\theta_0;Y_t) + \right]\right\}^{T}\left\{Var\left[u(\theta_0;Y_t)\right] + \right\}^{-1} E\left[u^{\prime}(\theta_0;Y_t) \right] \nonumber\\ + \Leftrightarrow M^{*}(\theta_0) &= V^{*}(\theta_0) \nonumber\\ + \Leftrightarrow Var(\hat\theta^{*}) &= \left[M^{*}(\theta_0)\right]^{-1} V^{*}(\theta_0) \left[\left[M^{*}(\theta_0)\right]^{-1}\right]^{T} \nonumber\\ + &= \left[V^{*}(\theta_0)\right]^{-1}. +\end{align} + +On considère la fonction $u(\theta;Y_t)$ de l'équation quadratique +\eqref{eq:generalquad} , dont on cherche le vecteur de pondération +optimal correspondant selon la proposition +\eqref{eq:Crowdervecteuroptimal}: +\begin{align} + \label{eq:15} + u(\theta;Y_t) &= \begin{bmatrix} + Y_t-\mu(\theta)\\ + (Y_t-\mu(\theta))^2-\sigma^2(\theta) + \end{bmatrix}. +\end{align} + +On évalue l'espérance, sous le vecteur de vrais paramètres $\theta_0$, +de la dérivée par rapport à $\theta$ de la fonction $u(\theta;Y)$: +\begin{align} + \label{eq:16} + E\left[u^{\prime}(\theta;Y_t)\right] &= \begin{bmatrix} + E \left[-\mu^{\prime}(\theta)\right] \\ + E + \left[-2(Y-\mu(\theta))\mu^{\prime}(\theta)-2\sqrt{\sigma^2(\theta)}\sigma^{\prime}(\theta)\right] + \end{bmatrix} \nonumber\\ + &= \begin{bmatrix} + -\mu^{\prime}(\theta) \\ + -2 \sqrt{\sigma^2(\theta)}\sigma^{\prime}(\theta). + \end{bmatrix}. +\end{align} + +Puis, on évalue la variance-covariance du vecteur $u(\theta)$. Pour ce +faire, on devra évaluer séparément chaque élément composant cette +matrice: +\begin{align} + \label{eq:matvcov-u-EE} + Var\left[u(\theta;Y_t) \right] &= + \begin{bmatrix} + Var\left[Y-\mu(\theta)\right] & Cov\left[Y-\mu(\theta),(Y-\mu(\theta))^2-\sigma^2(\theta) \right] \\ + Cov\left[Y-\mu(\theta),(Y-\mu(\theta))^2-\sigma^2(\theta) \right] + & Var\left[(Y-\mu(\theta))^2-\sigma^2(\theta)\right] + \end{bmatrix}. +\end{align} + +On définit les coefficients d'asymétrie et d'aplatissement normalisés, +ainsi qu'une constante $\gamma_3(\theta)$ qui en découle, afin de +simplifier les expressions qui seront obtenues conséquemment: +\begin{subequations}\label{eq:momentsupgamma} + \begin{align} + \gamma_1\left(\theta\right) &= \frac{m_3\left(\theta\right)}{m_2\left(\theta\right)^{3/2}} \nonumber\\ + &= \frac{E\left[\left(Y-\mu\left(\theta\right)\right)^3\right]}{\sigma^2\left(\theta\right)^{3/2}} \label{eq:momentsupgamma1}\\ + \gamma_2\left(\theta\right) &= \frac{m_4\left(\theta\right)}{m_2\left(\theta\right)^{2}} - 3 \nonumber\\ + &=\frac{E\left[\left(Y-\mu\left(\theta\right)\right)^4\right]}{\sigma^2\left(\theta\right)^2}-3 \label{eq:momentsupgamma2}\\ + \gamma_3(\theta) &= \gamma_2\left(\theta\right) + 2 - + \gamma_1\left(\theta\right)^2. \label{eq:momentsupgamma3} + \end{align} +\end{subequations} + +On développe les différentes composantes de la matrice: +\eqref{eq:matvcov-u-EE} +\begin{align} + \label{eq:18} + Var\left[Y-\mu(\theta)\right] &= E\left[(Y-\mu(\theta))^2\right] - + E\left[Y-\mu(\theta)\right]^2 \nonumber\\ + &=\sigma^2(\theta)\\ + Cov\left[Y-\mu(\theta),(Y-\mu(\theta))^2-\sigma^2(\theta) \right] &= + E\left[\left(Y-\mu(\theta)\right)\left((Y-\mu(\theta))^2-\sigma^2(\theta)\right) + \right] \nonumber\\ &\quad - E\left[Y-\mu(\theta) \right] + E\left[(Y-\mu(\theta))^2-\sigma^2(\theta) \right] \nonumber\\ + &= E\left[(Y-\mu(\theta))^3 - \sigma^2(\theta)(Y-\mu(\theta))\right] \nonumber\\ + &= \sigma^2(\theta)^{3/2}\gamma_1(\theta)\\ + Var\left[(Y-\mu(\theta))^2-\sigma^2(\theta)\right] &= + E\left[\left((Y-\mu(\theta))^2-\sigma^2(\theta) + \right)\left((Y-\mu(\theta))^2-\sigma^2(\theta) \right) \right]\nonumber\\ + &= E\left[(Y-\mu(\theta))^4 - 2\sigma^2(\theta)(Y-\mu(\theta))^2 + + \sigma^2(\theta)^2 \right] \nonumber\\ + &= \sigma^2(\theta)^2\left(\gamma_2(\theta)+3\right) - 2\sigma^2(\theta)^2 + \sigma^2(\theta)^2 \nonumber\\ + &=\sigma^2(\theta)^2\left(\gamma_2(\theta)+2 \right). +\end{align} + +On obtient alors la matrice de variance-covariance de la fonction +$u(\theta)$, dont on évalue par la suite l'inverse: +\begin{align} + \label{eq:19} + Var\left[u(\theta;Y_t) \right] &= \begin{bmatrix} + \sigma^2(\theta) & \sigma^2(\theta)^{3/2}\gamma_1(\theta)\\ + \sigma^2(\theta)^{3/2}\gamma_1(\theta) & + \sigma^2(\theta)^2\left(\gamma_2(\theta)+2 \right) + \end{bmatrix} \\ + Var^{-1}\left[u(\theta;Y_t) \right] &= + \frac{1}{\sigma^2(\theta)^3\gamma_3(\theta)}\begin{bmatrix} + \sigma^2(\theta)^2\left(\gamma_2(\theta)+2 \right) & -\sigma^2(\theta)^{3/2}\gamma_1(\theta)\\ + -\sigma^2(\theta)^{3/2}\gamma_1(\theta) & \sigma^2(\theta) + \end{bmatrix}. +\end{align} + +On peut enfin évaluer l'expression \eqref{eq:Crowdervecteuroptimal} à +l'aide des résultats précédents: +\begin{align*} + \mathbf{w}^{*}(\theta) &= \begin{bmatrix} -\mu^{\prime}(\theta) & -2 + \sqrt{\sigma^2(\theta)}\sigma^{\prime}(\theta). + \end{bmatrix} \times + \frac{1}{\sigma^2(\theta)^3\gamma_3(\theta)} \begin{bmatrix} + \sigma^2(\theta)^2\left(\gamma_2(\theta)+2 \right) & -\sigma^2(\theta)^{3/2}\gamma_1(\theta)\\ + -\sigma^2(\theta)^{3/2}\gamma_1(\theta) & \sigma^2(\theta) + \end{bmatrix}. +\end{align*} + +On obtient alors le vecteur de pondération optimal +\eqref{eq:vecteuroptimal.ab} formé des fonctions +$\mathbf{a}^{*}(\theta)$ et $\mathbf{b}^{*}(\theta)$: +\begin{subequations}\label{eq:coefficientscrowder} + \begin{align} + \mathbf{a}^{*}(\theta) &= \frac{\left\{ -\left( \gamma_2\left(\theta\right)+2 \right) \mu^{\prime}\left( \theta \right) + 2\gamma_1\left(\theta\right) \sigma^{\prime}\left( \theta \right) \right\}}{\sigma^2\left(\theta\right)\gamma_3(\theta)} \label{eq:acrowder}\\ + \mathbf{b}^{*}(\theta) &= + \frac{\gamma_1\left(\theta\right)\mu^{\prime}\left( \theta + \right)-2\sigma^{\prime}\left( \theta \right)}{\sigma^2\left( + \theta \right)^{3/2}\gamma_3(\theta)}. \label{eq:bcrowder} + \end{align} +\end{subequations} + +On obtient aisément les matrices $M^{\star}\left(\theta\right) \mbox{ + et } V^{\star}\left(\theta\right)$ à partir de leur définition +\eqref{eq:Crowder86-th3.3-def}: +\begin{align} + M^{\star}\left(\theta\right) &= V^{\star}\left(\theta\right) \nonumber\\ + &= T \cdot \sigma(\theta)^{-2} \left\{ \left(\mu^{\prime}\left( + \theta \right)\right)\left(\mu^{\prime}\left( \theta + \right)\right)^T+\gamma_3(\theta)^{-1}\left(\gamma_1\left(\theta\right)\mu^{\prime}\left( + \theta \right)-2\sigma^{\prime}\left( \theta + \right)\right)\left(\gamma_1\left(\theta\right)\mu^{\prime}\left( + \theta \right)-2\sigma^{\prime}\left( \theta \right)\right)^T + \right\}. \label{eq:Moptimalestimetheta} +\end{align} + +Afin d'obtenir la solution optimale $\theta^{\star}$, on définit la +fonction objectif $\Lambda\left(\theta;\mathbf{y}_T\right)$: +\begin{align} + \label{eq:eqnobjectifEE} + \Lambda\left(\theta;\mathbf{y}_T\right) &= + g\left(\theta;\mathbf{y}_T \right) W\left( \theta;\mathbf{y}_T + \right) g\left( \theta;\mathbf{y}_T \right)^T. +\end{align} + +$W\left( \theta ;\mathbf{y}_T \right)$ est une matrice définie +positive. En premier lieu, on suggère d'utiliser la matrice +identité. Puis, on peut raffiner l'estimation en lui substituant la +matrice $V^{\star}(\hat\theta)$ de manière itérative, de la même +manière qu'avec la méthode des moments généralisée itérative, à la +section \ref{sec:GMMtwostep}. + +\section{Équation d'estimation optimale modifiée} +\label{sec:eqoptmodif} + +Il est possible que, pour certaines distributions, les expressions +formant le vecteur de pondération optimal $\mathbf{w}^{*}(\theta)$ +soient particulièrement complexes. On pourra toujours les utiliser, +cependant, il peut être intéressant de développer une approximation de +celles-ci. Pour ce faire, on substitue les valeurs théoriques des +coefficients d'asymétrie et d'aplatissement par une valeur estimée à +partir de l'échantillon: +\begin{subequations}\label{eq:vecteurcrowdermod} + \begin{align} + \mathbf{a}_{mod}^{*}(\theta) &= \frac{\left\{ -\left( \hat\gamma_2\left(\mathbf{y}_T\right)+2 \right) \mu^{\prime}\left( \theta \right) + 2\hat\gamma_1\left(\mathbf{y}_T\right) \sigma^{\prime}\left( \theta \right) \right\}}{\sigma^2\left(\theta\right)\hat\gamma_3(\mathbf{y}_T)} \label{eq:acrowdermod}\\ + \mathbf{b}_{mod}^{*}(\theta) &= + \frac{\hat\gamma_1\left(\mathbf{y}_T\right)\mu^{\prime}\left( + \theta \right)-2\sigma^{\prime}\left( \theta + \right)}{\sigma^2\left( \theta + \right)^{3/2}\hat\gamma_3(\mathbf{y}_T)} \label{eq:bcrowdermod} + \end{align} + + où + \begin{align} + \label{eq:coeffemp} + \hat\gamma_1\left(\mathbf{y}_T\right) &= \frac{\sum_{t=1}^{T}(y_t-\overline{y})^3}{\left(\sum_{t=1}^{T}(y_t-\overline{y})^2\right)^{3/2}} \\ + \hat\gamma_2\left(\mathbf{y}_T\right) &= + \frac{\sum_{t=1}^{T}(y_t-\overline{y})^4}{\left(\sum_{t=1}^{T}(y_t-\overline{y})^2\right)^{2}} \\ + \hat\gamma_3\left(\mathbf{y}_T\right) &= + \hat\gamma_2\left(\mathbf{y}_T\right) + 2 - + \hat\gamma_1\left(\mathbf{y}_T\right)^2. + \end{align} +\end{subequations} + +On pourra ainsi réduire considérablement la taille des expressions à +évaluer tout en conservant les propriétés asymptotiques des +estimateurs, puisque l'espérance de chacune des statistiques utilisées +est égale à la valeur que l'on a remplacée. + +Au prochain chapitre, nous appliquerons la méthode des moments +généralisée et la méthode de l'équation d'estimation optimale à la +distribution de Laplace asymétrique généralisée. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/chapitre6.tex b/depotfinal/908144032/chapitre6.tex new file mode 100644 index 0000000..639d1a2 --- /dev/null +++ b/depotfinal/908144032/chapitre6.tex @@ -0,0 +1,396 @@ +\chapter{Estimation des paramètres de la distribution de Laplace + asymétrique généralisée} + +Ce chapitre présente les différents éléments qui permettent d'obtenir +des estimateurs convergents à l'aide de la méthode des moments +généralisée et de l'équation d'estimation optimale. Premièrement, une +méthode pour obtenir un point de départ efficace pour l'algorithme +d'optimisation sera détaillée. Enfin, les résultats découlant des deux +chapitres précédents seront présentés. + +\section{Vecteur de paramètres initiaux} +\label{sec:condinitGAL} + +Comme chacune des deux méthodes d'estimation étudiées requiert +l'utilisation d'un algorithme d'optimisation numérique, on doit être +en mesure de fournir un vecteur de paramètres initiaux qui favorisera +la convergence de celui-ci. Ce vecteur doit faire partie de l'espace +des paramètres $\Omega$, tel que défini à la table +\ref{tab:roleparamGAL}. Ce vecteur est plus facile à obtenir de +manière empirique, par exemple, avec la méthode des moments, lorsque +c'est possible. Cependant, pour la distribution de Laplace asymétrique +généralisée, le système d'équations formé à partir des quatre premiers +moments centraux empiriques et théoriques \eqref{eq:momentsGAL} n'a +pas de solution analytique. + +\cite{seneta2004fitting} propose une technique pour obtenir un vecteur +de paramètres initiaux basé sur la méthode des moments. En utilisant +la même démarche que ce dernier, on peut obtenir un vecteur pour la +forme en $\mu$ de la fonction caractéristique +\eqref{eq:fncaractGALmu}. On pose comme hypothèse dans les équations +des quatre premiers moments \eqref{eq:moments1GAL}, +\eqref{eq:moments2GAL}, \eqref{eq:moments5GAL} et +\eqref{eq:moments6GAL}, que le paramètre $\mu$ est significativement +nul, donc que la distribution est presque symétrique. On impose donc +que les puissances de ce paramètre plus grandes ou égales à deux +soient nulles: +\begin{align} + \label{eq:hypotheseMoM} + \mu^k=0, \quad k\in\{2,3,4\}. +\end{align} + +En posant l'égalité entre les moments théoriques et leur estimateur +correspondant, on obtient le système d'équations suivant: +\begin{align*} + \hat{m_1} &= \theta+\tau\,\mu \\ + \hat{m_2} &= {\sigma}^{2}\,\tau \\ + \hat{\gamma_1} &= \frac{3\,\mu}{\left| \sigma\right| \,\sqrt{\tau}} \\ + \hat{\gamma_2} &= \frac{3}{\tau}. +\end{align*} + +Ce système possède deux solutions analytiques, dont une respecte la +condition selon laquelle le paramètre $\sigma$ doit être positif. On +note que ce résultat ne constitue qu'un point de départ pour +l'algorithme d'optimisation numérique et qu'on ne peut, en aucun cas, +utiliser celui-ci à des fins statistiques. C'est un estimateur biaisé +et non convergent étant donné la condition \eqref{eq:hypotheseMoM}: +\begin{subequations}\label{eq:ptdepartGAL} + \begin{align} + \hat\theta &=-\frac{\hat{\gamma_1}\,\sqrt{\hat{m_2}}-\hat{\gamma_2}\,\hat{m_1}}{\hat{\gamma_2}} \label{eq:ptdepartGAL1}\\ + \hat\sigma &= \frac{\sqrt{\hat{\gamma_2}\,\hat{m_2}}}{\sqrt{3}} > 0 \label{eq:ptdepartGAL2}\\ + \hat\mu &= \frac{\hat{\gamma_1}\,\sqrt{\hat{m_2}}}{3} \label{eq:ptdepartGAL3}\\ + \hat\tau &= \frac{3}{\hat{\gamma_2}}. \label{eq:ptdepartGAL4} + \end{align} +\end{subequations} + +\section{Méthode des moments généralisée} + +En premier lieu, on utilise la méthode des moments généralisée afin +d'estimer les paramètres de la distribution de Laplace asymétrique +généralisée. Aux fins d'illustration, seulement deux conditions de +moments seront utilisées. Cependant, pour obtenir des résultats +optimaux en termes de convergence asymptotique, on doit utiliser au +moins autant de conditions de moment que de paramètres. Ainsi, en +considérant la différence entre les moyennes et variances empirique et +théorique, on obtient les conditions de moment suivantes, qui ont une +espérance nulle sous les vrais paramètres $\theta_0$: +\begin{align} + h(\theta;Y) &= \left[\begin{array}[]{c} + Y - m_1 \\ + \left(Y - m_1 \right)^2 - m_2 + \end{array}\right] \label{eq:condmom}\\ + &= \left[\begin{array}[]{c} + Y - \left(\theta+\mu\tau\right) \\ + \left(Y - \left(\theta+\mu\tau\right) \right)^2 - + \tau\left(\sigma^2+\mu^2\right) + \end{array}\right]. \label{eq:condmomGAL} +\end{align} + +On définit la contrepartie empirique de ces conditions de moment +comme suit: +\begin{align} + g(\theta;\mathbf{y}_T) &= \frac{1}{T} \left[\begin{array}[]{c} + g_1(\theta;\mathbf{y}_T)\\ + g_2(\theta;\mathbf{y}_T) + \end{array}\right] \nonumber\\ + &= \frac{1}{T} \left[\begin{array}[]{c} + \sum_{t=1}^T y_t - m_1 \\ + \sum_{t=1}^T \left(y_t - m_1 \right)^2 - m_2 + \end{array}\right] \label{eq:condmomEMP}\\ + &= \frac{1}{T} \left[\begin{array}[]{c} + \sum_{t=1}^Ty_t - \left(\theta+\mu\tau\right) \\ + \sum_{t=1}^T\left(y_t - \left(\theta+\mu\tau\right) \right)^2 - + \tau\left(\sigma^2+\mu^2\right) + \end{array}\right]. \label{eq:condmomEMPGAL} +\end{align} + +On doit définir la matrice de pondération optimale. + +\subsection{Matrice de pondération optimale} + +On s'intéresse à la forme analytique de la matrice de pondération +optimale \eqref{eq:matricevcov1}. Celle-ci est obtenue en prenant +l'espérance du produit extérieur du vecteur des conditions de moment +théoriques sur elles-mêmes. Pour des fins de simplification, on +définit le produit des conditions $i$ et $j$ par la notation +$H_{i,j}(\theta;Y) = h_i(\theta;Y)h_j(\theta;Y)$. +\begin{align} + S(\theta;Y) &= E \left[\begin{array}{cc} + H_{(1,1)}(\theta;Y) & H_{(1,2)}(\theta;Y) \\ + H_{(2,1)}(\theta;Y) & H_{(2,2)}(\theta;Y) + \end{array} \right] \nonumber\\ + &= \left[\begin{array}{cc} + E \left[H_{(1,1)}(\theta;Y)\right] & E \left[H_{(1,2)}(\theta;Y)\right] \\ + E \left[H_{(2,1)}(\theta;Y)\right] & E \left[H_{(2,2)}(\theta;Y)\right] + \end{array} \right] +\end{align} +où +\begin{align*} + H_{(1,1)}(\theta;Y) &=\left( Y-\theta-\mu\,\tau\right)^{2} \\ + H_{(1,2)}(\theta;Y) &=\left( {Y}^{2}-2\,\theta\,Y-2\,\mu\,\tau\,Y+{\theta}^{2}+2\,\mu\,\tau\,\theta+{\mu}^{2}\,{\tau}^{2}-{\sigma}^{2}\,\tau-{\mu}^{2}\,\tau\right)^{2} \\ + H_{(2,2)}(\theta;Y) &=\left( Y-\theta-\mu\,\tau\right) \,\left( + {Y}^{2}-2\,\theta\,Y-2\,\mu\,\tau\,Y+{\theta}^{2}+2\,\mu\,\tau\,\theta+{\mu}^{2}\,{\tau}^{2}-{\sigma}^{2}\,\tau-{\mu}^{2}\,\tau\right). +\end{align*} + +On évalue ensuite l'espérance de chacun des éléments de la matrice, où +l'on définit de manière analogue celle du produit des conditions de +moments, $W_{(i,j)}(\theta;Y) = E[H_{(i,j)}(\theta;Y)]$. +\begin{align} + W(\theta;Y) &= \left[\begin{array}{cc} + W_{(1,1)}(\theta;Y) & W_{(1,2)}(\theta;Y) \\ + W_{(2,1)}(\theta;Y) & W_{(2,2)}(\theta;Y) + \end{array} \right] +\end{align} +où +\begin{align*} + W_{(1,1)} &= {\mu}^{2}\,{\tau}^{2}-2\,{\mu}^{2}\,\nu\,\tau+\nu\,{\sigma}^{2}+{\mu}^{2}\,{\nu}^{2}+{\mu}^{2}\,\nu \\ + W_{(2,2)} &= {\mu}^{4}\,{\tau}^{4}+\left( -2\,{\mu}^{2}\,{\sigma}^{2}-4\,{\mu}^{4}\,\nu-2\,{\mu}^{4}\right) \,{\tau}^{3}+\left( {\sigma}^{4}+\left( 10\,{\mu}^{2}\,\nu+2\,{\mu}^{2}\right) \,{\sigma}^{2}+6\,{\mu}^{4}\,{\nu}^{2}+10\,{\mu}^{4}\,\nu+{\mu}^{4}\right) \,{\tau}^{2}\\ + &+\left( -2\,\nu\,{\sigma}^{4}+ \left( -14\,{\mu}^{2}\,{\nu}^{2}-16\,{\mu}^{2}\,\nu\right) \,{\sigma}^{2}-4\,{\mu}^{4}\,{\nu}^{3}-14\,{\mu}^{4}\,{\nu}^{2}-10\,{\mu}^{4}\,\nu\right) \,\tau+\left( 3\,{\nu}^{2}+3\,\nu\right) \,{\sigma}^{4}\\ + &+\left( 6\,{\mu}^{2}\,{\nu}^{3}+18\,{\mu}^{2}\,{\nu}^{2}+12\,{\mu}^{2}\,\nu\right) \,{\sigma}^{2}+{\mu}^{4}\,{\nu}^{4}+6\,{\mu}^{4}\,{\nu}^{3}+11\,{\mu}^{4}\,{\nu}^{2}+6\,{\mu}^{4}\,\nu \\ + W_{(1,2)} &= W_{(2,1)} = 3\,{\theta}^{4}+\left( 6\,\mu\,\nu-3\right) \,{\theta}^{3}+\left( 3\,\nu\,{\sigma}^{2}+3\,{\mu}^{2}\,{\nu}^{2}+\left( 3\,{\mu}^{2}-3\,\mu\right) \,\nu\right) \,{\theta}^{2}-{\mu}^{3}\,{\tau}^{3} \\ + &+\left( \mu\,{\sigma}^{2}+3\,{\mu}^{3}\,\nu+{\mu}^{3}\right) \,{\tau}^{2}+\left( -4\,\mu\,\nu\,{\sigma}^{2}-3\,{\mu}^{3}\,{\nu}^{2}-4\,{\mu}^{3}\,\nu\right) \,\tau+\left( 3\,\mu\,{\nu}^{2}+3\,\mu\,\nu\right) \,{\sigma}^{2}\\ + &+{\mu}^{3}\,{\nu}^{3}+3\,{\mu}^{3}\,{\nu}^{2}+2\,{\mu}^{3}\,\nu. +\end{align*} + +On évalue quelle serait la valeur de la variance-covariance sous des +paramètres optimaux. Ce résultat permet par la suite d'évaluer la +distribution asymptotique des estimateurs. + +On obtient l'estimateur de la matrice optimale \eqref{eq:matvcovGMM} +en effectuant le produit extérieur du vecteur de conditions de moment +empirique \eqref{eq:condmomGAL}, puis en évaluant la moyenne des +matrices résultantes de l'équation \eqref{eq:matponderationproduith}: +\begin{align} + \label{eq:matvcovGMMemp1} + \hat{S}(\theta;\mathbf{y}_T) &= \frac{1}{T} \sum_{t=1}^T + \left[\begin{array}{cc} + G_{(1,1)}(\theta;y_t) & G_{(1,2)}(\theta;y_t) \\ + G_{(2,1)}(\theta;y_t) & G_{(2,2)}(\theta;y_t) + \end{array} \right] +\end{align} +où +\begin{align*} + G_{(1,1)}(\theta;y_t) &=\left( y_t-\theta-\mu\,\tau\right)^{2} \\ + G_{(1,2)}(\theta;y_t) &=\left( {y_t}^{2}-2\,\theta\,y_t-2\,\mu\,\tau\,y_t+{\theta}^{2}+2\,\mu\,\tau\,\theta+{\mu}^{2}\,{\tau}^{2}-{\sigma}^{2}\,\tau-{\mu}^{2}\,\tau\right)^{2} \\ + G_{(2,2)}(\theta;y_t) &=\left( y_t-\theta-\mu\,\tau\right) \,\left( + {y_t}^{2}-2\,\theta\,y_t-2\,\mu\,\tau\,y_t+{\theta}^{2}+2\,\mu\,\tau\,\theta+{\mu}^{2}\,{\tau}^{2}-{\sigma}^{2}\,\tau-{\mu}^{2}\,\tau\right). +\end{align*} + + +\subsection{Variance-covariance des paramètres} + +On obtient la variance-covariance asymptotique des paramètres à partir +de la variance-covariance associée aux conditions de moment en +utilisant la méthode delta multivariée (Annexe +\ref{sec:deltamethod}). + +Pour ce faire, on évalue d'abord la valeur +théorique du gradient $D(\theta)$: +\begin{align} + D(\theta) &= E \left[ \begin{array}{cc} + -1 & -2\,\left( Y-\theta-\mu\,\tau\right) \\ + 0 & -2\,\sigma\,\tau \\ + -\tau & -2\,\tau\,\left( Y-\theta-\mu\,\tau\right) -2\,\mu\,\tau \\ + -\nu & -2\,\mu\,\left( Y-\theta-\mu\,\tau\right) + -{\sigma}^{2}-{\mu}^{2} + \end{array}\right] \nonumber\\ + &= \left[ \begin{array}{cc} + -1 & 0 \\ + 0 & -2\,\sigma\,\tau \\ + -\tau & -2\,\mu\,\tau \\ + -\nu & -{\sigma}^{2}-{\mu}^{2} + \end{array}\right]. +\end{align} + +Ensuite, on utilise le résultat \eqref{matricevcovparamGMMnc} afin +d'obtenir la valeur de la variance-covariance $\mathcal{J}_0^{-1}$. + +De la même manière, on évalue la variance-covariance des paramètres +optimaux en utilisant les valeurs empiriques de la variance-covariance +des conditions de moment \eqref{eq:matvcovGMMemp1} et du gradient +$\hat{D}(\theta,\mathbf{y}_T)$, défini par la matrice +\eqref{eq:gradientGMM}. On a essentiellement à calculer la moyenne +empirique des gradients obtenus en chaque point $y_t$, puis à utiliser +le résultat \eqref{matricevcovparamGMMnc} avec les estimateurs +$\hat{W}_T = \hat{S}^{-1}(\theta,\mathbf{y}_T)$ et +$\hat{D}(\theta,\mathbf{y}_T)$: +\begin{align} + \hat{D}(\theta,\mathbf{y}_T) &= \frac{1}{T} \sum_{t=1}^{T} + \left[ \begin{array}{cc} + -1 & -2\,\left( y_t-\theta-\mu\,\tau\right) \\ + 0 & -2\,\sigma\,\tau \\ + -\tau & -2\,\tau\,\left( y_t-\theta-\mu\,\tau\right) -2\,\mu\,\tau \\ + -\nu & -2\,\mu\,\left( y_t-\theta-\mu\,\tau\right) + -{\sigma}^{2}-{\mu}^{2} + \end{array}\right]. +\end{align} + +\subsection{Contraintes linéaires} + +Une fois l'estimation des paramètres effectuée, on peut tester si ces +derniers pourraient en fait correspondre à un cas particulier de la +distribution de Laplace asymétrique généralisée ayant le même support, +parmi celles figurant à la table \ref{tab:casspeciauxGAL}. Pour ce +faire, on doit déterminer les paramètres de la distribution sous un +ensemble de contraintes linéaires, sous la forme +\eqref{eq:contraintelin}, correspondant au cas particulier, tel +qu'énuméré dans la table \ref{tab:contraintesGAL}. +\begin{table}[h!] + \centering + \begin{tabular}{ccc} + & \multicolumn{2}{c}{\textbf{Paramètres}} \\ + \hline + $\text{\textbf{Distribution}}$ & $R$ & $r$ \\ + \hline + Laplace symétrique & $\begin{bmatrix} + 0&0&1&0\\ + 0&0&0&1 + \end{bmatrix}$ & $\begin{bmatrix} + 0\\ + 1 + \end{bmatrix}$ \\[1cm] + Laplace asymétrique & $\begin{bmatrix} 0&0&0&1 + \end{bmatrix}$ & $\begin{bmatrix} 1 + \end{bmatrix}$ \\[1cm] + Dégénérée à $\theta$ & $\begin{bmatrix} + 0&1&0&0\\ + 0&0&1&0\\ + 0&0&0&1 + \end{bmatrix}$ & $\begin{bmatrix} + 0\\ + 0\\ + 0\\ + \end{bmatrix}$\\\hline + \end{tabular} + \caption{Contraintes linéaires pour les cas particuliers de la distribution de Laplace asymétrique généralisée $GAL(\theta,\sigma,\mu,\tau)$} + \label{tab:contraintesGAL} +\end{table} + +On doit ensuite utiliser un algorithme d'optimisation numérique afin +de maximiser le lagrangien \eqref{eq:estimateurGMMlagrange} défini par +la fonction objectif $Q_T(\theta)$ \eqref{eq:objectifGMM2} et +l'ensemble de contraintes $a(\theta) = R\theta-r$ sélectionné +précédemment. Enfin, on peut effectuer les tests statistiques de la +section \ref{sec:testwald} afin de vérifier si les paramètres de la +distribution contrainte sont significativement différents de ceux de +la distribution non contrainte. Dans cette situation, l'hypothèse +selon laquelle le cas particulier serait approprié pour décrire la +distribution des données de l'échantillon serait rejetée. + +Notons qu'on peut aussi estimer les paramètres de chacun des cas +particuliers directement par la méthode des moments généralisée en +utilisant des conditions de moment basées sur la moyenne $m_1$ et la +variance $m_2$ de la forme \eqref{eq:condmom}. Par contre, dans ces +cas, on devra utiliser des tests d'adéquation non paramétriques afin +de décider laquelle représente mieux la distribution de l'échantillon +de données. Ces tests seront présentés au chapitre suivant, à la +section \ref{sec:testnonparam}. + +\section{Méthode de l'équation d'estimation optimale} +\label{sec:CrowderGAL} + +On utilise maintenant la méthode de l'équation d'estimation optimale +afin d’estimer les paramètres de la distribution de Laplace +asymétrique généralisée. On considère une équation d'estimation de la +forme quadratique \eqref{eq:generalquad}, basée sur les conditions de +moments \eqref{eq:condmomGAL} utilisées à la section précédente. + +On évalue d'abord les expressions des dérivées premières de la moyenne +et de l'écart-type par rapport au vecteur de paramètres +\eqref{eq:derivmom}: +\begin{subequations}\label{eq:derivmomGAL} + \begin{align} + \mu^{\prime}\left( \theta \right) &= \frac{d}{d\theta}\left(\theta+\tau\,\mu\right) \nonumber\\ + &= \left[ \begin{array}{c} 1\\ + 0\\ + \tau\\ + \mu\end{array} \right]^T \label{eq:derivmom1GAL}\\ + \sigma^{\prime}\left( \theta \right) &= \frac{d}{d\theta}\sqrt{\tau\sigma^2+\tau\mu^2} \nonumber\\ + &= \frac{1}{\sqrt {\tau\,{\sigma}^{2}+\tau\,{\mu}^{2}}} + \left[ \begin{array}{c} + 0\\ + {\tau\,\sigma}\\ + {\tau\,\mu}\\ + \frac{{\sigma}^{2}+{\mu}^{2}}{2} + \end{array} \right]^T.\label{eq:derivmom2GAL} + \end{align} +\end{subequations} + +On peut alors obtenir une forme analytique en utilisant les +expressions déterminées précédemment pour $\mathbf{a}(\theta)$ et +$\mathbf{b}(\theta)$ \eqref{eq:coefficientscrowder}, les moments de la +distribution \eqref{eq:momentsGAL} ainsi que les coefficients +d'asymétrie et d'aplatissement \eqref{eq:moments56GAL}. Étant donné la +longueur des expressions obtenues, elles ne seront pas présentées dans +ce texte. Cependant, on peut facilement les évaluer à l'aide d'un +logiciel de calcul symbolique. On combine ensuite celles-ci pour +obtenir l'équation d'estimation optimale $g\left(\theta;\mathbf{y}_T +\right)$ de forme quadratique. + +On obtient cependant des expressions plus faciles à manipuler pour +$\mathbf{a}(\theta)$ et $\mathbf{b}(\theta)$ en utilisant la version +modifiée du vecteur de pondération optimal +\eqref{eq:vecteurcrowdermod}: +\begin{subequations} + \label{eq:vecteurcrowdermodGAL} + \begin{align} + \mathbf{a}_{mod}^{*}(\theta;\mathbf{y}_T) &= \begin{bmatrix} + \frac{-\hat\gamma_2\left(\mathbf{y}_T\right)-2}{\left( {\sigma}^{2}+{\mu}^{2}\right) \,\tau\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + \frac{2\,\sigma\,\hat\gamma_1\left(\mathbf{y}_T\right)}{{\left( {\sigma}^{2}+{\mu}^{2}\right) }^{3/2}\,\sqrt{\tau}\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + \frac{\frac{2\,\mu\,\sqrt{\tau}\,\hat\gamma_1\left(\mathbf{y}_T\right)}{\sqrt{{\sigma}^{2}+{\mu}^{2}}}+\tau\,\left( -\hat\gamma_2\left(\mathbf{y}_T\right)-2\right) }{\left( {\sigma}^{2}+{\mu}^{2}\right) \,\tau\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + \frac{\frac{\sqrt{{\sigma}^{2}+{\mu}^{2}}\,\hat\gamma_1\left(\mathbf{y}_T\right)}{\sqrt{\tau}}+\mu\,\left( + -\hat\gamma_2\left(\mathbf{y}_T\right)-2\right) }{\left( + {\sigma}^{2}+{\mu}^{2}\right) + \,\tau\,\hat\gamma_3\left(\mathbf{y}_T\right) } + \end{bmatrix} \\ + \mathbf{b}_{mod}^{*}(\theta;\mathbf{y}_T) &= \begin{bmatrix} + \frac{\hat\gamma_1\left(\mathbf{y}_T\right)}{{\left( {\sigma}^{2}+{\mu}^{2}\right) }^{3/2}\,{\tau}^{3/2}\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + -\frac{2\,\sigma}{{\left( {\sigma}^{2}+{\mu}^{2}\right) }^{2}\,\tau\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + \frac{\tau\,\hat\gamma_1\left(\mathbf{y}_T\right)-\frac{2\,\mu\,\sqrt{\tau}}{\sqrt{{\sigma}^{2}+{\mu}^{2}}}}{{\left( {\sigma}^{2}+{\mu}^{2}\right) }^{3/2}\,{\tau}^{3/2}\,\hat\gamma_3\left(\mathbf{y}_T\right) } \\ + \frac{\mu\,\hat\gamma_1\left(\mathbf{y}_T\right)-\frac{\sqrt{{\sigma}^{2}+{\mu}^{2}}}{\sqrt{\tau}}}{{\left( + {\sigma}^{2}+{\mu}^{2}\right) + }^{3/2}\,{\tau}^{3/2}\,\hat\gamma_3\left(\mathbf{y}_T\right) } + \end{bmatrix}. + \end{align} +\end{subequations} + +On insère ensuite ces dernières dans la forme générale +\eqref{eq:generalquad} pour obtenir l'équation d'estimation optimale +modifiée $g_{mod}^{*}\left(\theta;Y \right)$: +\begin{align} + \label{eq:generalquadmod} + g_{mod}^{*}\left(\theta;Y \right) = \sum_{t=1}^n \left[ + \mathbf{a}_{mod}^{*}(\theta;\mathbf{y}_T)(y_t-\mu\left(\theta\right)) + + \mathbf{b}_{mod}^{*}(\theta;\mathbf{y}_T)\left((y_t-\mu\left(\theta\right))^2-\sigma^2\left(\theta\right) + \right)\right]. +\end{align} + +On note qu'afin d'éviter des irrégularités numériques, on suggère, +dans les deux cas, de faire l'estimation sur des données centrées et +réduites: +\begin{align} + X_t &= + \frac{Y_t-\hat{m_1}}{\sqrt{\hat{m_2}}}. \label{eq:defcentrereduite} +\end{align} + +Comme précédemment, on pourra utiliser la propriété d'invariance +\eqref{eq:GALscaletrans} de la paramétrisation en $\kappa$ afin de +retrouver les paramètres de la distribution de la variable aléatoire +$Y_t$ une fois l'estimation effectuée: +\begin{align} + Y_t &= \hat{\sigma}_{t} X_t + \hat{\mu}_{t} \sim + GAL(\hat{\sigma}_{t} \theta + + \hat{\mu}_{t},\hat{\sigma}_{t}\sigma,\kappa,\tau). \label{eq:paramnonreduit} +\end{align} + +Afin d'obtenir le vecteur de paramètres estimés $\hat\theta$, on +minimise la fonction objectif +$\Lambda\left(\theta;\mathbf{y}_T\right)$ +\eqref{eq:eqnobjectifEE}. Une fois les paramètres obtenus, on peut +ensuite évaluer la variance-covariance de ces estimateurs, à partir du +résultat \eqref{eq:Moptimalestimetheta}. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/chapitre7.tex b/depotfinal/908144032/chapitre7.tex new file mode 100644 index 0000000..61204b5 --- /dev/null +++ b/depotfinal/908144032/chapitre7.tex @@ -0,0 +1,336 @@ +\chapter{Tests statistiques} + +\section{Test de normalité} +\label{sec:testnormalite} + +On utilise un test de normalité afin de vérifier si l'on peut rejeter +l'hypothèse de normalité de la distribution des rendements. Comme +présenté au premier chapitre, c'est cette observation qui est à la +base de la recherche de modèles alternatifs, étant donné que cette +hypothèse est rejetée pour la plupart des séries financières. Il +existe plusieurs tests disponibles pour ce faire, dont ceux de +Shapiro-Wilk et d’Epps-Pulley. + +\subsection{Test de Shapiro-Wilk} +\label{sec:test-de-shapiro} + +Le test de Shapiro-Wilk \citep{shapiro1965analysis} est basé sur les +statistiques d'ordre de la distribution normale. Ce test est +particulièrement efficace même pour de petits échantillons $(T<20)$, +selon les auteurs. + +On considère l'échantillon ordonné +$y_{(1)},y_{(2)},\ldots,y_{(T)},$. On évalue la variance $S^2$ de +l'échantillon puis la statistique $b$, où les valeurs $a_{T-i+1}$ +proviennent de la \emph{Table 5} de \cite{shapiro1965analysis}: +\begin{enumerate} +\item Si la taille de l'échantillon $T$ est paire, alors + $k=\frac{T}{2}$ et + \begin{align} + \label{eq:shapirowilkb1} + b &= \sum_{i=1}^{k}a_{T-i+1}(y_{(T-i+1)}-y_{(i)}). + \end{align} +\item Si la taille de l'échantillon $T$ est impaire, alors + $k=\frac{T-1}{2}$ et + \begin{align} + \label{eq:shapirowilkb2} + b &= a_{T}(y_{(T)}-y_{(1)}) + \ldots + a_{k+2}(y_{(k+2)}-y_{(k)}). + \end{align} +\end{enumerate} + +La statistique de Shapiro-Wilk $W = \frac{b}{S^2}$ suit une +distribution particulière dont on retrouve différents quantiles à la +\emph{Table 6} de \cite{shapiro1965analysis}. On rejette l'hypothèse +de la normalité lorsque la statistique $W$ est inférieure au seuil +critique déterminé à partir de cette dernière table. + +\subsection{Test d’Epps-Pulley} +\label{sec:test-de-epps} + +Le test d’Epps-Pulley \citep{epps1983test} est basé sur le carré de la +différence entre les fonctions caractéristiques empirique et théorique +de la distribution normale. Ce test est considéré comme un des plus +puissants pour un seuil de tolérance donné, pour de larges +échantillons. La statistique de test prend la forme suivante, où +$\overline{X}$ et $S$ sont respectivement la moyenne et l'écart-type +échantillonnal: +\begin{align} + \label{eq:statistiqueEppsPulley} + EP_T &= T\int_{-\infty}^{\infty} \left\vert\frac{1}{T}\sum_{j=1}^T + \exp\left[it\frac{\left(X_j-\bar{X} \right)}{S} \right] - \exp + \left[-\frac{t^2}{2} \right]\right\vert^2w(t)dt. +\end{align} + +\cite{henze1990approximation} propose une procédure simple à +implémenter pour effectuer un test basé sur la statistique +d’Epps-Pulley $EP_T$, où la pondération de la différence quadratique +$w(\cdot)$ est remplacée par la densité de la loi normale centrée +réduite, ce qui permet d'accorder davantage d'importance aux +observations près de l'origine et aussi d'obtenir une forme +intégrable: +\begin{align} + \label{eq:EppsPulley} + EP_T &= \frac{2}{T} \sum_{1\leq\,j10$, on calcule une version modifiée de la statistique: +\begin{equation} + \label{eq:EppsPulleyMod} + EP_T^{*} = \left(EP_T-\frac{0.365}{T}+\frac{1.34}{T^2}\right)\left(1+\frac{1.3}{T}\right). +\end{equation} + +On obtient ensuite une approximation d'un quantile de la loi normale +centrée réduite: +\begin{align} + \label{eq:ZEppsPulley} + Z_T &= \gamma + \delta\ln\left(\left(EP_T^{*}-\xi + \right)\left(\xi+\lambda-EP_T^{*} \right) \right). +\end{align} + +Les constantes sont dérivées à l'équation \emph{4.1} de +\cite{henze1990approximation}: +\begin{align*} + \gamma &\approx 3.55295 \\ + \delta &\approx 1.23062 \\ + \lambda &\approx 2.26664 \\ + \xi &\approx -0.020682. +\end{align*} + +Puis, on compare cette statistique au quantile $Z_{1-\alpha}$ correspondant +au seuil de tolérance $\alpha$. Si $Z_n>Z_{1-\alpha}$ +alors on rejette la normalité des données. + +\section{Tests d'adéquation} +\label{sec:testnonparam} + +Les tests d'adéquation vérifient l'ajustement à l'échantillon de la +fonction de répartition estimée soit: +\begin{itemize} +\item à partir de l'échantillon ayant servi à l'estimation du vecteur + de paramètres +\item à partir d'un échantillon séparé, afin de vérifier, par exemple, + si un modèle estimé à partir d'anciennes données s'applique toujours + à de nouvelles informations. +\end{itemize} + +On cherche à rejeter ou non une hypothèse concernant l'échantillon $Y$ +et le vecteur de paramètres estimés: +\begin{itemize} +\item Si on considère le vecteur de paramètres comme étant estimé à + partir du même échantillon sur lequel on effectue le test, alors on + pose l'hypothèse composée suivante: + \begin{equation} + \label{eq:hypotheseadeqcomp} + H_0: Y \sim F_Y(\hat\theta). + \end{equation} +\item Si on considère le vecteur de paramètres comme étant connu ou + estimé à partir d'un autre échantillon, alors on pose l'hypothèse + simple suivante: + \begin{equation} + \label{eq:hypotheseadeq} + H_0: Y \sim F_Y(\theta_0). + \end{equation} +\end{itemize} + +La différence entre ces deux hypothèses est le nombre de degrés de +liberté de la statistique de test. Les deux premiers tests présentés +sont des classiques, largement utilisés, mais dont l'application est +basée sur la fonction de répartition. Étant donné qu'on ne dispose que +d'une approximation de celle-ci, on utilisera alors un test basé sur +la fonction génératrice des moments, qui a une forme analytique. + +\subsection{Test $\chi^2$ de Pearson} +\label{sec:testchi2} + +Le test $\chi^2$ de Pearson \citep[ch. 8]{hogg1978introduction} est +basé sur une approximation multinomiale de la fonction de répartition +$F_X(x;\theta)$ pour laquelle on veut vérifier l'ajustement des +données échantillonnales $X_1,\ldots,X_n$. Pour ce faire, on divise le +domaine de la variable aléatoire $X$ en $k$ intervalles, appelés +classes. On associe la probabilité $p_i(\theta)$ que $X$ prend une +valeur dans l'intervalle $\left[c_{i-1},c_{i}\right]$. Cette +probabilité est évaluée à l'aide de la fonction de répartition +$F_X(x;\theta)$ de la variable aléatoire $X$: +\begin{equation} + \label{eq:pmultinomiale} + p_i(\theta) = F_X(c_{i};\theta) - F_X(c_{i-1};\theta),\qquad i = 1,\ldots,k. +\end{equation} + +Soit $N_i$, le nombre de données de l'échantillon observées dans +l'intervalle défini précédemment. On définit alors la statistique $Q$ +comme étant la somme, pour chaque intervalle, du carré de la +différence entre le nombre d'observations obtenues et espérées, +pondérée par cette dernière valeur. Cette quantité a approximativement +une distribution asymptotique $\chi^2(k-1)$ avec $k-1$ degrés de +liberté: +\begin{equation} + \label{eq:statchi2} + Q_{k-1} = \sum_{i=1}^{k} \frac{\left(N_i-np_i(\theta_0)\right)^2}{np_i(\theta_0)} \sim \chi^2(k-1). +\end{equation} + +La statistique $Q_{k-1}$ peut être utilisée pour vérifier une +hypothèse simple, c'est-à-dire lorsque les paramètres sont connus: +\begin{align} + \label{eq:hyp0-1} + \mathcal{H}_0: \theta = \theta_0. +\end{align} + +Cependant, comme les $q$ paramètres de la distribution $f_X$ doivent +être estimés pour évaluer les probabilités $p_i(\theta)$, on retrouve +alors une hypothèse composée: +\begin{align} + \label{eq:hyp0-2} + \mathcal{H}_0: F \in \lbrace F_{\theta} \rbrace, \theta \in \Omega. +\end{align} + +On doit retrancher le même nombre de degrés de liberté à la +distribution asymptotique approximative de la statistique $Q$ qui +devient: +\begin{equation} + \label{eq:statchi2param} + Q_{k-q-1} = \sum_{i=1}^{k} \frac{\left(N_i-n{p}_i(\hat\theta)\right)^2}{n{p}_i(\hat\theta)} \sim \chi^2(k-q-1). +\end{equation} + +Il est important de noter que ce test est approximatif, car pour +obtenir asymptotiquement la distribution $\chi^2$, les paramètres +doivent être estimés en minimisant la statistique $Q_{k-q-1}$ +\eqref{eq:statchi2param}, qui devient alors une fonction objectif. + +On rejette l'hypothèse composée $\mathcal{H}_0$ lorsque la condition +$Q_{k-q-1} > \chi^2_{1-\alpha}(k-q-1)$ est respectée. On effectue le +même test pour l'hypothèse simple, en posant $q=0$. Le point critique +de la distribution $\chi^2(k-q-1)$ est sélectionné selon le critère +\begin{equation} + \label{eq:quantilechisq} + \operatorname{Pr}\left(Q_{k-q-1}\leq \chi^2_{1-\alpha}(k-q-1)\right)=1-\alpha. +\end{equation} + +\subsection{Test de Kolmogorov-Smirnov} +\label{sec:kolmogorovsmirnov} + +Le test de Kolmogorov-Smirnov vérifie l'hypothèse simple +\eqref{eq:hypotheseadeq}. On définit la statistique $D_n$ comme étant +la valeur maximale de la distance entre les fonctions de répartition +empiriques $F_n(x)$ et celle $F_X(x)$ de la distribution exacte +spécifiée par l'hypothèse nulle $H_0$: +\begin{equation} + \label{eq:statks} + D_n=\sup_x |F_n(x)-F_X(x)| +\end{equation} +où +\begin{equation} + \label{eq:repartemp} + F_n(x)={1 \over n}\sum_{i=1}^n I_{X_i\leq x}. +\end{equation} + +La statistique $\sqrt{n}D_n$ suit asymptotiquement une distribution de +Kolmogorov, selon \cite{wang2003evaluating}, dont la fonction de +répartition est définie comme suit: +\begin{align} + \label{eq:repartkolmogorov} + \operatorname{Pr}(\sqrt{n}D_n\leq x)&=1-2\sum_{k=1}^\infty (-1)^{k-1} e^{-2k^2 x^2}\\ + &=\frac{\sqrt{2\pi}}{x}\sum_{k=1}^\infty e^{-(2k-1)^2\pi^2/(8x^2)}. +\end{align} + +On rejette l'hypothèse nulle $H_0$ lorsque la condition +$\sqrt{n}D_n>K_{1-\alpha}$ est respectée. On sélectionne le quantile +de cette distribution selon le critère suivant: +\begin{equation} + \label{eq:quantilekolmogorov} + \operatorname{Pr}(\sqrt{n}D_n \leq K_{1-\alpha})=1-\alpha. +\end{equation} + +\subsection{Test de distance minimale basé sur la fonction génératrice + des moments} +\label{sec:test-de-distance} + +\cite{luong1987minimum} développent un ensemble de tests d'ajustement +pour l'hypothèse simple \eqref{eq:hypotheseadeq} basée sur une +transformation de la fonction de densité ou de répartition. Parmi +cette classe de tests, on retrouve celui d’Epps-Pulley, développé pour +vérifier l'hypothèse de normalité des données, tel que présenté à la +section \ref{sec:test-de-epps}. On retrouve aussi le test K-L de +\cite{feuerverger1981efficiency} basé sur les parties réelles et +imaginaires de la fonction caractéristique. Cependant, pour la +distribution de Laplace asymétrique généralisée, on ne peut effectuer +cette séparation. On préfèrera donc utiliser la fonction génératrice +des moments $M_Y(\xi)$ afin de construire la statistique de test +$D(F_T;f_{\theta_0})$. Comme on a déjà estimé les paramètres, on veut donc +vérifier une hypothèse simple. On se réfère à +\cite{KOUTROUVELIS01011980}, qui a développé le test K-L pour cette +situation. Cependant, on remplacera la fonction caractéristique par la +fonction génératrice des moments. + +On considère un ensemble de fonctions $h_j(y), j=1,\ldots,K$. La +transformée de la fonction de répartition $F_Y(y)$ est donnée par le +vecteur $\mathbf{z}(F)=[z_1(F),\ldots,z_K(F)]$ : +\begin{align} + \label{eq:4} + z_j(F) &= \int_{-\infty}^{\infty} h_j(y) dF_Y(y)\nonumber\\ + &= E[h_j(y)]\nonumber\\ + &= M_Y(t_j),\quad j=1,\ldots,K. +\end{align} + +En posant $h_j(y)=e^{t_j y}$, on définit $z_j(F)$ comme étant la +fonction génératrice des moments. On définit aussi la transformée de +la fonction de répartition empirique $F_T(y)$ par le vecteur +$\mathbf{z}(F_T)=[z_1(F_T),\ldots,z_K(F_T)]$, où +\begin{align} + \label{eq:5} + z_j(F_T) = \frac{1}{T} \sum_{t=1}^T e^{t_j y_t}. +\end{align} + +La statistique de distance quadratique prend alors la forme suivante: +\begin{align} + \label{eq:9} + Td(F_n,F_\theta) &= T\left\{\mathbf{z}(F_n)-\mathbf{z}(F_\theta) + \right\}^{\prime} \mathbf{Q}(F_\theta) + \left\{\mathbf{z}(F_n)-\mathbf{z}(F_\theta) \right\} \nonumber\\ + &= \mathbf{v}_n^{\prime }\mathbf{Q}(F_\theta)\mathbf{v}_n. +\end{align} + +On doit maintenant sélectionner une matrice $\mathbf{Q}(F_\theta)$ de +sorte que la distribution de la statistique $Td(F_n,F_\theta)$ soit +$\chi^2$. Comme $\mathbf{v}_n = +\sqrt{T}\left\{\mathbf{z}(F_n)-\mathbf{z}(F_\theta) \right\}$ suit +asymptotiquement une distribution normale multivariée de moyenne 0 et +de variance-covariance $\mathbf{\Sigma}$, on a, si la matrice +$\mathbf{\Sigma}$ est inversible et que l'on pose +$\mathbf{Q}=\mathbf{\Sigma}^{-1}$, que $\mathbf{v}_n^{\prime +}\mathbf{Q}(F_\theta)\mathbf{v}_n$ suit une distribution $\chi^2$ avec +$K$ degrés de liberté, dans le cadre d'une hypothèse simple. Afin +d'évaluer la variance-covariance $\mathbf{\Sigma}$, on définit +l'espérance du produit des deux fonctions $h_j(y)$ et $h_k(y)$ +:$z_{j,k}(F) = E[h_j(y)h_k(y)]$. + +On décrit la matrice de variance-covariance $\mathbf{\Sigma}$ comme +suit: +\begin{align} + \label{eq:14} + \mathbf{\Sigma} &= \begin{bmatrix} + \sigma_{1,1} & \cdots & \sigma_{1,k} \\ + \vdots & \ddots & \vdots \\ + \sigma_{j,1} & \cdots & \sigma_{j,k} + \end{bmatrix} +\end{align} +où +\begin{align} + \label{eq:10} + \sigma_{j,k} &= z_{j,k}(F) - z_j(F)z_k(F)\nonumber\\ + &= M_Y(t_j+t_k) - M_Y(t_j)M_Y(t_k),\quad j,k=1,\ldots,K. +\end{align} + +On peut maintenant évaluer la statistique $Td(F_n,F_\theta)$ et +effectuer le test. On rejette l'hypothèse $H_0$ lorsque la valeur de +celle-ci est supérieure au quantile correspondant au seuil $\alpha$ de +la distribution $\chi^2$ avec $K$ degrés de liberté: +$Td(F_n,F_\theta)>\chi^2_{1-\alpha}(K)$. + + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/chapitre8.tex b/depotfinal/908144032/chapitre8.tex new file mode 100644 index 0000000..bf8dd3c --- /dev/null +++ b/depotfinal/908144032/chapitre8.tex @@ -0,0 +1,506 @@ +\chapter{Évaluation d'options} +\label{chap:options} +Un des principaux intérêts de connaître la distribution des rendements +d'un actif est de pouvoir évaluer la valeur de différents produits +dérivés. + +\section{Définitions} +\label{sec:options} +En se référant à \cite{bingham2004risk}, on définit: + +\begin{itemize} +\item Un \textbf{produit dérivé} est un contrat financier dont la + valeur à la date d'échéance $T$ est déterminée par le prix de + l'actif sous-jacent au temps $T$ ou par celles prises au cours de + l'intervalle $\left[0,T \right]$. +\item Une \textbf{option} est un instrument financier qui donne le + droit (et non l'obligation) d'effectuer une transaction avant ou à + une date et pour un prix spécifiés. +\item Une \textbf{option d'achat (de vente) européenne} donne le + \textbf{droit} d'acheter (de vendre) un actif au \textbf{prix + d'exercice} $K$ au temps $T$. +\item Lorsque sa valeur actuelle est, par rapport au prix + d'exercice: + \begin{itemize} + \item supérieure $(S(t)>K)$, l'option d'achat est dite \textbf{dans le + cours}; + \item égale $(S(t)=K)$, l'option d'achat est dite \textbf{au cours}; + \item inférieure $(S(t) K \\ + 0 & S(T) \leq K. + \end{array} \right. +\end{align} + +La valeur du prix à l'échéance est appelée une \textbf{créance + éventuelle}. L'évaluation d'un produit dérivé équivaut à calculer la +valeur espérée actualisée de la réclamation contingente définie par le +contrat. Pour l'option d'achat européenne, on obtient la formule +suivante: +\begin{align} + C(t) &= B(t,T) E \left[C(T) \right] \nonumber\\ + &= B(t,T) \int_{K}^{\infty} (S(T)-K) + d\hat{F}_{S(T)} \label{eq:reclamationcall}. +\end{align} +$B(t,T)$ est la valeur au temps $t$ d'une obligation zéro-coupon au taux sans +risque $r$ d'échéance $T$. $\hat{F}_t(S(T))$ est la fonction de +répartition de la distribution neutre au risque de $S(T)$. + +Les déterminants de la valeur d'une option sont: +\begin{itemize} +\item le prix courant de l'actif $S(t)$ +\item le prix d'exercice $K$ +\item la volatilité de l'actif $\sigma$ +\item le temps d'ici l'échéance $T-t$ +\item le taux d'intérêt sans risque $r$. +\end{itemize} + +On peut aussi récrire \eqref{eq:reclamationcall} avec le logarithme du +prix $k=\ln{K}$. On utilise donc $s(t)=\ln{S(T)}$, dont la fonction de +répartition est définie par ${F}_{s(T)}(y) = Pr\left[\ln{S(T)} < y +\right]$: +\begin{align} + C(t) = B(t,T) \int_{k}^{\infty} (e^{s(T)}-e^{k}) d{F}_{s(T)}. \label{eq:reclamationcallintlog} +\end{align} + +Enfin, une relation fondamentale, appelée parité vente-achat, permet, +dans un scénario sans arbitrage, de calculer le prix d'une option de +vente $P(t)$ (d'achat $C(t)$) en connaissant la valeur: +\begin{itemize} +\item de l'autre lorsqu'elles sont de mêmes échéance et prix + d'exercice +\item de l'obligation $B(t,T)$ +\item initiale du titre sous-jacent $S(t)$ commun aux deux options. +\end{itemize} + +\begin{align} + \label{eq:pariteputcall} + P(t)+S(t) = C(t)+ B(t,T) K. +\end{align} + +Cette relation permettra d'évaluer les deux types d'option à l'aide +d'un seul calcul. + +\subsection{Équation martingale} +\label{sec:equationmartingale} +L'évaluation d'options se fait traditionnellement dans un univers ou +un espace de probabilités neutre au risque, noté $\mathbb{Q}$, +c'est-à-dire un point de vue selon lequel les investisseurs n'exigent +pas une prime de risque. Cette approche a été introduite par +\cite{black1973pricing}. Dans cette perspective, les rendements +espérés futurs sont escomptés au taux sans risque. La distribution des +rendements cumulés \eqref{eq:rendementcumL} répond à la propriété +martingale, selon laquelle la valeur actuelle d'un titre financier +reflète l'ensemble de l'information connue sur ce dernier. Cette +propriété permet de fixer un seul prix pour les +options. \textbf{L'équation martingale} établit l'égalité entre la +valeur espérée du titre au temps $t$ et celle d'un investissement de +même valeur dans un compte en banque créditant le taux sans risque: +\begin{align} + \label{eq:equationmartingale} + E\left[\exp(L_t) \right] = e^{rt}. +\end{align} + +\subsection{Paramètres neutres au risque} +\label{sec:GALrn} + +Afin de pouvoir utiliser les résultats de l'estimation paramétrique +des chapitres précédents pour évaluer le prix de produits dérivés, on +doit tout d'abord identifier les paramètres neutres au risque de la +distribution de Laplace asymétrique généralisée associant le +rendement cumulé $L_t$ au taux d'intérêt sans risque $r$. Pour ce +faire, on utilise l'équation martingale \eqref{eq:equationmartingale} +ainsi que la fonction génératrice des moments \eqref{eq:fgmGAL}. On +obtient alors une expression pour le paramètre de dérive neutre au +risque $\theta_{RN}$: +\begin{align} + e^{rt} &= E\left[\exp(L_t) \right] + = E\left[\exp(R_1+\ldots+R_t) \right] \nonumber\\ + &= M_{R_1+\ldots+R_t}(1) + = (M_{R}(1))^t \nonumber\\ + &= \left(\frac{e^{\theta_{RN}}}{(1-\mu-\sigma^2/2)^{\tau}} \right)^t \nonumber\\ + r &= \ln \left(\frac{e^{\theta_{RN}}}{(1-\mu-\sigma^2/2)^{\tau}} \right) \nonumber\\ + &= \theta_{RN} - \tau\ln(1-\mu-\frac{\sigma^2}{2}) \nonumber\\ + \theta_{RN} &= r + + \tau\ln(1-\mu-\frac{\sigma^2}{2}). \label{eq:martingaleGAL} +\end{align} + +On obtient ainsi le paramètre de correction de la dérive $\omega$ à +partir de l'équation martingale \eqref{eq:martingaleGAL}: +\begin{align} + \label{eq:omegaGAL} + \omega = \tau\ln(1-\mu-\frac{\sigma^2}{2}). +\end{align} + +Cette égalité impose une contrainte aux paramètres $\mu$ et $\sigma$ +qui doit respecter la condition +\begin{align} + \label{eq:inegaliteparamrn} + \mu+\frac{\sigma^2}{2} < 1. +\end{align} + +On obtient ensuite la fonction caractéristique de la mesure neutre au +risque $\mathbb{Q}$ pour $L_T$ en remplaçant $\theta$ par +$\theta_{RN}$: +\begin{align} + \label{eq:fncaractGALrn} + \phi(\xi,L_T) = \frac{\exp\left(i\xi*\left(\ln\left(S_{t}\right)+\left(T-t\right)\left(r+\omega\right)\right)\right)}{\left(1-i\mu\,\xi+\left(\sigma^2\xi^2\right)/2\right)^{\tau\left(T-t\right)}}. +\end{align} + +La distribution neutre au risque des rendements est donc $R_t \sim +GAL(r+\tau\ln(1-\mu-\sigma^2/2),\sigma,\mu,\tau)$. + +\section{Aperçu du modèle de Black-Scholes} +\label{sec:blackscholes} + +Étant donné l'importance du modèle de Black-Scholes en finance, on se +doit de le présenter comme outil de référence lorsque l'on veut le +remplacer. Ce modèle est basé sur les quatre hypothèses suivantes: +\begin{enumerate} +\item Les rendements ont une distribution normale de moyenne $\mu$ et + variance $\sigma^2$, ou, de manière équivalente, le processus de + prix $\lbrace S(t) \rbrace$ est un mouvement brownien géométrique de + dérive $\mu$ et de volatilité $\sigma$. +\item Une obligation zéro-coupon au taux d'intérêt sans risque $r$ + existe sur le marché. +\item Le marché est complet et sans friction. Une position prise sur + ce marché peut être couverte de manière continue. +\item La vente à découvert est autorisée, de plus tous les actifs sont + infiniment divisibles. +\end{enumerate} + +En utilisant l'équation martingale \eqref{eq:equationmartingale}, on +obtient la distribution neutre au risque des rendements, qui est +normale avec une moyenne $r-\frac{\sigma^2}{2}$ et une variance +$\sigma^2$. + +À partir de cette information, on peut dériver la formule de +Black-Scholes pour le prix d'une option d'achat ou de vente +\eqref{eq:pariteputcall} européenne: +\begin{align} + \label{eq:blackscholes} + C(S(t),K,T) &= \Phi(\Delta_1) ~ S(t) - \Phi(\Delta_2) ~ K B(t,T) \\ + \Delta_1 &= \frac{\ln\left(\frac{S(t)}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)}{\sigma\sqrt{T - t}} \nonumber\\ + \Delta_2 &= \Delta_1 - \sigma\sqrt{T - t} \nonumber\\ + P(S(t),K,T) &= \Phi(-\delta_2)K B(t,T) - + \Phi(-\delta_1)S(t). \label{eq:putblackscholes} +\end{align} + +L'avantage de ce modèle est que toutes les données requises pour +évaluer le prix d'options sur des actifs cotés sur les marchés +boursiers, à l'exception de la volatilité, sont accessibles auprès de +fournisseurs d'informations financières. Comme les options sont +négociées sur les marchés financiers, on a aussi accès à leur prix, ce +qui permet d'extrapoler la valeur de la volatilité implicite. C'est +pour cette raison que le modèle, bien qu'il soit basé sur des +hypothèses très restrictives, est toujours utilisé comme point de +référence. + +\section{Méthodes d'évaluation pour options européennes} +\label{sec:methodesoptions} + +Le modèle de Black-Scholes est un des seuls qui présentent une forme +analytique simple pour le prix des options. Bien qu'on puisse en +dériver une pour plusieurs modèles en utilisant les équations +différentielles stochastiques et le calcul d'ÃŽto, les résultats sont +souvent difficiles à utiliser. On préfèrera alors utiliser des +méthodes numériques pour calculer le prix des options. + +\subsection{Méthode de Heston} +\label{sec:heston1993} + +L'approche utilisée par \cite{heston1993closed} est de calculer +directement l'espérance de la créance éventuelle associée à l'option +de vente européenne $P(S(t),K,T-t)$ sous la mesure neutre au risque +$\mathbb{Q}$: +\begin{align} + P(S(t),K,T) &= B(t,T) \int_{0}^{\infty} \max\left(K-S(T),0\right)\cdot d{F}_{S(T)} \label{eq:PutHeston93-1}\\ + &= B(t,T)K \int_{0}^{K} dF_{S(T)} - B(t,T)\int_{0}^{K} S(T)\cdot d{F}_{S(T)} \nonumber\\ + &= B(t,T)K F_{S(t)}(K) - S(t) G_{S(t)}(K). \label{eq:PutHeston93} +\end{align} + +On définit la fonction de répartition $F_{S(t)}(K)$ de la variable +aléatoire $S(t) \leq K$ sous la mesure $\mathbb{Q}$. On définit aussi +la fonction de répartition $G_{S(t)}(K)$ sous une autre mesure +équivalant à $\mathbb{Q}$. Soit $\phi_F(\xi)$ la fonction +caractéristique de $S(T)$ sous $\mathbb{Q}$, celle de cette mesure est +alors définie comme suit: +\begin{align} + \label{eq:mesureGHeston} + \phi_G(\xi) = \frac{\phi_F(\xi-i)}{\phi_F(-i)}. +\end{align} +C'est une transformée d'Esscher de paramètre $h=1$ de la mesure +$\mathbb{Q}$. On remarquera la forme de \eqref{eq:PutHeston93} qui est +très similaire à la formule du prix de l'option de vente du modèle de +Black-Scholes \eqref{eq:putblackscholes}. + +Les fonctions de répartition $F_{S(t)}(K)$ et $G_{S(t)}(K)$ +peuvent être évaluées à partir de \eqref{eq:approxinvfncaract} ou de +la méthode du point de selle. + +\cite{carr1999option} notent qu'on ne peut pas inverser +l'approximation \eqref{eq:approxinvfncaract} en utilisant la méthode +de la transformée de Fourier rapide. Cependant, l'approche de la +méthode du point de selle n'a pas été considérée, bien qu'elle puisse +fournir une solution analytique dans plusieurs situations, en +particulier pour la distribution de Laplace asymétrique généralisée +\eqref{eq:saddlepointGAL}. + +\subsection{Méthode de Carr et Madan} +\label{sec:carrmadanfftoptions} + +La méthode de Heston nécessite l'évaluation de deux probabilités, donc +deux inversions de fonctions +caractéristiques. \cite{madan1998variance} ont développé une méthode +qui ne nécessite qu'une seule inversion, décrite par +\cite{epps2007pricing}. Cette méthode nécessite cependant la sélection +d'un paramètre d'amortissement $\theta_{D}$ par tâtonnement afin +d'obtenir de bons résultats. Elle utilise l'expression de la créance +éventuelle \emph{amortie} \footnote{traduction de l'anglais damped} de +l'option d'achat, construite à partir de l'expression +\eqref{eq:reclamationcallintlog}: +\begin{align} + \label{eq:callamortiCarr} + C_{\theta_D}(S(t),k,T) &= e^{\theta_{D}k} C(S(t),k,T) \\ + &= e^{\theta_{D}k} B(t,T) \int_{K}^{\infty} (e^{s(T)}-e^{k}) + d{F}_{s(T)}. +\end{align} + +On applique la transformée de Fourier: +\begin{align} + \label{eq:fouriercallamortiCarr} + \mathcal{F}C_{\theta_D}(\nu) = B(t,T) \frac{\phi\left( \nu - i(1+\theta_D) \ \right)}{(\theta_D+i\nu)(1+\theta_D+i\nu)}. +\end{align} + +Ce résultat est inversé et \emph{amplifié} \footnote{traduction de + l'anglais amplified} pour retrouver le prix de l'option: +\begin{align} + C(S(t),k,T) &= e^{-\theta_{D}k} \mathcal{F}^{-1}\left(\mathcal{F}C_{\theta_D}(\nu) \right) \nonumber\\ + &= \frac{e^{-\theta_{D}k}}{2\pi} \int_{-\infty}^{\infty} e^{-i\nu k} + (\mathcal{F}C_{\theta_D})(\nu)\cdot + d\nu \nonumber\\ + &= \frac{e^{-\theta_{D}k}}{\pi} \int_{0}^{\infty} e^{-i\nu k} + (\mathcal{F}C_{\theta_D})(\nu)\cdot d\nu \qquad \mbox{(par + symétrie)}. \label{eq:prixoptionCarr} +\end{align} + +On peut utiliser la méthode de la transformée de Fourier rapide +(section \ref{sec:methodeFFT}) pour retrouver les prix d'options en se +référant à \cite{carr1999option}. + +On doit donc discrétiser \eqref{eq:prixoptionCarr} et effectuer des +changements de variables pour obtenir la forme \eqref{eq:sommefft}. On +fixe le nombre de points $N$ (idéalement une puissance de deux) et +$\eta$ le pas de discrétisation. On définit l'incrément $\lambda$ pour le +vecteur $k_u$, $u=1,\ldots,N$ des logarithmes des prix d'exercice: +\begin{align} + \lambda = \frac{2\pi}{N\eta}. +\end{align} + +La valeur maximale $b$ de ce prix d'exercice sera: +\begin{align} + b=\frac{N\lambda}{2}. +\end{align} + +Enfin, on obtient la suite des points d'évaluation de la fonction +caractéristique $\nu_j = \eta (j-1)$. L'ensemble de ces substitutions +permet d'obtenir une expression qui est de la forme requise pour +appliquer la méthode de la transformée de Fourier rapide. +\begin{align} + C(S(t),k_u,T) &\approx \frac{e^{-\theta_D\,k}}{\pi} \sum_{j=1}^N e^{-i\nu_j\,k} (\mathcal{F}C_{\theta_D})(\nu_j) \nonumber\\ + &\approx \frac{e^{-\theta_D\,k}}{\pi} \sum_{j=1}^N + e^{-i\lambda\eta(j-1)(u-1)} e^{ib\nu_j} + (\mathcal{F}C_{\theta_D})(\nu_j) \eta. \label{eq:prixoptionFFT} +\end{align} + +C'est essentiellement une intégration par la méthode du trapèze. On +suggère d'implémenter la correction de Simpson, qui permet une +intégration plus précise tout en conservant le même nombre de points +de discrétisation. On obtiendra alors la formule suivante: +\begin{align} + C(S(t),k_u,T) &\approx \frac{e^{-\theta_D\,k}}{\pi} \sum_{j=1}^N + e^{-i\lambda\eta(j-1)(u-1)} e^{ib\nu_j} + (\mathcal{F}C_{\theta_D})(\nu_j) \frac{\eta}{3} \left(3 + (-1)^j + + \delta_{j-1}\right). \label{eq:prixoptionFFTsimpson} +\end{align} + +On peut par la suite calculer un prix d'option pour chaque prix +d'exercice en utilisant une méthode d'interpolation (splines cubiques +par exemple). + +\subsection{Prix d'exercice hors du cours} +\label{sec:outofmoneyfft} + +L'intégration numérique de \eqref{eq:prixoptionCarr} pose problème +quand l'échéance $T-t$ est petite, ou encore le prix d'exercice $K$ +est hors du cours ($k > \ln(S(t))$). Dans ce contexte particulier, +\cite{carr1999option} développent une formule alternative à l'équation +\eqref{eq:prixoptionCarr} pour évaluer le prix de l'option d'achat : +\begin{align} + C(S(t),k,T) &= \frac{1}{\pi\sinh(\theta^{*}_D k)} \int_{0}^{\infty} + e^{-i\nu k} (\mathcal{F}C^{*}_{\theta^{*}_D})(\nu)\cdot + d\nu. \label{eq:prixoptionCarrOOM} +\end{align} +La transformée de Fourier de l'expression du prix de l'option d'achat +européenne est exprimée sous la forme +\begin{align} + \mathcal{F}C^{*}_{\theta^{*}_D} &= \frac{\zeta_T(\nu-i\theta^{*}_D) - \zeta_T(\nu+i\theta^{*}_D)}{2}\\ + \zeta_T(\nu) &= e^{-r(T-t)}\bigg[\frac{1}{1+i\nu} - + \frac{e^{r(T-t)}}{iv} - \frac{\phi(\nu-i)}{\nu(\nu-i)} \bigg]. + \nonumber +\end{align} + +On note que ce paramètre d'amortissement $\theta^{*}_D$ peut être +différent du paramètre $\theta_D$ qui a été utilisé précédemment. + +L'expression à utiliser pour la méthode de la transformée de Fourier +rapide devient +\begin{align} + C^{*}(S(t),k_u,T) &\approx \frac{1}{\pi\sinh(\theta^{*}_D k)} + \sum_{j=1}^N e^{-i\lambda\eta(j-1)(u-1)} e^{ib\nu_j} + (\mathcal{F}C^{*}_{\theta^{*}_D})(\nu_j) \frac{\eta}{3} \left(3 + + (-1)^j + \delta_{j-1}\right). \label{eq:prixoptionFFTsimpsonOOM} +\end{align} +Les paramètres $N ,\eta, \lambda$ et $\nu_j$ prennent les valeurs +utilisées pour évaluer le prix dans la situation où le titre est dans +le cours \eqref{eq:prixoptionFFTsimpson}. + +\subsection{Critique de la méthode de Carr-Madan} +\label{sec:critiquecarrmadanfft} + +Un paramètre d'amortissement $\theta_{D}$ inférieur à la valeur +optimale (lorsque comparé avec la méthode de Heston ou d’Epps) aura +tendance à surestimer le prix des options d'achat +européennes. \cite{itkin2005pricing} démontre que pour certaines +régions de l'espace des paramètres, l'intégrale +\eqref{eq:prixoptionCarr} a un comportement très irrégulier. De plus, +il décrit plusieurs restrictions pour $\theta_{D}$ qui, dans plusieurs +cas, ne permettent pas d'estimation convergente. + +\subsection{Méthode d’Epps} +\label{sec:epps2007} + +\cite{epps2007pricing} propose une méthode qui, contrairement à celle +de Carr et Madan, ne nécessite pas de paramètre d'amortissement et qui +ne requiert aussi qu'une seule inversion de la fonction +caractéristique. Pour ce faire, on se base sur l'expression de +l'option de vente développée précédemment \eqref{eq:PutHeston93-1}, qui +est ensuite exprimée sous la forme du logarithme, à la manière de +\eqref{eq:reclamationcallintlog}. Après l'utilisation de la formule +d'inversion \eqref{eq:gilpelaez2}, on obtient le résultat suivant: +\begin{align} + \label{eq:putepps-1} + E\left[max(K-S_T;0) \right] &= \int_{-\infty}^{k} {F}_{S(T)}(s)e^s\cdot\,ds \nonumber\\ + &= \frac{K}{2}-\frac{1}{2}\int_{-\infty}^{k} \lim_{c\to\infty} + \underbrace{\int_{-c}^{c} \frac{e^{-i\nu s}}{\pi + i\nu}\phi(\nu)d\nu}_{a(c,s)}e^sds. +\end{align} + +La formule d'inversion implique que lorsque $|\lim_{c\to\infty} +a(c,s)| = |1-2{F}_{S(t)}(s)|\leq 1$, pour toute valeur $\epsilon>0$, +$c_{\epsilon}$ existe telle que l'on a le résultat suivant: +\begin{align} + \label{eq:putepps-resultat1} + |sup_s\left[1-2{F}_{S(t)}(s)-a(c_{\epsilon},s) \right]|<\epsilon. +\end{align} + +Dans cette situation, $|a(c,s)e^s|\leq e^s(1+\epsilon)$ lorsque la +constante $c$ est suffisamment grande. Comme la fonction $e^s$ est +intégrable sur le support $(-\infty,k)$, le théorème de convergence +dominée de Lebesgue (section \ref{sec:theor-de-conv}) implique que +l'on peut poser l'égalité suivante: +\begin{align} + \int_{-\infty}^{k} \lim_{c\to\infty} a(c,s)e^sds &= \lim_{c\to\infty}\int_{-\infty}^{k} a(c,s)e^sds \nonumber\\ + &= \lim_{c\to\infty} \int_{-\infty}^{k}\int_{-c}^{c} + \frac{e^{(1-i\nu) s}}{\pi i\nu}\phi(\nu)d\nu ds. \nonumber +\end{align} + +De plus, comme la limite de la double intégrale précédente est égale à +$K-2E\left[max(K-S_T;0) \right]$ qui appartient à l'intervalle +$\left[-K,K\right]$, le théorème de Fubini (section +\ref{sec:theoreme-de-fubini}) permet d'inverser l'ordre d'intégration: +\begin{align} + \lim_{c\to\infty} \int_{-\infty}^{k}\int_{-c}^{c} \frac{e^{-i\nu + s}}{\pi i\nu}\phi(\nu)d\nu ds &= \lim_{c\to\infty} \int_{-c}^{c} \int_{-\infty}^{k}e^{(1-i\nu)s} ds \frac{\phi(\nu)}{\pi i \nu} d\nu \nonumber\\ + &= \frac{K}{\pi} \lim_{c\to\infty} \int_{-c}^{c} + \frac{K^{-i\nu}}{i\nu+\nu^2}\phi(\nu)d\nu. \label{eq:fubini-integrale-a-EPPS} +\end{align} + +En remplaçant le résultat \eqref{eq:fubini-integrale-a-EPPS} dans +l'équation de départ \eqref{eq:putepps-1}, on obtient ainsi une +expression particulièrement simple pour le prix de l'option de vente: +\begin{align} + P(S(t),K,T) &= B(t,T)K\left[\frac{1}{2}-\frac{1}{2\pi} \int_{-c}^c + K^{-i\nu} \frac{\phi(\nu)}{\nu(i+\nu)} + d\nu\right]. \label{eq:PutEpps8.37} +\end{align} + +Cependant, puisque la forme ne se prête pas à l'utilisation de +l'algorithme de la transformée de Fourier rapide, on a recours à une +procédure d'intégration numérique. + +\section{Particularités} +\label{sec:monnaiescontrats} + +\subsection{Option sur actions avec dividendes} +\label{sec:dividentoptions} + +Lorsqu'une option a, pour titre sous-jacent, une action qui verse des +dividendes, on doit en tenir compte dans l'évaluation de son prix. Si +l'on considère que la valeur au marché de l'action $S^{*}(t)$ a été +évaluée avec la méthode de l'actualisation des flux financiers futurs, +on doit tenir compte de la valeur actualisée des dividendes qui seront +versés avant l'échéance de l'option et la soustraire de ce prix. + +Parfois, les dividendes ne sont pas fixes, mais proportionnels à la +valeur de l'action à la date ex-dividende avec un taux $\delta$. Soit +$n(T)$ le nombre de dates ex-dividende dans l'intervalle $\left] t,T +\right]$, le prix initial $S(t)$ considéré pour le calcul de la valeur +de l'option est alors défini comme suit: +\begin{align} + \label{eq:prixinitialdividende} + S(t) = S^{*}(t) (1-\delta)^{n(T)}. +\end{align} + +Lorsque l'on considère un indice composé de plusieurs titres, on peut +prendre un dividende versé de manière continue à un taux $q$. Dans ce +cas, le prix initial $S(t)$ sera +\begin{align} + \label{eq:pricinitialdivcontinu} + S(t) = S^{*}(t) e^{-q(T-t)}. +\end{align} + +\subsection{Options sur contrats à terme et taux de change} +\label{sec:futureoptions} + +\cite{black1976pricing} (p.177) démontre que le prix d'une option sur +un contrat à terme a le même prix que sur une action dont le taux de +dividende équivaut au taux sans risque. On peut donc utiliser les +résultats \eqref{eq:prixinitialdividende} et +\eqref{eq:pricinitialdivcontinu} en posant $q=r$. + +De même, on peut évaluer le prix d'une option sur une monnaie +étrangère en considérant le taux sans risque étranger $r_f$ comme un +taux de dividende $q=r_f$. Selon la Banque des Règlements +Internationaux, c'est le type d'options le plus transigé sur les +marchés non réglementés en date de 2005, même s'il est moins étudié +que les autres. + + + + + + + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/chapitre9.tex b/depotfinal/908144032/chapitre9.tex new file mode 100644 index 0000000..4f30cf2 --- /dev/null +++ b/depotfinal/908144032/chapitre9.tex @@ -0,0 +1,702 @@ +\chapter{Exemple d'application} +\label{chap:application}% + +On considère un échantillon $S_1$ formé de l'ensemble des prix +$S_1(t)$ à la fermeture du titre Abbey National entre le 31 juillet et +le 8 octobre 1991. La table \ref{prixabbeyn} présente l'ensemble des 50 +observations. Cet échantillon a notamment été étudié précédemment par +\cite{buckle1995bayesian}. + +\section{Description des données} +\label{sec:analysepA} + +On évalue tout d'abord les rendements quotidiens $R_1$ à l'aide de +l'équation \eqref{eq:rendementlogprix}. On obtient alors 49 +observations du processus des rendements $R(t)$, que l'on présente à +la figure \ref{fig:seriechronoR1} sous forme de série chronologique. + +\begin{figure}[!ht] + \centering + \includegraphics[height=4in, + width=4in]{./graphiques/ABBEYN-chronologie.pdf} + \caption{Représentation en série chronologique de l'échantillon + $R_1$} + \label{fig:seriechronoR1} +\end{figure} + +On énumère d'abord quelques propriétés de cet échantillon qui pourront +compléter l'analyse. À la table \ref{tab:statordreR1}, on présente +quelques statistiques d'ordre. De plus, à la table +\ref{tab:statmomentsR1}, on retrouve quelques valeurs relatives aux +premiers moments. + +\begin{table}[!ht] + \centering + \begin{tabular}{cccccc} + \hline + \textbf{Statistique d'ordre} & \textbf{Valeur} \\ + \hline + Minimum & -0.027500\\ + 1er quartile & -0.009790\\ + Médiane & -0.003260\\ + 3e quartile & 0.006620\\ + Maximum & 0.043400\\ + \hline + + \end{tabular} + \caption{Statistiques d'ordre de l'échantillon $R_1$} + \label{tab:statordreR1} +\end{table} + +\begin{table}[!ht] + \centering + \begin{tabular}{cccc} + \hline + \textbf{Statistique} & \textbf{Valeur} \\ + \hline + Moyenne & 0.000206\\ + Variance & 0.000169\\ + Coefficient d'asymétrie & 0.977563\\ + Coefficient d'aplatissement & 4.597114 \\ + \hline + \end{tabular} + \caption{Valeurs relatives aux premiers moments de l'échantillon $R_1$} + \label{tab:statmomentsR1} +\end{table} + +On présente maintenant la distribution des rendements sous la forme +d'une courbe de densité à la figure \ref{fig:distributionR1}. + +\begin{figure}[!ht] + \centering + \includegraphics[height=4in,width=4in]{./graphiques/ABBEYN-histogramme.pdf} + \caption{Distribution de la variable aléatoire $R_1$} + \label{fig:distributionR1} +\end{figure} + +À l'aide du test de normalité d'Epps-Pulley (Section +\ref{sec:test-de-epps}), on vérifie si la distribution des rendements +$R_1$ est significativement différente de la normale. On fixe le seuil +de tolérance $\alpha = 5\% $. + +On évalue d'abord la statistique $EP_T$ +\eqref{eq:statistiqueEppsPulley}, puis la statistique modifiée +$EP_T^{*}$ \eqref{eq:EppsPulleyMod} étant donné que $T>10$. La table +\ref{tab:eppspulleyR1} présente les résultats. + +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Statistique} & \textbf{Valeur} \\ + \hline + $EP_T$ & 0.626033 \\ + $EP_T^{*}$ & 0.635568 \\ + $Z_T$ & 2.44824 \\ + $p$ & 0.007178 \\ + \hline + \end{tabular} + \caption{Test de normalité d'Epps-Pulley pour $R_1$} + \label{tab:eppspulleyR1} +\end{table} + +Étant donné que la valeur $p$ est inférieure au seuil $\alpha$, on +rejette l'hypothèse de normalité de l'échantillon $R_1$. On peut +vérifier cette affirmation à l'aide du graphique de comparaison des +quantiles empiriques avec ceux de la loi normale présenté à la figure +\ref{fig:qqplotR1}. + +\begin{figure}[!ht] + \centering + \includegraphics[height=4in, + width=4in]{./graphiques/ABBEYN-qq.pdf} + \caption{Graphique Quantile-Quantile} + \label{fig:qqplotR1} +\end{figure} + +\section{Estimation} +\label{sec:estimation} + +Dans cette section, on estime les paramètres de la distribution de +Laplace asymétrique généralisée pour l'échantillon de données centrées +et réduites $R_1^{*}$, à l'aide des méthodes: +\begin{itemize} +\item des moments généralisée itérative (Section \ref{sec:GMMtwostep}) +\item d'estimation gaussienne de Whittle (table + \ref{tab:methodesquad}) +\item de l'équation d'estimation optimale de Crowder (Section + \ref{sec:equationsquadopt}) et +\item de l'équation d'estimation optimale modifiée (Section + \ref{sec:eqoptmodif}). +\end{itemize} + +On évalue d'abord le vecteur de paramètres initiaux $\theta_0$ à +l'aide des équations \eqref{eq:ptdepartGAL}. On effectue ensuite une +première optimisation à l'aide de chacune des trois méthodes afin de +pouvoir évaluer la matrice de pondération qui servira à la seconde. La +table \ref{tab:premiereoptimR1} présente les paramètres obtenus +$\theta_1$. + +\begin{table}[!ht] + \centering + \begin{tabular}{lcccc} + \hline + \textbf{Méthode} & $\theta$ & $\sigma$ & $\mu$ & $\tau$ \\ + \hline + Paramètres initiaux & -0.612081 & 0.515932 & 0.325854 & 1.878388 \\ + \hline + Moments généralisée & -0.641646 & 0.625908 & 0.326366 & 1.965995 \\ + Estimation gaussienne & -0.776204 & 0.581154 & 0.385120 & 2.015437 \\ + Équation d'estimation optimale & -0.660300 & 0.645678 & 0.376654 & 1.753069 \\ + Équation d'estimation optimale modifiée & -0.711439 & 0.606642 & 0.362932 & 1.960299 \\ + \hline + \end{tabular} + \caption{Paramètres $\theta_1$ de la première optimisation} + \label{tab:premiereoptimR1} +\end{table} + +On obtient la matrice de variance-covariance des paramètres pour les +méthodes basées sur une équation d'estimation +\eqref{eq:VarAsymptEstEE} en évaluant l'inverse de la +variance-covariance des éléments composant celle-ci +\eqref{eq:Crowder86-th3.3-def-2} et le gradient +\eqref{eq:Crowder86-th3.3-def-1}: +\begin{itemize} +\item Pour la méthode d'estimation gaussienne: + \begin{align} + \label{eq:vcov1gaussR1} + \mathbf{S}_T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} + 1.13756e-03 &-0.00207546& 0.000917316& 7.46064e-06\\ + -2.07546e-03 & 0.00984423& 0.002340623& 1.24328e-03\\ + 9.17316e-04 & 0.00234062& 0.003399880& 8.38932e-04\\ + 7.46064e-06 & 0.00124328& 0.000838932& 2.60840e-04 + \end{bmatrix} + \end{align} +\item Pour la méthode de l'équation d'estimation optimale: + \begin{align} + \label{eq:vcov1eeR1} + \mathbf{S}_T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} + 1.47029e-03 &-0.00212833& 0.00133597& 2.84688e-05\\ + -2.12833e-03 & 0.01115599& 0.00277669& 1.95192e-03\\ + 1.33597e-03 & 0.00277669& 0.00396182& 1.18855e-03\\ + 2.84688e-05 & 0.00195192& 0.00118855& 4.92502e-04 + \end{bmatrix} + \end{align} +\item Pour la méthode de l'équation d'estimation optimale modifiée: + \begin{align} + \label{eq:vcov1eemodR1} + \mathbf{S}_T^{*}(\hat\theta_1;R_1^{*}) = \begin{bmatrix} + 1.03165e-03& -0.00146603& 0.001145269& 6.63861e-05\\ + -1.46603e-03& 0.00812868& 0.001989247& 1.17588e-03\\ + 1.14527e-03& 0.00198925& 0.003435163& 8.33622e-04\\ + 6.63861e-05& 0.00117588& 0.000833622& 2.71162e-04 + \end{bmatrix} + \end{align} +\end{itemize} + +On peut ainsi construire des intervalles de confiance pour les +paramètres estimés. On utilise un seuil de tolérance de $\alpha=5\%$: +\begin{itemize} +\item Pour la méthode d'estimation gaussienne:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.8328 & -0.7762 & -0.7197 \\ + $\sigma$ & 0.4148 & 0.5812 & 0.7475 \\ + $\mu$ & 0.2874 & 0.3851 & 0.4829 \\ + $\tau$ & 1.9884 & 2.0154 & 2.0425 \\ + \hline + \end{tabular}\\ + +\item Pour la méthode de l'équation d'estimation optimale:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.7246 & -0.6603 & -0.5960 \\ + $\sigma$ & 0.4686 & 0.6457 & 0.8228 \\ + $\mu$ & 0.2711 & 0.3767 & 0.4822 \\ + $\tau$ & 1.7159 & 1.7531 & 1.7903 \\ + \hline + \end{tabular}\\ + +\item Pour la méthode de l'équation d'estimation optimale modifiée:\\ \\ +\begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.7653 & -0.7114 & -0.6576 \\ + $\sigma$ & 0.4555 & 0.6066 & 0.7578 \\ + $\mu$ & 0.2647 & 0.3629 & 0.4612 \\ + $\tau$ & 1.9327 & 1.9603 & 1.9879 \\ + \hline +\end{tabular}\\ + +\end{itemize} + +À l'aide de l'inverse de la variance-covariance des conditions de +moments et des équations d'estimation, on peut effectuer une seconde +optimisation afin d'obtenir des estimateurs convergents. Pour la +méthode des moments généralisée, on utilisera plutôt une procédure +itérative. Les paramètres présentés à la table +\ref{tab:secondeoptimR1} sont donc ceux obtenus avec la convergence de +l'algorithme itératif de la section \ref{sec:GMMtwostep} avec un +critère d'arrêt $\epsilon=10^{-7}$ \eqref{eq:criterearret}. + +\begin{table}[!ht] + \centering + \begin{tabular}{lcccc} + \hline + \textbf{Méthode} & $\theta$ & $\sigma$ & $\mu$ & $\tau$ \\ + \hline + Moments généralisée & -0.640067 & 0.625431 & 0.324311 & 1.973623 \\ + Estimation gaussienne & -0.775071 & 0.581413 & 0.384342 & 2.016619 \\ + Équation d'estimation optimale & -0.658697 & 0.646516 & 0.376251 & 1.750685 \\ + Équation d'estimation optimale modifiée & -0.712450 & 0.606193 & 0.363196 & 1.961614 \\ + \hline + \end{tabular} + \caption{Paramètres $\theta_1$ de la première optimisation} + \label{tab:secondeoptimR1} +\end{table} + +Pour la méthode des moments généralisée, puisque l'on utilise une +procédure itérative, présenter la première matrice de +variance-covariance des paramètres n'est pas pertinent. On obtient +celle à la convergence de l'algorithme \eqref{matricevcovparamGMMnc} +en évaluant l'inverse de la variance-covariance des conditions de +moments \eqref{eq:matponderationproduith} et le gradient +\eqref{eq:gradientGMM}. On obtient donc: +\begin{align} + \label{eq:vcov1gmmR1} + \mathbf{S}_T^{*}(\hat\theta_{OPT};R_1^{*}) = + \begin{bmatrix} + 0.00203708& 0.00438553& 0.00174636& 0.00154237\\ + 0.00438553& 0.01207044& 0.00239640& 0.00384905\\ + 0.00174636& 0.00239640& 0.00220402& 0.00104816\\ + 0.00154237& 0.00384905& 0.00104816& 0.00127406 + \end{bmatrix}. +\end{align} + +Pour les méthodes basées sur une équation d'estimation, on obtient, +pour la seconde optimisation, les variances-covariances suivantes: +\begin{itemize} +\item Pour la méthode d'estimation gaussienne: + \begin{align} + \label{eq:matvcov2R1-gauss} + \mathbf{S}_T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} + 1.13314e-03& -0.00206602& 0.000919367& 7.53773e-06\\ + -2.06602e-03 & 0.00983078& 0.002332240 &1.24238e-03\\ + 9.19367e-04 & 0.00233224& 0.003395736& 8.36472e-04\\ + 7.53773e-06 & 0.00124238& 0.000836472& 2.60254e-04 + \end{bmatrix} + \end{align} +\item Pour la méthode de l'équation d'estimation optimale: + \begin{align} + \label{eq:matvcov2R1-ee} + \mathbf{S}_T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} + 1.47327e-03& -0.00212557& 0.00134221& 2.89119e-05\\ + -2.12557e-03 & 0.01116770 &0.00277803& 1.96072e-03\\ + 1.34221e-03 & 0.00277803& 0.00396652& 1.19169e-03 \\ + 2.89119e-05 & 0.00196072& 0.00119169& 4.95537e-04 + \end{bmatrix} + \end{align} +\item Pour la méthode de l'équation d'estimation optimale modifiée: + \begin{align} + \label{eq:matvcov2R1-eemod} + \mathbf{S}_T^{*}(\hat\theta_2;R_1^{*}) = \begin{bmatrix} + 1.03121e-03& -0.00146899& 0.001142696& 6.60715e-05\\ + -1.46899e-03 & 0.00812703 &0.001987649& 1.17298e-03\\ + 1.14270e-03 & 0.00198765& 0.003432414& 8.32388e-04\\ + 6.60715e-05 & 0.00117298& 0.000832388& 2.70299e-04 + \end{bmatrix} + \end{align} +\end{itemize} + +On peut donc construire des intervalles de confiance: +\begin{itemize} +\item Pour la méthode des moments généralisée:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.715736& -0.640067& -0.564397\\ + $\sigma$ & 0.441236 & 0.625431 & 0.809627\\ + $\mu$ & 0.245602 & 0.324311 & 0.403020\\ + $\tau$ & 1.913780 & 1.973623 & 2.033466\\ + \hline + \end{tabular} \\ +\item Pour la méthode d'estimation gaussienne:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.8315 & -0.7751 & -0.7186 \\ + $\sigma$ & 0.4152 & 0.5814 & 0.7476 \\ + $\mu$ & 0.2866 & 0.3843 & 0.4820 \\ + $\tau$ & 1.9896 & 2.0166 & 2.0437 \\ + \hline + \end{tabular} \\ +\item Pour la méthode de l'équation d'estimation optimale:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.7230 & -0.6587 & -0.5943 \\ + $\sigma$ & 0.4693 & 0.6465 & 0.8237 \\ + $\mu$ & 0.2707 & 0.3763 & 0.4818 \\ + $\tau$ & 1.7134 & 1.7507 & 1.7880 \\ + \hline + \end{tabular} \\ +\item Pour la méthode de l'équation d'estimation optimale modifiée:\\ \\ + \begin{tabular}{rrrr} + \hline + & \textbf{Borne inférieure} & \textbf{Valeur estimée} & \textbf{Borne supérieure} \\ + \hline + $\theta$ & -0.7663 & -0.7124 & -0.6586 \\ + $\sigma$ & 0.4551 & 0.6062 & 0.7573 \\ + $\mu$ & 0.2650 & 0.3632 & 0.4614 \\ + $\tau$ & 1.9341 & 1.9616 & 1.9892 \\ + \hline + \end{tabular} \\ +\end{itemize} + +En utilisant la propriété \eqref{eq:transparamGALNS}, on retrouve les +paramètres, correspondants aux données $R(t)= \sqrt{Var[R_1]} R^{*}(t) ++ E[R_1], t=1,\ldots,50$, présentés à la table +\ref{tab:parametresdonneesorigineR1}. + +\begin{table}[!ht] \centering + \begin{tabular}{lcccc} + \hline + \textbf{Méthode} & $\theta$ & $\sigma$ & $\mu$ & $\tau$ \\ + \hline + Moments généralisée& -0.008119 & 0.008134 & 0.002983 & 1.973623 \\ + Estimation gaussienne& -0.009875 & 0.007562 & 0.003535 & 2.016619 \\ + Équation d'estimation optimale& -0.008361 & 0.008409 & 0.003460 & 1.750685 \\ + Équation d'estimation optimale modifiée& -0.009060 & 0.007884 & 0.003340 & 1.961614 \\ + \hline + \end{tabular} + \caption{Paramètres des données $R_1$} + \label{tab:parametresdonneesorigineR1} +\end{table} + +\section{Approximation} +\label{sec:approximationR1} + +On effectue l'approximation des fonctions de densité et de répartition +pour un ensemble de points $r \in +\left\{-0.03,-0.02,-0.01,0,0.01,0.02,0.03 \right\}$ à l'aide de la +méthode du point de selle. On utilise les paramètres obtenus par la +méthode des moments généralisée. + +On résout d'abord l'équation du point de selle +\eqref{eq:saddlepoint}. On illustre graphiquement cette équation pour +$r=0.01$ à la figure \ref{fig:equationptselle0.01R1} : +\begin{figure}[!ht] + \centering + \includegraphics[scale=0.5]{./graphiques/pointdeselleGMM.png} + \caption{Équation du point de selle pour $r=0.01$} + \label{fig:equationptselle0.01R1} +\end{figure} + +Le point de selle prend alors une valeur de $\hat{s}=56.050951$ dans +cette situation. En utilisant l'approximation de premier ordre +$\hat{f}_{R,1}(r)$ +\eqref{eq:approximationsaddlepointordre1} % et celle +% de second ordre $\hat{f}_{R,2}(r)$ +% \eqref{eq:approximationsaddlepointordre2} +de la fonction de densité, et en comparant le résutat avec la fonction +de densité ${f}_{R}(r)$ \eqref{eq:densitekotz2001}, on obtient les +résultats présentés à la table \ref{tab:approximationdensiteR1}. Dans +les deux cas, la fonction de densité a été normalisée à l'aide de +l'intégrale \eqref{eq:normalisationsaddle1}. + +% latex table generated in R 2.15.2 by xtable 1.7-1 package Mon Aug 19 +% 16:36:59 2013 +\begin{table}[ht] + \centering + \begin{tabular}{cccc} + & \multicolumn{2}{c}{\textbf{Densité}} & \multicolumn{1}{c}{\textbf{Erreur relative}} \\ + \hline + $r$ & $\hat{f}_{R,1}(r)$ & ${f}_{R}(r)$ & $(\hat{f}_{R,1}(r)-{f}_{R}(r))/{{f}_{R}(r)}$ \\ + \hline + -0.03 & 1.561365 & 2.065547 & -0.244091 \\ + -0.02 & 9.038616 & 10.702896 & -0.155498 \\ + -0.01 & 31.216317 & 37.581108 & -0.169361 \\ + 0.00 & 33.015869 & 31.073988 & 0.062492 \\ + 0.01 & 15.941633 & 12.450742 & 0.280376 \\ + 0.02 & 5.988373 & 4.120017 & 0.453483 \\ + 0.03 & 2.026838 & 1.246205 & 0.626409 \\ + \hline + \end{tabular} + \caption{Approximation de la densité de $R_1$} + \label{tab:approximationdensiteR1} +\end{table} + +On évalue ensuite la valeur de la fonction de répartition à l'aide de +l'approximation de premier ordre $\hat{F}_{R,1}(r)$ +\eqref{eq:approximationsaddlepointREPordre1} % et de second ordre +% $\hat{F}_{R,2}(r)$ \eqref{eq:approximationsaddlepointREPordre2} +. On compare celle-ci à la valeur de la fonction de répartition +obtenue en intégrant numériquement la fonction de densité. Les +résultats sont présentés à la table \ref{tab:approximationrepartR1}. + +% latex table generated in R 2.15.2 by xtable 1.7-1 package Mon Aug 19 +% 16:49:26 2013 +\begin{table}[ht] + \centering + \begin{tabular}{cccc} + & \multicolumn{2}{c}{\textbf{Fonction de répartition}} & \multicolumn{1}{c}{\textbf{Erreur relative}} \\ + \hline + $r$ & $\hat{F}_{R,1}(r)$ & ${F}_{R}(r)$ & $(\hat{F}_{R,1}(r)-{F}_{R}(r))/{{F}_{R}(r)}$ \\ + \hline + -0.03 & 0.007402 & 0.011577 & -0.360606 \\ + -0.02 & 0.048942 & 0.064944 & -0.246402 \\ + -0.01 & 0.249911 & 0.292034 & -0.144241 \\ + 0.00 & 0.625225 & 0.681167 & -0.082126 \\ + 0.01 & 0.857830 & 0.889832 & -0.035965 \\ + 0.02 & 0.951508 & 0.965916 & -0.014916 \\ + 0.03 & 0.984393 & 0.990080 & -0.005744 \\ + \hline + \end{tabular} + \caption{Approximation de la fonction de répartition de $R_1$} + \label{tab:approximationrepartR1} +\end{table} + +\section{Graphiques} +\label{sec:graphiques} + +On illustre graphiquement, aux figures \ref{fig:densite1R1} et +\ref{fig:densite3R1}, la fonction de densité de la distribution de +Laplace asymétrique généralisée avec les paramètres estimés par la +méthode des moments généralisée et la méthode de l'équation +d'estimation optimale. La table \ref{tab:courbesdensite} décrit +chacune des courbes. +\begin{table}[!ht] + \centering + \begin{tabular}{lp{12cm}} + \hline + \textbf{Abbréviation} & \textbf{Description} \\ + \hline + Emp. & Données empiriques. \\ + Norm. & Densité de la distribution normale ayant les mêmes moyenne et variance que la variable aléatoire $R$. \\ + Estim. & Densité de la distribution de Laplace asymétrique généralisée avec les paramètres estimés.\\ + Pt selle o.1 & Approximation de premier ordre avec la méthode du point de selle. \\ + FFT & Transformée de Fourier rapide. \\ + \hline + \end{tabular} + \caption{Courbes de densité} + \label{tab:courbesdensite} +\end{table} + +On remarquera au passage que la méthode du point de selle nécessite +une normalisation. La valeur de l'intégrale $c$ +\eqref{eq:normalisationsaddle1} pour les quatre méthodes se trouve à +la table \ref{tab:intapproxpointselleR1}. +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Méthode} & \textbf{Valeur de l'intégrale $c$}\\ + \hline + Moments généralisée & 0.920148 \\ + Estimation gaussienne & 0.932246 \\ + Équation d'estimation optimale & 0.917326 \\ + Équation d'estimation optimale modifiée & 0.925781 \\ + \hline + \end{tabular} + \caption{Valeur de l'intégrale de l'approximation de la densité par la méthode du point de selle } + \label{tab:intapproxpointselleR1} +\end{table} + +\begin{figure}[!ht] + \centering + \includegraphics[height=6in, + width=6in]{./graphiques/ABBEYN-densiteGALmu-7.pdf} + \caption{Densité de $R_1^{*}$ selon la méthode des moments + généralisée} + \label{fig:densite1R1} +\end{figure} + +\begin{figure}[!ht] + \centering + \includegraphics[height=6in, + width=6in]{./graphiques/ABBEYN-densiteGALmu-5.pdf} + \caption{Densité de $R_1^{*}$ selon la méthode de l'équation + d'estimation optimale} + \label{fig:densite3R1} +\end{figure} + +\clearpage +\section{Tests statistiques} +\label{sec:tests-statistiques} + +On effectue le test du $\chi^2$ en utilisant sept classes optimales +déterminées par l'algorithme du logiciel GNU R. On obtient la fonction +de répartition à partir de la fonction caractéristique en utilisant la +formule d'inversion \eqref{eq:approxinvfncaract}. On se rappelle que +cet algorithme produit surtout des erreurs aux extrémités de la +distribution. Par contre, ce test donne davantage d'importance aux +classes ayant un plus grand nombre de données par construction. On +utilisera donc l'inversion plutôt que la méthode du point de selle. On +obtient la statistique $Q_{6}$ présentée à la table +\ref{tab:testchi2R1} à partir de la définition \eqref{eq:statchi2}. On +utilise un seuil de tolérance $\alpha = 5\%$. +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Méthode} & \textbf{Valeur de la statistique $Q_{6}$} \\ + \hline + Moments généralisée & 0.484919 \\ + Estimation gaussienne & 0.473527 \\ + Équation d'estimation optimale & 0.531888 \\ + Équation d'estimation optimale modifiée & 0.494769 \\ + \hline + \end{tabular} + \caption{Test du $\chi^2$} + \label{tab:testchi2R1} +\end{table} + +Ce test ne permet pas de rejeter l'hypothèse de la distribution de +Laplace asymétrique généralisée avec les paramètres obtenus pour +aucune des méthodes d'estimation. + +On effectue maintenant le test de Kolmogorov-Smirnov. Comme celui-ci +est basé sur la fonction de répartition, on utilisera la même méthode +que précédemment, impliquant l'inversion de la fonction +caractéristique. On évalue la fonction de répartition en chaque point +de l'échantillon $R_1^{*}$ et l'on obtient la statistique $D_{49}$ +présentée à la table \ref{tab:testKSR1} à l'aide de la définition +\eqref{eq:statks}. En utilisant un seuil de tolérance de $\alpha=5\%$, +on obtient une valeur critique de $0.194286$. + +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Méthode} & \textbf{Valeur de la statistique $D_{49}$} \\ + \hline + Moments généralisée & 0.0684329 \\ + Estimation gaussienne & 0.0784436 \\ + Équation d'estimation optimale & 0.0758317 \\ + Équation d'estimation optimale modifiée & 0.074668 \\ + \hline + \end{tabular} + \caption{Test de Kolmogorov-Smirnov} + \label{tab:testKSR1} +\end{table} + +Encore une fois, on ne peut pas rejeter l'hypothèse de la distribution +de Laplace asymétrique généralisée avec les paramètres estimés pour +chacune des méthodes. + +Ces deux tests sont approximatifs, puisque les paramètres n'ont pas +été estimés en minimisant la statistique utilisée. Cependant, on +obtient un test asymptotiquement exact en utilisant une statistique de +distance minimale basée sur la fonction génératrice des moments, tel +que développé à la section \ref{sec:test-de-distance}. Avec un seuil +de tolérance de $\alpha=5\%$, on obtient une valeur critique de +67.5048. Les statistiques obtenues sont présentées à la table +\ref{tab:testDMR1}. + +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Méthode} & \textbf{Valeur de la statistique $Td(F_{49},F_{\theta})$} \\ + \hline + Moments généralisée & 0.226891 \\ + Estimation gaussienne & 0.110573 \\ + Équation d'estimation optimale & 0.074020 \\ + Équation d'estimation optimale modifiée & 0.108625 \\ + \hline + \end{tabular} + \caption{Test de distance minimale basé sur la fonction génératrice des moments} + \label{tab:testDMR1} +\end{table} + +On ne peut pas rejeter l'hypothèse de la distribution de Laplace +asymétrique généralisée avec les paramètres estimés pour chacune des +méthodes avec ce test. + +\section{Évaluation d'options} +\label{sec:evaluation-doptions} + +À titre d'exemple, on évaluera une option européenne dont les +différentes caractéristiques figurent à la table +\ref{tab:caracteristiqueoptionR1}. + +\begin{table}[!ht] + \centering + \begin{tabular}{ll} + \hline + \textbf{Caractéristique} & \textbf{Valeur} \\ + \hline + Type & Option de vente \\ + Échéance ($T$) & 30 jours \\ + Valeur actuelle du titre ($S(0)$) & 299 \\ + Prix d'exercice ($K$) & entre 95\% et 105\% de la valeur actuelle \\ + Taux sans risque annuel ($r_f$) & 5\% \\ + \hline + \end{tabular} + \caption{Caractéristiques de l'option} + \label{tab:caracteristiqueoptionR1} +\end{table} + +À partir des paramètres de la table +\ref{tab:parametresdonneesorigineR1}, en utilisant l'équation +martingale appliquée à la distribution de Laplace asymétrique +généralisée \eqref{eq:martingaleGAL}, on obtient l'ensemble +correspondant pour la mesure neutre au risque. On rappelle que +celle-ci n'est pas unique étant donné que le processus de Laplace est +un processus de sauts. La table \ref{tab:paramrisqueneutreR1} présente +ces paramètres. +\begin{table}[!ht] + \centering + \begin{tabular}{lcccc} + \hline + & $\theta$ & $\sigma$ & $\mu$ & $\tau$ \\ + \hline + Moments généralisée & -0.005824 & 0.008134 & 0.002983 & 1.973623 \\ + Estimation gaussienne & -0.007062 & 0.007562 & 0.003535 & 2.016619 \\ + Équation d'estimation optimale & -0.005993 & 0.008409 & 0.003460 & 1.750685 \\ + Équation d'estimation optimale modifiée & -0.006487 & 0.007884 & 0.003340 & 1.961614 \\ + \hline + \end{tabular} + \caption{Paramètres neutres au risque} + \label{tab:paramrisqueneutreR1} +\end{table} + +On présente les graphiques de la valeur du prix de l'option de vente +pour les méthodes des moments généralisée et de l'équation d'estimation +optimale aux figures \ref{fig:prix1R1-1} et \ref{fig:prix1R1-3}. On +peut facilement remarquer le manque de précision de l'approche de +Carr-Madan, qui s'approche de la courbe de Black-Scholes lorsque le +titre est dans le cours et qui se met à osciller dès que le titre +est hors le cours. Les méthodes de Epps et de Heston donnent des +résultats très similaires, et l'approximation du point de selle +d'ordre 1 est très précise dans ce contexte. +\begin{figure}[!ht] + \centering + \includegraphics[height=6in, + width=6in]{./graphiques/ABBEYN-callGAL-7.pdf} + \caption{Prix de l'option selon les paramètres estimés avec la + méthode des moments généralisée} + \label{fig:prix1R1-1} +\end{figure} + +\begin{figure}[!ht] + \centering + \includegraphics[height=6in, + width=6in]{./graphiques/ABBEYN-callGAL-5.pdf} + \caption{Prix de l'option selon les paramètres estimés avec la + méthode de l'équation d'estimation optimale} + \label{fig:prix1R1-3} +\end{figure} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/conclusion.tex b/depotfinal/908144032/conclusion.tex new file mode 100644 index 0000000..942a0ea --- /dev/null +++ b/depotfinal/908144032/conclusion.tex @@ -0,0 +1,47 @@ +\chapter*{Conclusion} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Conclusion} % dans TdM + +%% Revenir sur le papier de Derman, résumé des avantages et +%% inconvénients du modèle, des méthodes d'estimation et d'évaluation +%% d'options + +En guise de conclusion, j'aimerais tout d'abord effectuer un retour +sur les différents éléments introduits au début du premier chapitre, +concernant le risque de modélisation. Le modèle présenté n'en est pas +exempt, bien au contraire. Cependant, il +nécessite moins d'hypothèses restrictives que les autres modèles +présentés pour être valide, bien qu'il exige toujours l'indépendance +des observations. Il tient compte de la possibilité de sauts tout en +conservant une composante de mouvement aléatoire, ce qui décrit +adéquatement les observations empiriques à ce jour. Comme tout modèle +paramétrique, il reste dépendant du nombre et de la qualité des +données disponibles. Étant donné que l'utilisation d'algorithmes +d'optimisation numérique est inévitable, il subsiste un risque +important autour de l'estimation des paramètres et de l'approximation de la distribution. De plus, étant donné qu'il n'existe pas +de mesure neutre au risque unique, l'arbitrage de modèle reste +possible et doit être considéré. L'utilisation d'un échantillon de +données instables à travers le temps peut produire des résultats +inattendus, surtout au niveau de la distribution de la volatilité +historique, un aspect qui pourra être approfondi ultérieurement. +Enfin, subsiste toujours le risque d'erreurs de nature informatique +qui pourraient produire de faux résultats. + +Ce retour permet de constater qu'il y a toujours place à +l'amélioration des outils développés. Entre autres, il pourrait être +pertinent d'étudier les différences entre le comportement à court et à long terme du modèle. En se basant sur la théorie de +l'utilité, on pourrait développer une meilleure approche pour +déterminer les paramètres de la distribution neutre au risque. Il +pourrait aussi être intéressant de développer des mesures de risque +cohérentes pour les processus de Lévy, notamment +avec les avancées de celles basées sur l'entropie. L'extension +multivariée de ce modèle n'a toujours pas été développée dans la +littérature, alors il pourrait être pertinent de s'y attarder, entre +autres pour étudier les titres indiciels et optimiser la composition +de portefeuilles. Enfin, il pourrait être intéressant d'aborder le +problème inverse de l'estimation des paramètres à partir des prix des +produits dérivés observés sur les marchés financiers. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/deed.tex b/depotfinal/908144032/deed.tex new file mode 100644 index 0000000..093a15e --- /dev/null +++ b/depotfinal/908144032/deed.tex @@ -0,0 +1,46 @@ +\chapter*{Contrat de partage} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Contrat de + partage} % dans TdM + +Cette création est mise à disposition selon le contrat +\emph{Attribution - Partage dans les Mêmes Conditions 4.0 International} +de Creative Commons disponible à l'adresse +\url{http://creativecommons.org/licenses/by-sa/4.0/deed.fr} + +En vertu de ce contrat, vous êtes autorisé à: +\begin{itemize} +\item \textbf{Partager} --- copier, distribuer et communiquer le + matériel par tous moyens et sous tous formats +\item \textbf{Adapter} --- remixer, transformer et créer à partir du + matériel pour toute utilisation, y compris commerciale. +\end{itemize} + +L'Offrant ne peux retirer les autorisations concédées par la +licence tant que vous appliquez les termes de cette licence. \\ + +Selon les conditions suivantes:\\ + + \begin{tabularx}{\linewidth}{@{}lX@{}} + \raisebox{-9mm}[0mm][13mm]{% + \includegraphics[height=11mm,keepaspectratio=true]{by}} & + \textbf{Attribution} --- Vous devez créditer l'Oeuvre, intégrer un lien vers la licence et indiquer si des modifications ont été effectuées à l'Oeuvre. Vous devez indiquer ces informations par tous les moyens possibles mais vous ne pouvez pas suggérer que l'Offrant vous soutient ou soutient la façon dont vous avez utilisé son Oeuvre. \\ \\ + \raisebox{-9mm}{\includegraphics[height=11mm,keepaspectratio=true]{sa}} + & \textbf{Partage à l'identique} --- Dans le cas où vous effectuez + un remix, que vous transformez, ou créez à partir du matériel + composant l'Oeuvre originale, vous devez diffuser l'Oeuvre + modifiée dans les même conditions, c'est-à-dire avec la même + licence avec laquelle l'Oeuvre originale a été diffusée. + \end{tabularx} \\ \\ + \textbf{Aucune autre restriction} --- Vous n'êtes pas autorisé à + appliquer des conditions légales ou des mesures techniques qui + restreindraient légalement autrui à utiliser l'Oeuvre dans les + conditions décrites par la licence. + + \begin{center} + \includegraphics[height=7mm,keepaspectratio=true]{by-sa}\\% + \end{center} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/graphiques/ABBEYN-callGAL-5.pdf b/depotfinal/908144032/graphiques/ABBEYN-callGAL-5.pdf similarity index 100% rename from graphiques/ABBEYN-callGAL-5.pdf rename to depotfinal/908144032/graphiques/ABBEYN-callGAL-5.pdf diff --git a/graphiques/ABBEYN-callGAL-7.pdf b/depotfinal/908144032/graphiques/ABBEYN-callGAL-7.pdf similarity index 100% rename from graphiques/ABBEYN-callGAL-7.pdf rename to depotfinal/908144032/graphiques/ABBEYN-callGAL-7.pdf diff --git a/graphiques/ABBEYN-chronologie.pdf b/depotfinal/908144032/graphiques/ABBEYN-chronologie.pdf similarity index 100% rename from graphiques/ABBEYN-chronologie.pdf rename to depotfinal/908144032/graphiques/ABBEYN-chronologie.pdf diff --git a/graphiques/ABBEYN-densiteGALmu-5.pdf b/depotfinal/908144032/graphiques/ABBEYN-densiteGALmu-5.pdf similarity index 100% rename from graphiques/ABBEYN-densiteGALmu-5.pdf rename to depotfinal/908144032/graphiques/ABBEYN-densiteGALmu-5.pdf diff --git a/graphiques/ABBEYN-densiteGALmu-7.pdf b/depotfinal/908144032/graphiques/ABBEYN-densiteGALmu-7.pdf similarity index 100% rename from graphiques/ABBEYN-densiteGALmu-7.pdf rename to depotfinal/908144032/graphiques/ABBEYN-densiteGALmu-7.pdf diff --git a/graphiques/ABBEYN-histogramme.pdf b/depotfinal/908144032/graphiques/ABBEYN-histogramme.pdf similarity index 100% rename from graphiques/ABBEYN-histogramme.pdf rename to depotfinal/908144032/graphiques/ABBEYN-histogramme.pdf diff --git a/graphiques/ABBEYN-qq.pdf b/depotfinal/908144032/graphiques/ABBEYN-qq.pdf similarity index 100% rename from graphiques/ABBEYN-qq.pdf rename to depotfinal/908144032/graphiques/ABBEYN-qq.pdf diff --git a/graphiques/CH3-SIMGAMMAGAUSS.pdf b/depotfinal/908144032/graphiques/CH3-SIMGAMMAGAUSS.pdf similarity index 100% rename from graphiques/CH3-SIMGAMMAGAUSS.pdf rename to depotfinal/908144032/graphiques/CH3-SIMGAMMAGAUSS.pdf diff --git a/graphiques/CH3-SIMULGAL0121.pdf b/depotfinal/908144032/graphiques/CH3-SIMULGAL0121.pdf similarity index 100% rename from graphiques/CH3-SIMULGAL0121.pdf rename to depotfinal/908144032/graphiques/CH3-SIMULGAL0121.pdf diff --git a/graphiques/dGAL-exemples.pdf b/depotfinal/908144032/graphiques/dGAL-exemples.pdf similarity index 100% rename from graphiques/dGAL-exemples.pdf rename to depotfinal/908144032/graphiques/dGAL-exemples.pdf diff --git a/graphiques/increment1.tex b/depotfinal/908144032/graphiques/increment1.tex similarity index 100% rename from graphiques/increment1.tex rename to depotfinal/908144032/graphiques/increment1.tex diff --git a/graphiques/mitchell1.pdf b/depotfinal/908144032/graphiques/mitchell1.pdf similarity index 100% rename from graphiques/mitchell1.pdf rename to depotfinal/908144032/graphiques/mitchell1.pdf diff --git a/graphiques/normaletronque.tex b/depotfinal/908144032/graphiques/normaletronque.tex similarity index 100% rename from graphiques/normaletronque.tex rename to depotfinal/908144032/graphiques/normaletronque.tex diff --git a/graphiques/pointdeselleGMM.png b/depotfinal/908144032/graphiques/pointdeselleGMM.png similarity index 100% rename from graphiques/pointdeselleGMM.png rename to depotfinal/908144032/graphiques/pointdeselleGMM.png diff --git a/graphiques/probdroite.tex b/depotfinal/908144032/graphiques/probdroite.tex similarity index 100% rename from graphiques/probdroite.tex rename to depotfinal/908144032/graphiques/probdroite.tex diff --git a/depotfinal/908144032/introduction.tex b/depotfinal/908144032/introduction.tex new file mode 100644 index 0000000..4bcd4a0 --- /dev/null +++ b/depotfinal/908144032/introduction.tex @@ -0,0 +1,75 @@ +\chapter*{Introduction} +\phantomsection\addcontentsline{toc}{chapter}{Introduction} % inclure dans TdM + +L'utilisation de la distribution de Laplace asymétrique généralisée +dans le cadre de la modélisation des rendements financiers est de plus +en plus courante dans les milieux académiques et pratiques. Entre +autres, elle présente une alternative intéressante au modèle de +Black-Scholes pour l'évaluation des options de type européennes. De +plus, elle peut être utilisée pour modéliser les rendements +obligataires à long terme et les variations du taux de +change. Cependant, la littérature actuelle présente peu d'outils qui +facilitent l'utilisation en pratique de cette distribution. La +principale motivation derrière le travail de recherche et de synthèse +présenté dans ce texte a été de développer certains outils qui +permettent l'estimation des paramètres de la distribution et +l'approximation de celle-ci. De plus, plusieurs tests statistiques +sont présentés afin d'évaluer la validité du modèle estimé et de +certaines contraintes linéaires qu'on pourrait lui appliquer. + +Au chapitre 1, on présente les différents types de modèles financiers +ainsi que le risque associé à la modélisation. On introduit aussi +différents concepts théoriques entourant les rendements +financiers. Puis, on dresse un historique des modèles financiers ayant +mené à l'utilisation de la distribution de Laplace asymétrique +généralisée. + +Au chapitre 2, on introduit les processus gamma et de Wiener qui sont +les deux composantes essentielles du processus de Laplace. Puis, on +présente les principales caractéristiques de la distribution de +Laplace asymétrique généralisée. Enfin, on présente quelques cas +particuliers de celle-ci et on fait le lien avec le modèle +variance-gamma de Madan et Seneta. + +Au chapitre 3, on présente la méthode du point de selle qui permet +d'effectuer l'approximation de la densité et de la fonction de +répartition lorsqu'elles n'ont pas de forme analytique. On applique +ensuite cette méthode à la distribution de Laplace asymétrique +généralisée. + +Au chapitre 4, on présente la méthode des moments généralisée ainsi +que son utilisation dans le cadre de l'estimation sous contraintes. On +présente ensuite certains tests d'hypothèse paramétriques. + +Au chapitre 5, on présente la méthode des équations d'estimation +optimales ainsi qu'une version légèrement modifiée qui diminue la +quantité de calcul nécessaire. + +Au chapitre 6, on applique les méthodes d'estimation des deux +chapitres précédents à la distribution de Laplace asymétrique +généralisée. On présente en premier lieu une méthode qui permet +d'obtenir un point de départ pour l'algorithme d'optimisation. Puis, +on détaille les résultats obtenus. + +Au chapitre 7, on présente les tests de normalité de Shapiro-Wilk et +d'Epps-Pulley, puis ceux d'adéquation du $\chi^2$ de Pearson, celui de +Kolmogorov-Smirnov ainsi qu'un autre basé sur la fonction +génératrice des moments. + +Au chapitre 8, on présente différentes méthodes pour l'évaluation +d'options. On présente d'abord quelques notions liées aux produits +dérivés, puis on détaille trois méthodes pouvant être utilisées avec +la distribution de Laplace asymétrique généralisée. Enfin, on présente +quelques particularités liées à certains titres financiers. + +Au chapitre 9, on présente un exemple d'application des outils +développés aux chapitres 3 à 8 avec un ensemble de données. + +Enfin, on présente en annexe certaines notions de théorie des +probabilités et de statistique. + + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/memoire.bib b/depotfinal/908144032/memoire.bib new file mode 100644 index 0000000..d55accd --- /dev/null +++ b/depotfinal/908144032/memoire.bib @@ -0,0 +1,640 @@ +@Comment{x-kbibtex-personnameformatting=<%f ><%l><, %s>} + +@Article{KOUTROUVELIS01011980, + Author = "Ioannis A. Koutrouvelis", + Journal = "{B}iometrika", + Number = 1, + Pages = "238--240", + Publisher = "Biometrika Trust", + Title = "{{A} {G}oodness-{O}f-{F}it {T}est of {S}imple {H}ypotheses {B}ased on the {E}mpirical {C}haracteristic {F}unction}", + Volume = 67, + Year = 1980 +} + +@Manual{RpackageVarianceGamma, + Author = "David Scott and Christine Yang Dong", + Note = "Consulté le 24 janvier 2014", + Title = "{{v}ariancegamma: {T}he {V}ariance {G}amma {D}istribution}", + Url = "http://CRAN.R-project.org/package=VarianceGamma", + Year = 2012 +} + +@Manual{Rsoftware, + Address = "Vienna, Austria", + Author = "{R Core Team}", + Organization = "R Foundation for Statistical Computing", + Note = "Consulté le 24 janvier 2014", + Title = "{{R}: A {L}anguage and {E}nvironment for {S}tatistical {C}omputing}", + Url = "http://www.R-project.org", + Year = 2014 +} + +@Article{Singer:2009, + Author = "Sa\v{s}a Singer and John Nelder", + Journal = "Scholarpedia", + Number = 7, + Pages = 2928, + Publisher = "Scholarpedia", + Title = "{N}elder-{M}ead {A}lgorithm", + Volume = 4, + Year = 2009 +} + +@Book{abramowitz1965handbook, + Author = "Milton Abramowitz and Irene A. 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Barndorff-Nielsen", + Publisher = "Birkh{\"a}user", + Title = "{L}{\'e}vy {P}rocesses: {T}heory and {A}pplications", + Year = 2001 +} + +@Article{berkson1980minimum, + Author = "Joseph Berkson", + Journal = "The Annals of Statistics", + Pages = "457--487", + Publisher = "Institute of Mathematical Statistics", + Title = "{M}inimum {C}hi-Square, not {M}aximum {L}ikelihood!", + Year = 1980 +} + +@Book{bingham2004risk, + Author = "Nicholas H. Bingham and R{\"u}diger Kiesel", + Publisher = "Springer", + Title = "{R}isk-Neutral {V}aluation: {P}ricing and {H}edging of {F}inancial {D}erivatives", + Year = 2004 +} + +@Article{black1973pricing, + Author = "Fischer Black and Myron Scholes", + Journal = "The Journal of Political Economy", + Pages = "637--654", + Publisher = "University of Chicago Press", + Title = "{T}he {P}ricing of {O}ptions and {C}orporate {L}iabilities", + Year = 1973 +} + +@Article{black1976pricing, + Author = "Fischer Black", + Journal = "Journal of Financial Economics", + Number = 1, + Pages = "167--179", + Publisher = "Elsevier", + Title = "{T}he {P}ricing of {C}ommodity {C}ontracts", + Volume = 3, + Year = 1976 +} + +@Article{buckle1995bayesian, + Author = "D. J. Buckle", + Journal = "Journal of the American Statistical Association", + Number = 430, + Pages = "605--613", + Publisher = "Taylor \& Francis Group", + Title = "{B}ayesian {I}nference for {S}table {D}istributions", + Volume = 90, + Year = 1995 +} + +@Book{butler2007saddlepoint, + Author = "Ronald W. Butler", + Publisher = "Cambridge University Press", + Title = "{S}addlepoint {A}pproximations with {A}pplications", + Volume = 22, + Year = 2007 +} + +@Article{carr1999option, + Author = "Peter P. Carr and Dilip B. Madan", + Journal = "Journal of Computational Finance", + Number = 4, + Pages = "61--73", + Publisher = "Risk Waters Group", + Title = "{O}ption {V}aluation {U}sing the {F}ast {F}ourier {T}ransform", + Volume = 2, + Year = 1999 +} + +@Article{crowder1986consistency, + Author = "Martin Crowder", + Journal = "Econometric Theory", + Pages = "305--330", + Publisher = "Cambridge University Press", + Title = "{O}n {C}onsistency and {I}nconsistency of {E}stimating {E}quations", + Year = 1986 +} + +@Article{crowder1987linear, + Author = "Martin Crowder", + Journal = "Biometrika", + Number = 3, + Pages = "591--597", + Publisher = "Biometrika Trust", + Title = "{O}n {L}inear and {Q}uadratic {E}stimating {F}unctions", + Volume = 74, + Year = 1987 +} + +@Article{daniels1954saddlepoint, + Author = "Henry E. Daniels", + Journal = "The Annals of Mathematical Statistics", + Pages = "631--650", + Publisher = "Institute of Mathematical Statistics", + Title = "{S}addlepoint {A}pproximations in {S}tatistics", + Year = 1954 +} + +@Article{daniels1987tail, + Author = "Henry E. Daniels", + Journal = "International Statistical Review/Revue internationale de Statistique", + Pages = "37--48", + Publisher = "Wiley Online Library", + Title = "{T}ail {P}robability {A}pproximations", + Year = 1987 +} + +@Techreport{derman1996modelrisk, + Author = "Emanuel Derman", + Institution = "Goldman Sachs", + Title = "{M}odel {R}isk", + Year = 1996 +} + +@Book{dodge2004statistique, + Author = "Yadolah Dodge", + Publisher = "Springer Verlag France", + Title = "{S}tatistique: {D}ictionnaire Encyclop{\'e}dique", + Year = 2004 +} + +@Article{epps1983test, + Author = "Thomas W. Epps and Lawrence B. Pulley", + Journal = "Biometrika", + Number = 3, + Pages = "723--726", + Publisher = "Biometrika Trust", + Title = "{A} {T}est for {N}ormality {B}ased on the {E}mpirical {C}haracteristic {F}unction", + Volume = 70, + Year = 1983 +} + +@Book{epps2007pricing, + Author = "Thomas W. Epps", + Publisher = "World Scientific Publishing Company incorporated", + Title = "{P}ricing {D}erivative {S}ecurities", + Year = 2007 +} + +@Book{everitt2006cambridge, + Author = "Brian Everitt and Anders Skrondal", + Publisher = "Cambridge University Press", + Title = "{T}he {C}ambridge {D}ictionary of {S}tatistics", + Volume = 4, + Year = 2006 +} + +@Article{fama1993common, + Author = "Eugene F. Fama and Kenneth R. French", + Journal = "Journal of Financial Economics", + Number = 1, + Pages = "3--56", + Publisher = "Elsevier", + Title = "{C}ommon {R}isk {F}actors in the {R}eturns on {S}tocks and {B}onds", + Volume = 33, + Year = 1993 +} + +@Article{feuerverger1981efficiency, + Author = "Andrey Feuerverger and Philip McDunnough", + Journal = "Journal of the Royal Statistical Society. Series B (methodological)", + Pages = "20--27", + Publisher = "Blackwell Publishing", + Title = "{O}n the {E}fficiency of {E}mpirical {C}haracteristic {F}unction {P}rocedures", + Year = 1981 +} + +@Article{fox1986large, + Author = "Robert Fox and Murad S. Taqqu", + Journal = "The Annals of Statistics", + Number = 2, + Pages = "517--532", + Publisher = "Institute of Mathematical Statistics", + Title = "{L}arge-Sample {P}roperties of {P}arameter {E}stimates for {S}trongly {D}ependent {S}tationary {G}aussian {T}ime {S}eries", + Volume = 14, + Year = 1986 +} + +@Article{gil1951note, + Author = "J. Gil-Pelaez", + Journal = "Biometrika", + Number = "3-4", + Pages = "481--482", + Publisher = "Biometrika Trust", + Title = "{N}ote on the {I}nversion {T}heorem", + Volume = 38, + Year = 1951 +} + +@Book{gourieroux1989statistique, + Author = "Christian Gourieroux and Alain Monfort", + Publisher = "Economica", + Title = "{S}tatistique et Mod{\`e}les {\'E}conom{\'e}triques: Notions G{\'e}n{\'e}rales, Estimation, Pr{\'e}vision, Algorithmes", + Volume = 1, + Year = 1989 +} + +@Book{hall2005generalized, + Author = "Alastair R. Hall", + Publisher = "Oxford University Press", + Title = "{G}eneralized {M}ethod of {M}oments", + Year = 2005 +} + +@Book{hamilton1994time, + Author = "James Douglas Hamilton", + Publisher = "Cambridge University Press", + Title = "{T}ime {S}eries {A}nalysis", + Volume = 2, + Year = 1994 +} + +@Article{hansen1982large, + Author = "Lars Peter Hansen", + Journal = "Econometrica", + Pages = "1029--1054", + Publisher = "Wiley-Blackwell", + Title = "{L}arge {S}ample {P}roperties of {G}eneralized {M}ethod of {M}oments {E}stimators", + Year = 1982 +} + +@Article{henze1990approximation, + Author = "Norbert Henze", + Journal = "Metrika", + Number = 1, + Pages = "7--18", + Publisher = "Springer", + Title = "{A}n {A}pproximation to the {L}imit {D}istribution of the {E}pps-{P}ulley {T}est {S}tatistic for {N}ormality", + Volume = 37, + Year = 1990 +} + +@Article{heston1993closed, + Author = "Steven L. Heston", + Journal = "Review of Financial Studies", + Number = 2, + Pages = "327--343", + Publisher = "Sociery for Financial Studies", + Title = "{A} {C}losed-{F}orm {S}olution for {O}ptions with {S}tochastic {V}olatility with {A}pplications to {B}ond and {C}urrency {O}ptions", + Volume = 6, + Year = 1993 +} + +@Article{hinkley1977estimation, + Author = "David V. Hinkley and Nagesh S. Revankar", + Journal = "Journal of Econometrics", + Number = 1, + Pages = "1--11", + Publisher = "Elsevier", + Title = "{E}stimation of the {P}areto {L}aw from {U}nderreported {D}ata: {A} {F}urther {A}nalysis", + Volume = 5, + Year = 1977 +} + +@Book{hogg1978introduction, + Author = "Robert V. Hogg and Allen Craig", + Publisher = "Macmillan", + Title = "{I}ntroduction to {M}athematical {S}tatistics", + Year = 1978 +} + +@Book{hull1999options, + Author = "John C. Hull", + Publisher = "Pearson Education", + Title = "{O}ptions, {F}utures, and {O}ther {D}erivatives", + Year = 1999 +} + +@Article{itkin2005pricing, + Author = "Andrey Itkin", + Journal = "arXiv preprint physics/0503137", + Publisher = "arXiv", + Title = "{P}ricing {O}ptions with {VG} {M}odel {U}sing {FFT}", + Year = 2005 +} + +@Book{kotz2001laplace, + Author = "Samuel Kotz and Tomasz J. Kozubowski and Krzystof Podg{\'o}rski", + Publisher = "Birkh{\"a}user", + Series = "Progress in Mathematics Series", + Title = "{T}he {L}aplace {D}istribution and {G}eneralizations: {A} {R}evisit with {A}pplications to {C}ommunications, {E}conomics, {E}ngineering, and {F}inance", + Year = 2001 +} + +@Article{kozubowski1999class, + Author = "Tomasz J. Kozubowski and Krzysztof Podg{\'o}rski", + Journal = "Actuarial Research Clearing House", + Pages = "113--134", + Publisher = "SOA Education and Research", + Title = "{A} {C}lass of {A}symmetric {D}istributions", + Volume = 1, + Year = 1999 +} + +@Article{kozubowski2001asymmetric, + Author = "Tomasz J. Kozubowski and Krzysztof Podg{\'o}rski", + Journal = "{M}athematical and {C}omputer {M}odeling", + Number = 9, + Pages = "1003--1021", + Publisher = "Elsevier", + Title = "{A}symmetric {L}aplace {L}aws and {M}odeling {F}inancial {D}ata", + Volume = 34, + Year = 2001 +} + +@Book{kyprianou2007introductory, + Author = "Andreas E. Kyprianou", + Publisher = "Springer", + Title = "{I}ntroductory {L}ectures on {F}luctuations of {L}{\'e}vy {P}rocesses with {A}pplications", + Year = 2007 +} + +@Article{lugannani1980saddle, + Author = "Robert Lugannani and Stephen Rice", + Journal = "Advances in Applied Probability", + Pages = "475--490", + Publisher = "Applied Probability Trust", + Title = "{S}addle {P}oint {A}pproximation for the {D}istribution of the {S}um of {I}ndependent {R}andom {V}ariables", + Year = 1980 +} + +@Book{lukacs1960characteristic, + Author = "Eugene Lukacs", + Publisher = "Griffin London", + Title = "{C}haracteristic {F}unctions", + Volume = 4, + Year = 1960 +} + +@Article{luong1987minimum, + Author = "Andrew Luong and Mary E. Thompson", + Journal = "Canadian Journal of Statistics", + Number = 3, + Pages = "239--251", + Publisher = "Wiley Online Library", + Title = "{M}inimum-{D}istance {M}ethods {B}ased on {Q}uadratic {D}istances for {T}ransforms", + Volume = 15, + Year = 1987 +} + +@Article{madan1990variance, + Author = "Dilip B. Madan and Eugene Seneta", + Journal = "Journal of Business", + Pages = "511--524", + Publisher = "University of Chicago Press", + Title = "{T}he {V}ariance {G}amma {M}odel for {S}hare {M}arket {R}eturns", + Year = 1990 +} + +@Article{madan1998variance, + Author = "Peter P. Carr and Eric C. Chang and Dilip B. Madan", + Journal = "European Finance Review", + Number = 1, + Pages = "79--105", + Publisher = "Kluwer Academic Publishers", + Title = "{T}he {V}ariance {G}amma {P}rocess and {O}ption {P}ricing", + Volume = 2, + Year = 1998 +} + +@Article{mandelbrot1963variation, + Author = "Benoit Mandelbrot", + Journal = "Journal of Business", + Pages = "394--419", + Publisher = "University of Chicago Press", + Title = "{T}he {V}ariation of {C}ertain {S}peculative {P}rices", + Year = 1963 +} + +@Article{merton1976option, + Author = "Robert C. Merton", + Journal = "Journal of Financial Economics", + Number = 1, + Pages = "125--144", + Publisher = "Elsevier", + Title = "{O}ption {P}ricing when {U}nderlying {S}tock {R}eturns {A}re {D}iscontinuous", + Volume = 3, + Year = 1976 +} + +@Article{mitchell1916critique, + Author = "Wesley C. Mitchell", + Journal = "The Journal of Political Economy", + Pages = "625--693", + Publisher = "University of Chicago Press", + Title = "{A} {C}ritique of {I}ndex {N}umbers of the {P}rices of {S}tocks", + Year = 1916 +} + +@Book{musiela2005martingale, + Author = "Marek Musiela and Marek Rutkowski", + Publisher = "Springer", + Title = "{M}artingale {M}ethods in {F}inancial {M}odelling", + Volume = 36, + Year = 2005 +} + +@Article{newey1987hypothesis, + Author = "Whitney K. Newey and Kenneth D. West", + Journal = "International Economic Review", + Number = 3, + Pages = "777--787", + Publisher = "Blackwell Publishing ", + Title = "{H}ypothesis {T}esting with {E}fficient {M}ethod of {M}oments {E}stimation", + Volume = 28, + Year = 1987 +} + +@Article{newey1994large, + Author = "Whitney K. Newey and Daniel McFadden", + Journal = "Handbook of Econometrics", + Pages = "2111--2245", + Publisher = "Elsevier", + Title = "{L}arge {S}ample {E}stimation and {H}ypothesis {T}esting", + Volume = 4, + Year = 1994 +} + +@Article{praetz1972distribution, + Author = "Peter D. Praetz", + Journal = "Journal of Business", + Pages = "49--55", + Publisher = "University of Chicago Press", + Title = "{T}he {D}istribution of {S}hare {P}rice {C}hanges", + Year = 1972 +} + +@Article{press1967compound, + Author = "S. James Press", + Journal = "Journal of Business", + Pages = "317--335", + Publisher = "University of Chicago Press", + Title = "{A} {C}ompound {E}vents {M}odel for {S}ecurity {P}rices", + Year = 1967 +} + +@Article{randal2004non, + Author = "John A. Randal and Peter J. Thomson and Martin T. Lally", + Journal = "Quantitative Finance", + Number = 4, + Pages = "427--440", + Publisher = "Taylor \& Francis Group", + Title = "{N}on-parametric {E}stimation of {H}istorical {V}olatility", + Volume = 4, + Year = 2004 +} + +@Book{sato1999levy, + Author = "{Ken-Iti} {Sato}", + Publisher = "Cambridge University Press", + Series = "Cambridge Studies in Advanced Mathematics", + Title = "{L}{\'e}vy {P}rocesses and {I}nfinitely {D}ivisible {D}istributions", + Year = 1999 +} + +@Book{schoutens2003levy, + Author = "Wim Schoutens", + Publisher = "Wiley", + Title = "{L}{\'e}vy {P}rocesses in {F}inance", + Year = 2003 +} + +@Article{seneta2004fitting, + Author = "Eugene Seneta", + Journal = "Journal of Applied Probability", + Pages = "177--187", + Publisher = "Applied Probability Trust", + Title = "{F}itting the {V}ariance {G}amma {M}odel to {F}inancial {D}ata", + Year = 2004 +} + +@Article{shapiro1965analysis, + Author = "Samuel Sanford Shapiro and Martin B. Wilk", + Journal = "Biometrika", + Number = "3/4", + Pages = "591--611", + Publisher = "Biometrika Trust", + Title = "{A}n {A}nalysis of {V}ariance {T}est for {N}ormality ({C}omplete {S}amples)", + Volume = 52, + Year = 1965 +} + +@Article{shephard1991characteristic, + Author = "Neil G. Shephard", + Journal = "Econometric Theory", + Number = 04, + Pages = "519--529", + Publisher = "Cambridge University Press", + Title = "{F}rom {C}haracteristic {F}unction to {D}istribution {F}unction: {A} {S}imple {F}ramework for the {T}heory", + Volume = 7, + Year = 1991 +} + +@Book{spiegel1999schaum, + Author = "Murray R. Spiegel and John Liu", + Publisher = "McGraw-Hill", + Title = "{S}chaum's {M}athematical {H}andbook of {F}ormulas and {T}ables", + Volume = 1000, + Year = 1999 +} + +@Book{stuart1987kendall, + Author = "Alan Stuart and J. Keith Ord", + Publisher = "Oxford University Press", + Title = "{K}endall{\rq}s {A}dvanced {T}heory of {S}tatistics, {V}ol. 1", + Year = 1987 +} + +@Manual{teschl2004topics, + Author = "Gerald Teschl", + Title = "{T}opics in {R}eal and {F}unctional {A}nalysis", + Note = "Consulté le 24 janvier 2014", + Url = "http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html", + Year = 2004 +} + +@Phdthesis{torczon1989multi, + Author = "Virginia Joanne Torczon", + Title = "{M}ulti-directional {S}earch: {A} {D}irect {S}earch {A}lgorithm for {P}arallel {M}achines", + Year = 1989 +} + +@Inproceedings{walterlevy, + Author = "Christian Walter", + Booktitle = "Proceedings of the 5th AFIR colloquium", + Publisher = "International Actuarial Association ", + Title = "{L}{\'e}vy-{S}tability {U}nder {A}ddition and {F}ractal {S}tructure of {M}arkets: {I}mplications for the {A}ctuaries and {E}mphasized {E}xamination of {MATIF} {N}ational {C}ontract", + Year = 1995 +} + +@Article{wang2003evaluating, + Author = "Wai Wan Tsang {Jingbo Wang} and George Marsaglia", + Journal = "Journal of Statistical Software", + Number = 18, + Pages = "1--4", + Publisher = "American Statistical Association", + Title = "{E}valuating {K}olmogorov's {D}istribution", + Volume = 8, + Year = 2003 +} + +@Article{wendel1961non, + Author = "J. G. Wendel", + Journal = "The Annals of Mathematical Statistics", + Number = 1, + Pages = "338--339", + Publisher = "Institute of Mathematical Statistics", + Title = "{T}he {N}on-{A}bsolute {C}onvergence of {G}il-{P}elaez' {I}nversion {I}ntegral", + Volume = 32, + Year = 1961 +} + +@Article{wolfowitz1957minimum, + Author = "Jacob Wolfowitz", + Journal = "The Annals of Mathematical Statistics", + Pages = "75--88", + Publisher = "Institute of Mathematical Statistics", + Title = "{T}he {M}inimum {D}istance {M}ethod", + Year = 1957 +} + +@Book{wooldridge2001econometric, + Author = "Jeffrey M. Wooldridge", + Publisher = "MIT press", + Title = "{E}conometric {A}nalysis of {C}ross {S}ection and {P}anel {D}ata", + Year = 2001 +} + diff --git a/memoire/plainnatmod.bst b/depotfinal/908144032/plainnat-fr.bst similarity index 72% rename from memoire/plainnatmod.bst rename to depotfinal/908144032/plainnat-fr.bst index cc74f1f..76e3707 100644 --- a/memoire/plainnatmod.bst +++ b/depotfinal/908144032/plainnat-fr.bst @@ -1,1436 +1,1503 @@ -%% File: `plainnat.bst' -%% A modification of `plain.bst' for use with natbib package -%% -%% Copyright 1993-2007 Patrick W Daly -%% Max-Planck-Institut f\"ur Sonnensystemforschung -%% Max-Planck-Str. 2 -%% D-37191 Katlenburg-Lindau -%% Germany -%% E-mail: daly@mps.mpg.de -%% -%% This program can be redistributed and/or modified under the terms -%% of the LaTeX Project Public License Distributed from CTAN -%% archives in directory macros/latex/base/lppl.txt; either -%% version 1 of the License, or any later version. -%% - % Version and source file information: - % \ProvidesFile{natbst.mbs}[2007/11/26 1.93 (PWD)] - % - % BibTeX `plainnat' family - % version 0.99b for BibTeX versions 0.99a or later, - % for LaTeX versions 2.09 and 2e. - % - % For use with the `natbib.sty' package; emulates the corresponding - % member of the `plain' family, but with author-year citations. - % - % With version 6.0 of `natbib.sty', it may also be used for numerical - % citations, while retaining the commands \citeauthor, \citefullauthor, - % and \citeyear to print the corresponding information. - % - % For version 7.0 of `natbib.sty', the KEY field replaces missing - % authors/editors, and the date is left blank in \bibitem. - % - % Includes field EID for the sequence/citation number of electronic journals - % which is used instead of page numbers. - % - % Includes fields ISBN and ISSN. - % - % Includes field URL for Internet addresses. - % - % Includes field DOI for Digital Object Idenfifiers. - % - % Works best with the url.sty package of Donald Arseneau. - % - % Works with identical authors and year are further sorted by - % citation key, to preserve any natural sequence. - % -ENTRY - { address - author - booktitle - chapter - doi - eid - edition - editor - howpublished - institution - isbn - issn - journal - key - month - note - number - organization - pages - publisher - school - series - title - type - url - volume - year - } - {} - { label extra.label sort.label short.list } - -INTEGERS { output.state before.all mid.sentence after.sentence after.block } - -FUNCTION {init.state.consts} -{ #0 'before.all := - #1 'mid.sentence := - #2 'after.sentence := - #3 'after.block := -} - -STRINGS { s t } - -FUNCTION {output.nonnull} -{ 's := - output.state mid.sentence = - { ", " * write$ } - { output.state after.block = - { add.period$ write$ - newline$ - "\newblock " write$ - } - { output.state before.all = - 'write$ - { add.period$ " " * write$ } - if$ - } - if$ - mid.sentence 'output.state := - } - if$ - s -} - -FUNCTION {output} -{ duplicate$ empty$ - 'pop$ - 'output.nonnull - if$ -} - -FUNCTION {output.check} -{ 't := - duplicate$ empty$ - { pop$ "empty " t * " in " * cite$ * warning$ } - 'output.nonnull - if$ -} - -FUNCTION {fin.entry} -{ add.period$ - write$ - newline$ -} - -FUNCTION {new.block} -{ output.state before.all = - 'skip$ - { after.block 'output.state := } - if$ -} - -FUNCTION {new.sentence} -{ output.state after.block = - 'skip$ - { output.state before.all = - 'skip$ - { after.sentence 'output.state := } - if$ - } - if$ -} - -FUNCTION {not} -{ { #0 } - { #1 } - if$ -} - -FUNCTION {and} -{ 'skip$ - { pop$ #0 } - if$ -} - -FUNCTION {or} -{ { pop$ #1 } - 'skip$ - if$ -} - -FUNCTION {new.block.checka} -{ empty$ - 'skip$ - 'new.block - if$ -} - -FUNCTION {new.block.checkb} -{ empty$ - swap$ empty$ - and - 'skip$ - 'new.block - if$ -} - -FUNCTION {new.sentence.checka} -{ empty$ - 'skip$ - 'new.sentence - if$ -} - -FUNCTION {new.sentence.checkb} -{ empty$ - swap$ empty$ - and - 'skip$ - 'new.sentence - if$ -} - -FUNCTION {field.or.null} -{ duplicate$ empty$ - { pop$ "" } - 'skip$ - if$ -} - -FUNCTION {emphasize} -{ duplicate$ empty$ - { pop$ "" } - { "\emph{" swap$ * "}" * } - if$ -} - -INTEGERS { nameptr namesleft numnames } - -FUNCTION {format.names} -{ 's := - #1 'nameptr := - s num.names$ 'numnames := - numnames 'namesleft := - { namesleft #0 > } - { s nameptr "{ff~}{vv~}{ll}{, jj}" format.name$ 't := - nameptr #1 > - { namesleft #1 > - { ", " * t * } - { numnames #2 > - { "," * } - 'skip$ - if$ - t "others" = - { " et~al." * } - { " et " * t * } - if$ - } - if$ - } - 't - if$ - nameptr #1 + 'nameptr := - namesleft #1 - 'namesleft := - } - while$ -} - -FUNCTION {format.key} -{ empty$ - { key field.or.null } - { "" } - if$ -} - -FUNCTION {format.authors} -{ author empty$ - { "" } - { author format.names } - if$ -} - -FUNCTION {format.editors} -{ editor empty$ - { "" } - { editor format.names - editor num.names$ #1 > - { ", editors" * } - { ", editor" * } - if$ - } - if$ -} - -FUNCTION {format.isbn} -{ isbn empty$ - { "" } - { new.block "ISBN " isbn * } - if$ -} - -FUNCTION {format.issn} -{ issn empty$ - { "" } - { new.block "ISSN " issn * } - if$ -} - -FUNCTION {format.url} -{ url empty$ - { "" } - { new.block "URL \url{" url * "}" * } - if$ -} - -FUNCTION {format.doi} -{ doi empty$ - { "" } - { new.block "\doi{" doi * "}" * } - if$ -} - -FUNCTION {format.title} -{ title empty$ - { "" } - { title "t" change.case$ } - if$ -} - -FUNCTION {format.full.names} -{'s := - #1 'nameptr := - s num.names$ 'numnames := - numnames 'namesleft := - { namesleft #0 > } - { s nameptr - "{vv~}{ll}" format.name$ 't := - nameptr #1 > - { - namesleft #1 > - { ", " * t * } - { - numnames #2 > - { "," * } - 'skip$ - if$ - t "others" = - { " et~al." * } - { " et " * t * } - if$ - } - if$ - } - 't - if$ - nameptr #1 + 'nameptr := - namesleft #1 - 'namesleft := - } - while$ -} - -FUNCTION {author.editor.full} -{ author empty$ - { editor empty$ - { "" } - { editor format.full.names } - if$ - } - { author format.full.names } - if$ -} - -FUNCTION {author.full} -{ author empty$ - { "" } - { author format.full.names } - if$ -} - -FUNCTION {editor.full} -{ editor empty$ - { "" } - { editor format.full.names } - if$ -} - -FUNCTION {make.full.names} -{ type$ "book" = - type$ "inbook" = - or - 'author.editor.full - { type$ "proceedings" = - 'editor.full - 'author.full - if$ - } - if$ -} - -FUNCTION {output.bibitem} -{ newline$ - "\bibitem[" write$ - label write$ - ")" make.full.names duplicate$ short.list = - { pop$ } - { * } - if$ - "]{" * write$ - cite$ write$ - "}" write$ - newline$ - "" - before.all 'output.state := -} - -FUNCTION {n.dashify} -{ 't := - "" - { t empty$ not } - { t #1 #1 substring$ "-" = - { t #1 #2 substring$ "--" = not - { "--" * - t #2 global.max$ substring$ 't := - } - { { t #1 #1 substring$ "-" = } - { "-" * - t #2 global.max$ substring$ 't := - } - while$ - } - if$ - } - { t #1 #1 substring$ * - t #2 global.max$ substring$ 't := - } - if$ - } - while$ -} - -FUNCTION {format.date} -{ year duplicate$ empty$ - { "empty year in " cite$ * warning$ - pop$ "" } - 'skip$ - if$ - month empty$ - 'skip$ - { month - " " * swap$ * - } - if$ - extra.label * -} - -FUNCTION {format.btitle} -{ title emphasize -} - -FUNCTION {tie.or.space.connect} -{ duplicate$ text.length$ #3 < - { "~" } - { " " } - if$ - swap$ * * -} - -FUNCTION {either.or.check} -{ empty$ - 'pop$ - { "can't use both " swap$ * " fields in " * cite$ * warning$ } - if$ -} - -FUNCTION {format.bvolume} -{ volume empty$ - { "" } - { "volume" volume tie.or.space.connect - series empty$ - 'skip$ - { " of " * series emphasize * } - if$ - "volume and number" number either.or.check - } - if$ -} - -FUNCTION {format.number.series} -{ volume empty$ - { number empty$ - { series field.or.null } - { output.state mid.sentence = - { "number" } - { "Number" } - if$ - number tie.or.space.connect - series empty$ - { "there's a number but no series in " cite$ * warning$ } - { " in " * series * } - if$ - } - if$ - } - { "" } - if$ -} - -FUNCTION {format.edition} -{ edition empty$ - { "" } - { output.state mid.sentence = - { edition "l" change.case$ " edition" * } - { edition "t" change.case$ " edition" * } - if$ - } - if$ -} - -INTEGERS { multiresult } - -FUNCTION {multi.page.check} -{ 't := - #0 'multiresult := - { multiresult not - t empty$ not - and - } - { t #1 #1 substring$ - duplicate$ "-" = - swap$ duplicate$ "," = - swap$ "+" = - or or - { #1 'multiresult := } - { t #2 global.max$ substring$ 't := } - if$ - } - while$ - multiresult -} - -FUNCTION {format.pages} -{ pages empty$ - { "" } - { pages multi.page.check - { "pages" pages n.dashify tie.or.space.connect } - { "page" pages tie.or.space.connect } - if$ - } - if$ -} - -FUNCTION {format.eid} -{ eid empty$ - { "" } - { "art." eid tie.or.space.connect } - if$ -} - -FUNCTION {format.vol.num.pages} -{ volume field.or.null - number empty$ - 'skip$ - { "\penalty0 (" number * ")" * * - volume empty$ - { "there's a number but no volume in " cite$ * warning$ } - 'skip$ - if$ - } - if$ - pages empty$ - 'skip$ - { duplicate$ empty$ - { pop$ format.pages } - { ":\penalty0 " * pages n.dashify * } - if$ - } - if$ -} - -FUNCTION {format.vol.num.eid} -{ volume field.or.null - number empty$ - 'skip$ - { "\penalty0 (" number * ")" * * - volume empty$ - { "there's a number but no volume in " cite$ * warning$ } - 'skip$ - if$ - } - if$ - eid empty$ - 'skip$ - { duplicate$ empty$ - { pop$ format.eid } - { ":\penalty0 " * eid * } - if$ - } - if$ -} - -FUNCTION {format.chapter.pages} -{ chapter empty$ - 'format.pages - { type empty$ - { "chapter" } - { type "l" change.case$ } - if$ - chapter tie.or.space.connect - pages empty$ - 'skip$ - { ", " * format.pages * } - if$ - } - if$ -} - -FUNCTION {format.in.ed.booktitle} -{ booktitle empty$ - { "" } - { editor empty$ - { "In " booktitle emphasize * } - { "In " format.editors * ", " * booktitle emphasize * } - if$ - } - if$ -} - -FUNCTION {empty.misc.check} -{ author empty$ title empty$ howpublished empty$ - month empty$ year empty$ note empty$ - and and and and and - key empty$ not and - { "all relevant fields are empty in " cite$ * warning$ } - 'skip$ - if$ -} - -FUNCTION {format.thesis.type} -{ type empty$ - 'skip$ - { pop$ - type "t" change.case$ - } - if$ -} - -FUNCTION {format.tr.number} -{ type empty$ - { "Technical Report" } - 'type - if$ - number empty$ - { "t" change.case$ } - { number tie.or.space.connect } - if$ -} - -FUNCTION {format.article.crossref} -{ key empty$ - { journal empty$ - { "need key or journal for " cite$ * " to crossref " * crossref * - warning$ - "" - } - { "In \emph{" journal * "}" * } - if$ - } - { "In " } - if$ - " \citet{" * crossref * "}" * -} - -FUNCTION {format.book.crossref} -{ volume empty$ - { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ - "In " - } - { "Volume" volume tie.or.space.connect - " of " * - } - if$ - editor empty$ - editor field.or.null author field.or.null = - or - { key empty$ - { series empty$ - { "need editor, key, or series for " cite$ * " to crossref " * - crossref * warning$ - "" * - } - { "\emph{" * series * "}" * } - if$ - } - 'skip$ - if$ - } - 'skip$ - if$ - " \citet{" * crossref * "}" * -} - -FUNCTION {format.incoll.inproc.crossref} -{ editor empty$ - editor field.or.null author field.or.null = - or - { key empty$ - { booktitle empty$ - { "need editor, key, or booktitle for " cite$ * " to crossref " * - crossref * warning$ - "" - } - { "In \emph{" booktitle * "}" * } - if$ - } - { "In " } - if$ - } - { "In " } - if$ - " \citet{" * crossref * "}" * -} - -FUNCTION {article} -{ output.bibitem - format.authors "author" output.check - author format.key output - new.block - format.title "title" output.check - new.block - crossref missing$ - { journal emphasize "journal" output.check - eid empty$ - { format.vol.num.pages output } - { format.vol.num.eid output } - if$ - format.date "year" output.check - } - { format.article.crossref output.nonnull - eid empty$ - { format.pages output } - { format.eid output } - if$ - } - if$ - format.issn output - format.doi output - format.url output - new.block - note output - fin.entry -} - -FUNCTION {book} -{ output.bibitem - author empty$ - { format.editors "author and editor" output.check - editor format.key output - } - { format.authors output.nonnull - crossref missing$ - { "author and editor" editor either.or.check } - 'skip$ - if$ - } - if$ - new.block - format.btitle "title" output.check - crossref missing$ - { format.bvolume output - new.block - format.number.series output - new.sentence - publisher "publisher" output.check - address output - } - { new.block - format.book.crossref output.nonnull - } - if$ - format.edition output - format.date "year" output.check - format.isbn output - format.doi output - format.url output - new.block - note output - fin.entry -} - -FUNCTION {booklet} -{ output.bibitem - format.authors output - author format.key output - new.block - format.title "title" output.check - howpublished address new.block.checkb - howpublished output - address output - format.date output - format.isbn output - format.doi output - format.url output - new.block - note output - fin.entry -} - -FUNCTION {inbook} -{ output.bibitem - author empty$ - { format.editors "author and editor" output.check - editor format.key output - } - { format.authors output.nonnull - crossref missing$ - { "author and editor" editor either.or.check } - 'skip$ - if$ - } - if$ - new.block - format.btitle "title" output.check - crossref missing$ - { format.bvolume output - format.chapter.pages "chapter and pages" output.check - new.block - format.number.series output - new.sentence - publisher "publisher" output.check - address output - } - { format.chapter.pages "chapter and pages" output.check - new.block - format.book.crossref output.nonnull - } - if$ - format.edition output - format.date "year" output.check - format.isbn output - format.doi output - format.url output - new.block - note output - fin.entry -} - -FUNCTION {incollection} -{ output.bibitem - format.authors "author" output.check - author format.key output - new.block - format.title "title" output.check - new.block - crossref missing$ - { format.in.ed.booktitle "booktitle" output.check - format.bvolume output - format.number.series output - format.chapter.pages output - new.sentence - publisher "publisher" output.check - address output - format.edition output - format.date "year" output.check - } - { format.incoll.inproc.crossref output.nonnull - format.chapter.pages output - } - if$ - format.isbn output - format.doi output - format.url output - new.block - note output - fin.entry -} - -FUNCTION {inproceedings} -{ output.bibitem - format.authors "author" output.check - author format.key output - new.block - format.title "title" output.check - new.block - crossref missing$ - { format.in.ed.booktitle "booktitle" output.check - format.bvolume output - format.number.series output - format.pages output - address empty$ - { organization publisher new.sentence.checkb - organization output - publisher output - format.date "year" output.check - } - { address output.nonnull - format.date "year" output.check - new.sentence - organization output - publisher output - } - if$ - } - { format.incoll.inproc.crossref output.nonnull - format.pages output - } - if$ - format.isbn output - format.doi output - format.url output - new.block - note output - fin.entry -} - -FUNCTION {conference} { inproceedings } - -FUNCTION {manual} -{ output.bibitem - format.authors output - author format.key output - new.block - format.btitle "title" output.check - organization address new.block.checkb - organization output - address output - format.edition output - format.date output - format.url output - new.block - note output - fin.entry -} - -FUNCTION {mastersthesis} -{ output.bibitem - format.authors "author" output.check - author format.key output - new.block - format.title "title" output.check - new.block - "Master's thesis" format.thesis.type output.nonnull - school "school" output.check - address output - format.date "year" output.check - format.url output - new.block - note output - fin.entry -} - -FUNCTION {misc} -{ output.bibitem - format.authors output - author format.key output - title howpublished new.block.checkb - format.title output - howpublished new.block.checka - howpublished output - format.date output - format.issn output - format.url output - new.block - note output - fin.entry - empty.misc.check -} - -FUNCTION {phdthesis} -{ output.bibitem - format.authors "author" output.check - author format.key output - new.block - format.btitle "title" output.check - new.block - "PhD thesis" format.thesis.type output.nonnull - school "school" output.check - address output - format.date "year" output.check - format.url output - new.block - note output - fin.entry -} - -FUNCTION {proceedings} -{ output.bibitem - format.editors output - editor format.key output - new.block - format.btitle "title" output.check - format.bvolume output - format.number.series output - address output - format.date "year" output.check - new.sentence - organization output - publisher output - format.isbn output - format.doi output - format.url output - new.block - note output - fin.entry -} - -FUNCTION {techreport} -{ output.bibitem - format.authors "author" output.check - author format.key output - new.block - format.title "title" output.check - new.block - format.tr.number output.nonnull - institution "institution" output.check - address output - format.date "year" output.check - format.url output - new.block - note output - fin.entry -} - -FUNCTION {unpublished} -{ output.bibitem - format.authors "author" output.check - author format.key output - new.block - format.title "title" output.check - new.block - note "note" output.check - format.date output - format.url output - fin.entry -} - -FUNCTION {default.type} { misc } - - -MACRO {jan} {"January"} - -MACRO {feb} {"February"} - -MACRO {mar} {"March"} - -MACRO {apr} {"April"} - -MACRO {may} {"May"} - -MACRO {jun} {"June"} - -MACRO {jul} {"July"} - -MACRO {aug} {"August"} - -MACRO {sep} {"September"} - -MACRO {oct} {"October"} - -MACRO {nov} {"November"} - -MACRO {dec} {"December"} - - - -MACRO {acmcs} {"ACM Computing Surveys"} - -MACRO {acta} {"Acta Informatica"} - -MACRO {cacm} {"Communications of the ACM"} - -MACRO {ibmjrd} {"IBM Journal of Research and Development"} - -MACRO {ibmsj} {"IBM Systems Journal"} - -MACRO {ieeese} {"IEEE Transactions on Software Engineering"} - -MACRO {ieeetc} {"IEEE Transactions on Computers"} - -MACRO {ieeetcad} - {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"} - -MACRO {ipl} {"Information Processing Letters"} - -MACRO {jacm} {"Journal of the ACM"} - -MACRO {jcss} {"Journal of Computer and System Sciences"} - -MACRO {scp} {"Science of Computer Programming"} - -MACRO {sicomp} {"SIAM Journal on Computing"} - -MACRO {tocs} {"ACM Transactions on Computer Systems"} - -MACRO {tods} {"ACM Transactions on Database Systems"} - -MACRO {tog} {"ACM Transactions on Graphics"} - -MACRO {toms} {"ACM Transactions on Mathematical Software"} - -MACRO {toois} {"ACM Transactions on Office Information Systems"} - -MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"} - -MACRO {tcs} {"Theoretical Computer Science"} - - -READ - -FUNCTION {sortify} -{ purify$ - "l" change.case$ -} - -INTEGERS { len } - -FUNCTION {chop.word} -{ 's := - 'len := - s #1 len substring$ = - { s len #1 + global.max$ substring$ } - 's - if$ -} - -FUNCTION {format.lab.names} -{ 's := - s #1 "{vv~}{ll}" format.name$ - s num.names$ duplicate$ - #2 > - { pop$ " et~al." * } - { #2 < - 'skip$ - { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = - { " et~al." * } - { " et " * s #2 "{vv~}{ll}" format.name$ * } - if$ - } - if$ - } - if$ -} - -FUNCTION {author.key.label} -{ author empty$ - { key empty$ - { cite$ #1 #3 substring$ } - 'key - if$ - } - { author format.lab.names } - if$ -} - -FUNCTION {author.editor.key.label} -{ author empty$ - { editor empty$ - { key empty$ - { cite$ #1 #3 substring$ } - 'key - if$ - } - { editor format.lab.names } - if$ - } - { author format.lab.names } - if$ -} - -FUNCTION {author.key.organization.label} -{ author empty$ - { key empty$ - { organization empty$ - { cite$ #1 #3 substring$ } - { "The " #4 organization chop.word #3 text.prefix$ } - if$ - } - 'key - if$ - } - { author format.lab.names } - if$ -} - -FUNCTION {editor.key.organization.label} -{ editor empty$ - { key empty$ - { organization empty$ - { cite$ #1 #3 substring$ } - { "The " #4 organization chop.word #3 text.prefix$ } - if$ - } - 'key - if$ - } - { editor format.lab.names } - if$ -} - -FUNCTION {calc.short.authors} -{ type$ "book" = - type$ "inbook" = - or - 'author.editor.key.label - { type$ "proceedings" = - 'editor.key.organization.label - { type$ "manual" = - 'author.key.organization.label - 'author.key.label - if$ - } - if$ - } - if$ - 'short.list := -} - -FUNCTION {calc.label} -{ calc.short.authors - short.list - "(" - * - year duplicate$ empty$ - short.list key field.or.null = or - { pop$ "" } - 'skip$ - if$ - * - 'label := -} - -FUNCTION {sort.format.names} -{ 's := - #1 'nameptr := - "" - s num.names$ 'numnames := - numnames 'namesleft := - { namesleft #0 > } - { - s nameptr "{vv{ } }{ll{ }}{ ff{ }}{ jj{ }}" format.name$ 't := - nameptr #1 > - { - " " * - namesleft #1 = t "others" = and - { "zzzzz" * } - { numnames #2 > nameptr #2 = and - { "zz" * year field.or.null * " " * } - 'skip$ - if$ - t sortify * - } - if$ - } - { t sortify * } - if$ - nameptr #1 + 'nameptr := - namesleft #1 - 'namesleft := - } - while$ -} - -FUNCTION {sort.format.title} -{ 't := - "A " #2 - "An " #3 - "The " #4 t chop.word - chop.word - chop.word - sortify - #1 global.max$ substring$ -} - -FUNCTION {author.sort} -{ author empty$ - { key empty$ - { "to sort, need author or key in " cite$ * warning$ - "" - } - { key sortify } - if$ - } - { author sort.format.names } - if$ -} - -FUNCTION {author.editor.sort} -{ author empty$ - { editor empty$ - { key empty$ - { "to sort, need author, editor, or key in " cite$ * warning$ - "" - } - { key sortify } - if$ - } - { editor sort.format.names } - if$ - } - { author sort.format.names } - if$ -} - -FUNCTION {author.organization.sort} -{ author empty$ - { organization empty$ - { key empty$ - { "to sort, need author, organization, or key in " cite$ * warning$ - "" - } - { key sortify } - if$ - } - { "The " #4 organization chop.word sortify } - if$ - } - { author sort.format.names } - if$ -} - -FUNCTION {editor.organization.sort} -{ editor empty$ - { organization empty$ - { key empty$ - { "to sort, need editor, organization, or key in " cite$ * warning$ - "" - } - { key sortify } - if$ - } - { "The " #4 organization chop.word sortify } - if$ - } - { editor sort.format.names } - if$ -} - - -FUNCTION {presort} -{ calc.label - label sortify - " " - * - type$ "book" = - type$ "inbook" = - or - 'author.editor.sort - { type$ "proceedings" = - 'editor.organization.sort - { type$ "manual" = - 'author.organization.sort - 'author.sort - if$ - } - if$ - } - if$ - " " - * - year field.or.null sortify - * - " " - * - cite$ - * - #1 entry.max$ substring$ - 'sort.label := - sort.label * - #1 entry.max$ substring$ - 'sort.key$ := -} - -ITERATE {presort} - -SORT - -STRINGS { longest.label last.label next.extra } - -INTEGERS { longest.label.width last.extra.num number.label } - -FUNCTION {initialize.longest.label} -{ "" 'longest.label := - #0 int.to.chr$ 'last.label := - "" 'next.extra := - #0 'longest.label.width := - #0 'last.extra.num := - #0 'number.label := -} - -FUNCTION {forward.pass} -{ last.label label = - { last.extra.num #1 + 'last.extra.num := - last.extra.num int.to.chr$ 'extra.label := - } - { "a" chr.to.int$ 'last.extra.num := - "" 'extra.label := - label 'last.label := - } - if$ - number.label #1 + 'number.label := -} - -FUNCTION {reverse.pass} -{ next.extra "b" = - { "a" 'extra.label := } - 'skip$ - if$ - extra.label 'next.extra := - extra.label - duplicate$ empty$ - 'skip$ - { "{\natexlab{" swap$ * "}}" * } - if$ - 'extra.label := - label extra.label * 'label := -} - -EXECUTE {initialize.longest.label} - -ITERATE {forward.pass} - -REVERSE {reverse.pass} - -FUNCTION {bib.sort.order} -{ sort.label 'sort.key$ := -} - -ITERATE {bib.sort.order} - -SORT - -FUNCTION {begin.bib} -{ preamble$ empty$ - 'skip$ - { preamble$ write$ newline$ } - if$ - "\begin{thebibliography}{" number.label int.to.str$ * "}" * - write$ newline$ - "\providecommand{\natexlab}[1]{#1}" - write$ newline$ - "\providecommand{\url}[1]{\texttt{#1}}" - write$ newline$ - "\expandafter\ifx\csname urlstyle\endcsname\relax" - write$ newline$ - " \providecommand{\doi}[1]{doi: #1}\else" - write$ newline$ - " \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi" - write$ newline$ -} - -EXECUTE {begin.bib} - -EXECUTE {init.state.consts} - -ITERATE {call.type$} - -FUNCTION {end.bib} -{ newline$ - "\end{thebibliography}" write$ newline$ -} - -EXECUTE {end.bib} +%% +%% Bib. style "plainnat-fr" (version francisee de plainnat.bst) +%% +%% NM, 2005/10/16 +%% markey@lsv.ens-cachan.fr +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Original Copyright (Patrick Daly): +%% +%% Copyright 1993-1999 Patrick W Daly +%% Max-Planck-Institut f\"ur Aeronomie +%% Max-Planck-Str. 2 +%% D-37191 Katlenburg-Lindau +%% Germany +%% E-mail: daly@linmpi.mpg.de +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN +%% archives in directory macros/latex/base/lppl.txt; either +%% version 1 of the License, or any later version. +%% + % Version and source file information: + % \ProvidesFile{natbst.mbs}[1999/05/11 1.6 (PWD)] + % + % BibTeX `plainnat' family + % version 0.99b for BibTeX versions 0.99a or later, + % for LaTeX versions 2.09 and 2e. + % + % For use with the `natbib.sty' package; emulates the corresponding + % member of the `plain' family, but with author-year citations. + % + % With version 6.0 of `natbib.sty', it may also be used for numerical + % citations, while retaining the commands \citeauthor, \citefullauthor, + % and \citeyear to print the corresponding information. + % + % For version 7.0 of `natbib.sty', the KEY field replaces missing + % authors/editors, and the date is left blank in \bibitem. + % + % Includes fields ISBN and ISSN. + % + % Includes field URL for Internet addresses (best used with + % with the url.sty package of Donald Arseneau + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +ENTRY + { address + author + booktitle + chapter + edition + editor + howpublished + institution + isbn + issn + journal + key + month + note + number + organization + pages + publisher + school + series + title + type + url + volume + year + } + {} + { label extra.label sort.label short.list } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% +%% C'est ici que je definis les "traductions". Normalement, y a que ca a changer +%% pour franciser le style... +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +FUNCTION{fr.and}{ %% le "et" entre les deux derniers auteurs +" et " +} + +FUNCTION{fr.editeur}{ %% ", {\'e}diteur" au singulier, ou " ({\'e}diteur)"... +", {\'e}diteur" +} + +FUNCTION{fr.editeurs}{ %% ", {\'e}diteurs" au pluriel, ou " ({\'e}diteurs)"... +", {\'e}diteurs" +} + +FUNCTION{fr.et.al}{ %% " et~al." +" \emph{et~al.}" +} + +FUNCTION{fr.in}{ %% "Dans " +"\emph{In} " +} + +FUNCTION{fr.in.spc}{ %% " dans " +" \emph{in} " +} + +FUNCTION{fr.of}{ %% " de " +" de " +} + +FUNCTION{fr.number}{ %% "num{\'e}ro " +"num{\'e}ro " +} + +FUNCTION{fr.number.maj}{ %% "Num{\'e}ro " +"Num{\'e}ro " +} + +FUNCTION{fr.volume}{ %% "volume" +"volume" +} + +FUNCTION{fr.volume.maj}{ %% "Volume" +"Volume" +} + +FUNCTION{fr.edition}{ %% " {\'e}dition" +" {\'e}dition" +} + +FUNCTION{fr.pages}{ %% "pages" +"pages" +} + +FUNCTION{fr.page}{ %% "page" +"page" +} + +FUNCTION{fr.chapter}{ %% "chapitre" +"chapitre" +} + +FUNCTION{fr.tech.rep}{ %% "Rapport Technique" +"Rapport technique" +} + +FUNCTION{fr.master}{ %% "M{\'e}moire de D.E.A." +"M{\'e}moire de D.E.A." +} + +FUNCTION{fr.phd}{ %% "Th{\`e}se de doctorat" +"Th{\`e}se de doctorat" +} + +FUNCTION{fr.auteurs.style}{ + %% si on veut des small caps sur le LASTNAME de l'auteur + %% Cette fonction est utilisee dans la definition d'une + %% fonction LaTeX. +"\scshape" +} + +FUNCTION{fr.ponctuation.apres.auteurs}{ %% Comme son nom l'indique... +" : " +} + +FUNCTION{fr.deuxpoints}{ %% Pour eviter l'espace avant les ":". +"\string:\penalty500\relax " +} + +MACRO {jan} {"janvier"} +MACRO {feb} {"f\'evrier"} +MACRO {mar} {"mars"} +MACRO {apr} {"avril"} +MACRO {may} {"mai"} +MACRO {jun} {"juin"} +MACRO {jul} {"juillet"} +MACRO {aug} {"ao\^ut"} +MACRO {sep} {"septembre"} +MACRO {oct} {"octobre"} +MACRO {nov} {"novembre"} +MACRO {dec} {"d\'ecembre"} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% +%% La suite, normalement, y a rien a changer... +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +INTEGERS { output.state before.all mid.sentence after.sentence after.block after.authors } + +FUNCTION {init.state.consts} +{ #0 'before.all := + #1 'mid.sentence := + #2 'after.sentence := + #3 'after.block := + #0 'after.authors := +} + +STRINGS { s t } + +FUNCTION{ fr.add.period }{ +after.authors + { fr.ponctuation.apres.auteurs empty$ + { add.period$ } + { fr.ponctuation.apres.auteurs * #0 'after.authors := } + if$} + { add.period$ } +if$ +} + +FUNCTION {output.nonnull} +{ 's := + output.state mid.sentence = + { ", " * write$ } + { output.state after.block = + { fr.add.period write$ + newline$ + "\newblock " write$ + } + { output.state before.all = + 'write$ + { add.period$ " " * write$ } + if$ + } + if$ + mid.sentence 'output.state := + } + if$ + s +} + +FUNCTION {output} +{ duplicate$ empty$ + 'pop$ + 'output.nonnull + if$ +} + +FUNCTION {output.check} +{ 't := + duplicate$ empty$ + { pop$ "empty " t * " in " * cite$ * warning$ } + 'output.nonnull + if$ +} + +FUNCTION {fin.entry} +{ add.period$ + write$ + newline$ +} + +FUNCTION {new.block} +{ output.state before.all = + 'skip$ + { after.block 'output.state := } + if$ +} + +FUNCTION {new.sentence} +{ output.state after.block = + 'skip$ + { output.state before.all = + 'skip$ + { after.sentence 'output.state := } + if$ + } + if$ +} + +FUNCTION {not} +{ { #0 } + { #1 } + if$ +} + +FUNCTION {and} +{ 'skip$ + { pop$ #0 } + if$ +} + +FUNCTION {or} +{ { pop$ #1 } + 'skip$ + if$ +} + +FUNCTION {new.block.checka} +{ empty$ + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.block.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.sentence.checka} +{ empty$ + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {new.sentence.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {field.or.null} +{ duplicate$ empty$ + { pop$ "" } + 'skip$ + if$ +} + +FUNCTION {emphasize} +{ duplicate$ empty$ + { pop$ "" } + { "{\em " swap$ * "}" * } + if$ +} + +INTEGERS { nameptr namesleft numnames } + +FUNCTION {format.names} +{ 's := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr "{ff~}{vv~}\bgroup\fonteauteurs\bgroup {ll}\egroup\egroup{{}}{, jj}" format.name$ 't := + nameptr #1 > + { namesleft #1 > + { ", " * t * } + { %numnames #2 > + % { "," * } + % 'skip$ + %if$ + t "\bgroup\fonteauteurs\bgroup others\egroup\egroup{}" = + { fr.et.al * } + { fr.and * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {format.key} +{ empty$ + { key field.or.null } + { "" } + if$ +} + +FUNCTION {format.authors} +{ author empty$ + { "" } + { author format.names #1 'after.authors := } + if$ +} + +FUNCTION {format.editors} +{ editor empty$ + { "" } + { editor format.names + editor num.names$ #1 > + { fr.editeurs * } + { fr.editeur * } + if$ + } + if$ +} + +FUNCTION {format.isbn} +{ isbn empty$ + { "" } + { new.block "ISBN " isbn * } + if$ +} + +FUNCTION {format.issn} +{ issn empty$ + { "" } + { new.block "ISSN " issn * } + if$ +} + +FUNCTION {format.url} +{ url empty$ + { "" } + { new.block "URL \url{" url * "}" * } + if$ +} + +FUNCTION {format.title} +{ title empty$ + { "" } + { title "t" change.case$ } + if$ +} + +FUNCTION {format.full.names} +{% + % Ici, on laisse les noms en minuscules, on ne les mets pas en small caps. + % + 's := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr + "{vv~}{ll}" format.name$ 't := + nameptr #1 > + { + namesleft #1 > + { ", " * t * } + { + numnames #2 > + { "," * } + 'skip$ + if$ + t "\bgroup\fonteauteurs\bgroup others\egroup\egroup{}" = + { fr.et.al * } + { fr.and * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {author.editor.full} +{ author empty$ + { editor empty$ + { "" } + { editor format.full.names } + if$ + } + { author format.full.names } + if$ +} + +FUNCTION {author.full} +{ author empty$ + { "" } + { author format.full.names } + if$ +} + +FUNCTION {editor.full} +{ editor empty$ + { "" } + { editor format.full.names } + if$ +} + +FUNCTION {make.full.names} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.full + { type$ "proceedings" = + 'editor.full + 'author.full + if$ + } + if$ +} + +FUNCTION {output.bibitem} +{ newline$ + "\bibitem[" write$ + label write$ + ")" make.full.names duplicate$ short.list = + { pop$ } + { * } + if$ + "]{" * write$ + cite$ write$ + "}" write$ + newline$ + "" + before.all 'output.state := +} + +FUNCTION {n.dashify} +{ 't := + "" + { t empty$ not } + { t #1 #1 substring$ "-" = + { t #1 #2 substring$ "--" = not + { "--" * + t #2 global.max$ substring$ 't := + } + { { t #1 #1 substring$ "-" = } + { "-" * + t #2 global.max$ substring$ 't := + } + while$ + } + if$ + } + { t #1 #1 substring$ * + t #2 global.max$ substring$ 't := + } + if$ + } + while$ +} + +FUNCTION {format.date} +{ year duplicate$ empty$ + { "empty year in " cite$ * warning$ + pop$ "" } + 'skip$ + if$ + month empty$ + 'skip$ + { month + " " * swap$ * + } + if$ + extra.label * +} + +FUNCTION {format.btitle} +{ title emphasize +} + +FUNCTION {tie.or.space.connect} +{ duplicate$ text.length$ #3 < + { "~" } + { " " } + if$ + swap$ * * +} + +FUNCTION {either.or.check} +{ empty$ + 'pop$ + { "can't use both " swap$ * " fields in " * cite$ * warning$ } + if$ +} + +FUNCTION {format.bvolume} +{ volume empty$ + { "" } + { fr.volume volume tie.or.space.connect + series empty$ + 'skip$ + { fr.of * series emphasize * } + if$ + "volume and number" number either.or.check + } + if$ +} + +FUNCTION {format.number.series} +{ volume empty$ + { number empty$ + { series field.or.null } + { output.state mid.sentence = + { fr.number } + { fr.number.maj } + if$ + number tie.or.space.connect + series empty$ + { "there's a number but no series in " cite$ * warning$ } + { fr.in.spc * series * } + if$ + } + if$ + } + { "" } + if$ +} + +FUNCTION {format.edition} +{ edition empty$ + { "" } + { output.state mid.sentence = + { edition "l" change.case$ fr.edition * } + { edition "t" change.case$ fr.edition * } + if$ + } + if$ +} + +INTEGERS { multiresult } + +FUNCTION {multi.page.check} +{ 't := + #0 'multiresult := + { multiresult not + t empty$ not + and + } + { t #1 #1 substring$ + duplicate$ "-" = + swap$ duplicate$ "," = + swap$ "+" = + or or + { #1 'multiresult := } + { t #2 global.max$ substring$ 't := } + if$ + } + while$ + multiresult +} + +FUNCTION {format.pages} +{ pages empty$ + { "" } + { pages multi.page.check + { fr.pages pages n.dashify tie.or.space.connect } + { fr.page pages tie.or.space.connect } + if$ + } + if$ +} + +FUNCTION {format.vol.num.pages} +{ volume field.or.null + number empty$ + 'skip$ + { "\penalty0 (" number * ")" * * + volume empty$ + { "there's a number but no volume in " cite$ * warning$ } + 'skip$ + if$ + } + if$ + pages empty$ + 'skip$ + { duplicate$ empty$ + { pop$ format.pages } + { fr.deuxpoints * pages n.dashify * } + if$ + } + if$ +} + +FUNCTION {format.chapter.pages} +{ chapter empty$ + 'format.pages + { type empty$ + { fr.chapter } + { type "l" change.case$ } + if$ + chapter tie.or.space.connect + pages empty$ + 'skip$ + { ", " * format.pages * } + if$ + } + if$ +} + +FUNCTION {format.in.ed.booktitle} +{ booktitle empty$ + { "" } + { editor empty$ + { fr.in booktitle emphasize * } + { fr.in format.editors #1 'after.authors := fr.add.period " " * * booktitle emphasize * } + if$ + } + if$ +} + +FUNCTION {empty.misc.check} +{ author empty$ title empty$ howpublished empty$ + month empty$ year empty$ note empty$ + and and and and and + key empty$ not and + { "all relevant fields are empty in " cite$ * warning$ } + 'skip$ + if$ +} + +FUNCTION {format.thesis.type} +{ type empty$ + 'skip$ + { pop$ + type "t" change.case$ + } + if$ +} + +FUNCTION {format.tr.number} +{ type empty$ + { fr.tech.rep } + 'type + if$ + number empty$ + { "t" change.case$ } + { number tie.or.space.connect } + if$ +} + +FUNCTION {format.article.crossref} +{ key empty$ + { journal empty$ + { "need key or journal for " cite$ * " to crossref " * crossref * + warning$ + "" + } + { fr.in "{\em " * journal * "\/}" * } + if$ + } + { fr.in key * } + if$ + " \citep{" * crossref * "}" * +} + +FUNCTION {format.book.crossref} +{ volume empty$ + { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ + fr.in + } + { fr.volume.maj volume tie.or.space.connect + fr.of * + } + if$ + editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { series empty$ + { "need editor, key, or series for " cite$ * " to crossref " * + crossref * warning$ + "" * + } + { "{\em " * series * "\/}" * } + if$ + } + { key * } + if$ + } + 'skip$ + if$ + ", \citet{" * crossref * "}" * +} + +FUNCTION {format.incoll.inproc.crossref} +{ editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { booktitle empty$ + { "need editor, key, or booktitle for " cite$ * " to crossref " * + crossref * warning$ + "" + } + { fr.in "{\em " * booktitle * "\/}" * } + if$ + } + { fr.in key * } + if$ + } + { fr.in } + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {article} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { journal emphasize "journal" output.check + format.vol.num.pages output + format.date "year" output.check + } + { format.article.crossref output.nonnull + format.pages output + } + if$ + format.issn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {book} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {booklet} +{ output.bibitem + format.authors output + author format.key output + new.block + format.title "title" output.check + howpublished address new.block.checkb + howpublished output + address output + format.date output + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inbook} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + format.chapter.pages "chapter and pages" output.check + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { format.chapter.pages "chapter and pages" output.check + new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {incollection} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.chapter.pages output + new.sentence + publisher "publisher" output.check + address output + format.edition output + format.date "year" output.check + } + { format.incoll.inproc.crossref output.nonnull + format.chapter.pages output + } + if$ + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inproceedings} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.pages output + address empty$ + { organization publisher new.sentence.checkb + organization output + publisher output + format.date "year" output.check + } + { address output.nonnull + format.date "year" output.check + new.sentence + organization output + publisher output + } + if$ + } + { format.incoll.inproc.crossref output.nonnull + format.pages output + } + if$ + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {conference} { inproceedings } + +FUNCTION {manual} +{ output.bibitem + format.authors output + author format.key output + new.block + format.btitle "title" output.check + organization address new.block.checkb + organization output + address output + format.edition output + format.date output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {mastersthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + fr.master format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {misc} +{ output.bibitem + format.authors output + author format.key output + title howpublished new.block.checkb + format.title output + howpublished new.block.checka + howpublished output + format.date output + format.issn output + format.url output + new.block + note output + fin.entry + empty.misc.check +} + +FUNCTION {phdthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.btitle "title" output.check + new.block + fr.phd format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {proceedings} +{ output.bibitem + format.editors output + editor format.key output + new.block + format.btitle "title" output.check + format.bvolume output + format.number.series output + address output + format.date "year" output.check + new.sentence + organization output + publisher output + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {techreport} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + format.tr.number output.nonnull + institution "institution" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {unpublished} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + format.url output + new.block + note "note" output.check + format.date output + fin.entry +} + +FUNCTION {default.type} { misc } + +MACRO {acmcs} {"ACM Computing Surveys"} +MACRO {acta} {"Acta Informatica"} +MACRO {cacm} {"Communications of the ACM"} +MACRO {ibmjrd} {"IBM Journal of Research and Development"} +MACRO {ibmsj} {"IBM Systems Journal"} +MACRO {ieeese} {"IEEE Transactions on Software Engineering"} +MACRO {ieeetc} {"IEEE Transactions on Computers"} +MACRO {ieeetcad} {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"} +MACRO {ipl} {"Information Processing Letters"} +MACRO {jacm} {"Journal of the ACM"} +MACRO {jcss} {"Journal of Computer and System Sciences"} +MACRO {scp} {"Science of Computer Programming"} +MACRO {sicomp} {"SIAM Journal on Computing"} +MACRO {tocs} {"ACM Transactions on Computer Systems"} +MACRO {tods} {"ACM Transactions on Database Systems"} +MACRO {tog} {"ACM Transactions on Graphics"} +MACRO {toms} {"ACM Transactions on Mathematical Software"} +MACRO {toois} {"ACM Transactions on Office Information Systems"} +MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"} +MACRO {tcs} {"Theoretical Computer Science"} + + +READ + +FUNCTION {sortify} +{ purify$ + "l" change.case$ +} + +INTEGERS { len } + +FUNCTION {chop.word} +{ 's := + 'len := + s #1 len substring$ = + { s len #1 + global.max$ substring$ } + 's + if$ +} + +FUNCTION {format.lab.names} +% +% Ici, on laisse les noms en minuscules, on ne les mets pas +% en small caps. +% +{ 's := + s #1 "{vv~}{ll}" format.name$ + s num.names$ duplicate$ + #2 > + { pop$ fr.et.al * } + { #2 < + 'skip$ + { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = + { fr.et.al * } + { fr.and * s #2 "{vv~}{ll}" format.name$ * } + if$ + } + if$ + } + if$ +} + +FUNCTION {author.key.label} +{ author empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.editor.key.label} +{ author empty$ + { editor empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { editor format.lab.names } + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.key.organization.label} +{ author empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "Le " #3 + "La " #3 + "Les " #4 + "The " #4 organization chop.word + chop.word + chop.word + chop.word + #3 text.prefix$ } + if$ + } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {editor.key.organization.label} +{ editor empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "Le " #3 + "La " #3 + "Les " #4 + "The " #4 organization chop.word + chop.word + chop.word + chop.word + #3 text.prefix$ } + if$ + } + 'key + if$ + } + { editor format.lab.names } + if$ +} + +FUNCTION {calc.short.authors} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.key.label + { type$ "proceedings" = + 'editor.key.organization.label + { type$ "manual" = + 'author.key.organization.label + 'author.key.label + if$ + } + if$ + } + if$ + 'short.list := +} + +FUNCTION {calc.label} +{ calc.short.authors + short.list + "(" + * + year duplicate$ empty$ + short.list key field.or.null = or + { pop$ "" } + 'skip$ + if$ + * + 'label := +} + +FUNCTION {sort.format.names} +{ 's := + #1 'nameptr := + "" + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { nameptr #1 > + { " " * } + 'skip$ + if$ + s nameptr "{vv{ } }{ll{ }}{ ff{ }}{ jj{ }}" format.name$ 't := + nameptr numnames = t "others" = and + { fr.et.al * } %% sera "purify"e ensuite... + { t sortify * } + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {sort.format.title} +{ 't := + "Un " #3 + "Une " #4 + "Des " #4 + "Le " #3 + "La " #3 + "Les " #4 + "A " #2 + "An " #3 + "The " #4 t chop.word + chop.word + chop.word + chop.word + chop.word + chop.word + chop.word + chop.word + chop.word + sortify + #1 global.max$ substring$ +} + +FUNCTION {author.sort} +{ author empty$ + { key empty$ + { "to sort, need author or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.editor.sort} +{ author empty$ + { editor empty$ + { key empty$ + { "to sort, need author, editor, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { editor sort.format.names } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.organization.sort} +{ author empty$ + { organization empty$ + { key empty$ + { "to sort, need author, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "Le " #3 + "La " #3 + "Les " #4 + "The " #4 organization chop.word + chop.word + chop.word + chop.word + sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {editor.organization.sort} +{ editor empty$ + { organization empty$ + { key empty$ + { "to sort, need editor, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "Le " #3 + "La " #3 + "Les " #4 + "The " #4 organization chop.word + chop.word + chop.word + chop.word + sortify } + if$ + } + { editor sort.format.names } + if$ +} + +FUNCTION {presort} +{ calc.label + label sortify + " " + * + type$ "book" = + type$ "inbook" = + or + 'author.editor.sort + { type$ "proceedings" = + 'editor.organization.sort + { type$ "manual" = + 'author.organization.sort + 'author.sort + if$ + } + if$ + } + if$ + " " + * + year field.or.null sortify + * + " " + * + title field.or.null + sort.format.title + * + #1 entry.max$ substring$ + 'sort.label := + sort.label * + #1 entry.max$ substring$ + 'sort.key$ := +} + +ITERATE {presort} + +SORT + +STRINGS { longest.label last.label next.extra } + +INTEGERS { longest.label.width last.extra.num number.label } + +FUNCTION {initialize.longest.label} +{ "" 'longest.label := + #0 int.to.chr$ 'last.label := + "" 'next.extra := + #0 'longest.label.width := + #0 'last.extra.num := + #0 'number.label := +} + +FUNCTION {forward.pass} +{ last.label label = + { last.extra.num #1 + 'last.extra.num := + last.extra.num int.to.chr$ 'extra.label := + } + { "a" chr.to.int$ 'last.extra.num := + "" 'extra.label := + label 'last.label := + } + if$ + number.label #1 + 'number.label := +} + +FUNCTION {reverse.pass} +{ next.extra "b" = + { "a" 'extra.label := } + 'skip$ + if$ + extra.label 'next.extra := + extra.label + duplicate$ empty$ + 'skip$ + { "{\natexlab{" swap$ * "}}" * } + if$ + 'extra.label := + label extra.label * 'label := +} + +EXECUTE {initialize.longest.label} + +ITERATE {forward.pass} + +REVERSE {reverse.pass} + +FUNCTION {bib.sort.order} +{ sort.label 'sort.key$ := +} + +ITERATE {bib.sort.order} + +SORT + +FUNCTION {begin.bib} +{ preamble$ empty$ + 'skip$ + { preamble$ write$ newline$ } + if$ + "\begin{thebibliography}{" number.label int.to.str$ * "}" * + write$ newline$ + "\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi" + write$ newline$ + "\expandafter\ifx\csname fonteauteurs\endcsname\relax" + write$ newline$ + "\def\fonteauteurs{" fr.auteurs.style * "}\fi" * write$ newline$ + "\expandafter\ifx\csname url\endcsname\relax" write$ newline$ + " \def\url#1{{\tt #1}}% " write$ newline$ + " \message{You should include the url package}\fi" write$ newline$ +} + +EXECUTE {begin.bib} + +EXECUTE {init.state.consts} + +ITERATE {call.type$} + +FUNCTION {end.bib} +{ newline$ + "\end{thebibliography}" write$ newline$ +} + +EXECUTE {end.bib} diff --git a/depotfinal/908144032/remerciements.tex b/depotfinal/908144032/remerciements.tex new file mode 100644 index 0000000..99b5697 --- /dev/null +++ b/depotfinal/908144032/remerciements.tex @@ -0,0 +1,17 @@ +\chapter*{Remerciements} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Remerciements} % inclure dans TdM + +Je dois avouer que je ne savais pas trop à quoi m'attendre au moment où j'ai commencé la rédaction de ce mémoire. En essayant de ne pas oublier personne, j'aimerais tout d'abord remercier celui qui m'a accompagné durant ces deux dernières années et qui m'a énormément appris, le professeur Andrew Luong. + +J'aimerais aussi remercier mes évaluateurs, les professeurs Claire Bilodeau et Julien Trufin. J'aimerais aussi remercier le professeur Vincent Goulet, qui m'a fortement encouragé de m'inscrire à la maîtrise, et pour les nombreuses références qu'il a pu me fournir, pour la rédaction entre autres, qui m'ont grandement facilité la tâche. Je remercie tout autant le professeur Ghislain Léveillé pour son soutien et pour m'avoir offert la possibilité de présenter mes résultats. + +J'aimerais aussi remercier l'ensemble des étudiants gradués, des professeurs et du personnel de l'École d'Actuariat de l'Université Laval, ainsi que mes collègues de travail de la CARRA, pour leur soutien et leurs encouragements. + +Je suis reconnaissant envers l'École d'Actuariat et la Chaire d'Actuariat de l'Université Laval pour le soutien financier en m'offrant des postes d'auxiliaires d'enseignement et de recherche. + +Enfin, j'aimerais remercier mes parents, ma soeur et l'ensemble de ma famille et de mes amis pour avoir accepté que je passe plus de temps à «écrire des formules bizarres sur l'ordinateur» qu'à être avec eux. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/resume.tex b/depotfinal/908144032/resume.tex new file mode 100644 index 0000000..048ba9b --- /dev/null +++ b/depotfinal/908144032/resume.tex @@ -0,0 +1,19 @@ +\chapter*{Résumé} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Résumé} % inclure dans TdM + +\begin{otherlanguage*}{francais} + Les modèles classiques en finance sont basés sur des + hypothèses qui ne sont pas toujours vérifiables + empiriquement. L'objectif de ce mémoire est de présenter + l'utilisation de la distribution de Laplace asymétrique généralisée + comme une alternative intéressante à ces derniers. Pour ce faire, on + utilise ses différentes propriétés afin de développer des méthodes + d'estimation paramétrique, d'approximation et de test, en plus d'élaborer quelques principes d'évaluation de produits dérivés. On + présente enfin un exemple d'application numérique afin d'illustrer ces + différents concepts. +\end{otherlanguage*} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "gabarit-maitrise" +%%% End: diff --git a/depotfinal/908144032/sa.pdf b/depotfinal/908144032/sa.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f8a641fb5e59dfd60dee8903162abd6ea67fbc85 GIT binary patch literal 1732 zcmY!laB;*OT3jMEMO5Qfp37DU>G2%^m<-~H$KW^vu*V{jB|1&lJzP(Rf{gbHTwa(!! zE*aW-`Inq_-xjS7?N(m(&FpCVuQfZjTzaaO<6r#HPg6Pm$C>8`?@2b;zCJVi_Zl(N z?Jl$TcACD;ZvM6_W!3t?u0qS=XW-q^ZT z&vE~$+R64o>L#;Ut}pnmCVIok`P>aFmfH`FmkO_rns{$^pVSnih}c^}(!oK=Os}Mt z+&;LOF+IcT3}4(u{Ry&+4JLOE23`0t?e^R+&dE_lk*s+u?B!g_8+z7VyIwfYQE|eI z@O1IHd;z|ToZ9C3<<8ZK&Az{tb&n%|wOe%FEzhTwvwpDFf7tY}Z7E--R{X~`_LmpE zjg^hFDX|H)eW$hToS2nbn^SXc$qp0m`l*eQ?ce;8BMWxu&fRFF-g_`UM3DV$9ojqaAJH2V&yEkgA^8{bq z+jKs>=VQZz)bor{Ax8WR-GX1|vT!k!otHE?vh(J)&yW|#nx26y zP&NRieGqMKUCO3f=#Foxuopw#00(xT+lVg)mZfOCFc 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\renewcommand*{\UL@extensionat}{Universit\'e du Qu\'ebec \`a Chicoutimi} + \renewcommand*{\UL@extensionloc}{Chicoutimi, Qu\'ebec}} +\newcommand{\faculteUL}[1]{\renewcommand*{\UL@facUL}{#1}} +\newcommand*{\faculteUdeS}[1]{\renewcommand*{\UL@facother}{#1}} +\newcommand*{\faculteUQO}[1]{\renewcommand*{\UL@facother}{#1}} +\newcommand*{\faculteUQAC}[1]{\renewcommand*{\UL@facother}{#1}} +\newcommand{\UL@title}{% + {\UL@fonttitle\UL@maintitle\par} + {\UL@fontsubtitle + \ifthenelse{\boolean{UL@hassubtitle}}{\UL@subtitle}{% + \vspace*{\baselineskip}}\par}} +\newcommand{\UL@docid}{% + {\UL@fontprogram\UL@typeofdoc\par + \ifnum\UL@typenum=2 \UL@program\par \fi}} +\newcommand{\UL@details}{% + \ifcase\UL@typenum\relax% 0 standard + \vspace{96pt} + {\UL@fontprogram\UL@program}\par + \UL@degree\par + \vspace{112pt} + Qu\'ebec, Canada\par + \or% 1 multifacultaire + \vspace{96pt} + {\UL@fontprogram\UL@program}\par + \UL@degree\par + \vspace{36pt} + \UL@facUL\par + \vspace{48pt} + Qu\'ebec, Canada\par + \or% 2 cotutelle + \vspace{72pt} + Universit\'e Laval\par Qu\'ebec, Canada\par + \UL@degree\par + \vspace{\baselineskip} et\par \vspace{\baselineskip} + \UL@nameother\par + \UL@degreeother\par + \or% 3 extension + \vspace{48pt} + {\UL@fontprogram\UL@program\ de l'Universit\'e Laval\par + \ifthenelse{\boolean{UL@isprogmasc}}{offert}{offerte} + en extension \`a l'\UL@extensionat}\par + \vspace{36pt} + \UL@degree\par + \vspace{36pt} + \UL@facother\par \UL@extensionat\par \UL@extensionloc\par + \vspace{\baselineskip} + \UL@facUL\par Universit\'e Laval\par Qu\'ebec, Canada\par + \fi} +\newcommand{\pagetitre}{{% + \clearpage + \thispagestyle{empty} + \SingleSpacing\setlength{\parskip}{0pt} + \centering + \UL@fontbase + \ifthenelse{\UL@typenum > 1}{\vspace*{0pt}\par}{% + \includegraphics[height=40px,keepaspectratio=true]{ul_p}\par} + \vspace{72pt} + \UL@title + \vspace{48pt} + \UL@docid + \vspace{72pt} + {\UL@fontauthor\UL@author}\par + \UL@details + \vfill + {\textcopyright} \UL@author, \UL@year\par + \cleardoublepage + }} +\addto\captionsfrench{\renewcommand{\listfigurename}{Liste des figures}} +\newcommand{\dedicace}[1]{{% + \clearpage + \pagestyle{empty} + \setlength{\beforeepigraphskip}{10\baselineskip} + \setlength{\epigraphrule}{0pt} + \epigraphtextposition{flushright} + \mbox{}\epigraph{\itshape #1}{} + }} +\newcommand{\epigraphe}[2]{{% + \clearpage + \pagestyle{empty} + \setlength{\beforeepigraphskip}{10\baselineskip} + \mbox{}\epigraph{#1}{#2} + }} +\renewenvironment{quote}{% + \list{}{\rightmargin 10mm \leftmargin 10mm}% + \item[]}{\endlist} +\renewenvironment{quotation}{% + \list{}{% + \SingleSpacing + \listparindent 0em + \itemindent \listparindent + \leftmargin 10mm + \rightmargin \leftmargin + \parsep 6\p@ \@plus\p@}% + \item[]}{\endlist} +\setsecnumdepth{subsection} +\endinput +%% +%% End of file `ulthese.cls'. diff --git a/depotfinal/908144032/ulthese.dtx b/depotfinal/908144032/ulthese.dtx new file mode 100644 index 0000000..2e88725 --- /dev/null +++ b/depotfinal/908144032/ulthese.dtx @@ -0,0 +1,1608 @@ +% \iffalse meta-comment +% +% Copyright (C) 2012 Universite Laval +% +% This file may be distributed and/or modified under the conditions +% of the LaTeX Project Public License, either version 1.3c of this +% license or (at your option) any later version. The latest version +% of this license is in: +% +% http://www.latex-project.org/lppl.txt +% +% and version 1.3c or later is part of all distributions of LaTeX +% version 2006/05/20 or later. +% +% This work has the LPPL maintenance status `maintained'. +% +% The Current Maintainer of this work is Universite Laval +% . +% +% This work consists of the files ulthese.dtx and ulthese.ins +% and the derived files listed in the README file. +% +% \fi +% +% \iffalse +%<*dtx> +\ProvidesFile{ulthese.dtx} +% +%\NeedsTeXFormat{LaTeX2e}[2009/09/24] +%\ProvidesClass{ulthese}% +%<*class> + [2012/09/30 v1.0 Classe pour les theses et memoires de l'Universite Laval] +% +%<*driver> +\documentclass[11pt]{ltxdoc} + \usepackage[utf8]{inputenc} + \usepackage[T1]{fontenc} + \usepackage{natbib} + \usepackage[francais]{babel} + \usepackage[autolanguage]{numprint} + \usepackage{mathpazo} + \usepackage[scaled=0.92]{helvet} + \usepackage{color,booktabs} + \EnableCrossrefs + \CodelineIndex + \RecordChanges + + \definecolor{doclinkcolor}{rgb}{0,0,0.5} + + \usepackage{hyperref} + \hypersetup{colorlinks,allcolors=doclinkcolor} + + \frenchbsetup{% + CompactItemize=false, % ne pas compacter les listes + ThinSpaceInFrenchNumbers=true % espace fine dans les nombres + } +\begin{document} + \DocInput{ulthese.dtx} +\end{document} +% +% \fi +% \CheckSum{568} +% \DoNotIndex{\',\^,\`,\ ,\ae} +% \DoNotIndex{\RequirePackage,\ExecuteOptions,\ifthenelse,\ProcessOptions} +% \DoNotIndex{\newcommand,\newcommand*,\newboolean} +% \DoNotIndex{\setlength,\setboolean} +% \changes{1.0}{2012-09-30}{Version initiale} +% \GetFileInfo{ulthese.dtx} +% \title{\textsf{ulthese}: une classe pour les thèses et mémoires de +% l'Université Laval\thanks{Ce document décrit la classe +% \textsf{ulthese}~\fileversion, datée du \filedate.}} +% \author{Faculté des études supérieures et postdoctorales\thanks{% +% Cette classe et sa documentation ont été rédigés par Vincent +% Goulet~(Faculté des sciences et de génie) avec la collaboration de +% Koassi D'Almeida~(Faculté des études supérieures et postdoctorales) et +% Pierre Lasou~(Bibliothèque).}} +% \maketitle +% \tableofcontents +% +% \section{Introduction} +% +% La classe \textsf{ulthese} permet de composer avec {\LaTeX} des thèses +% et mémoires immédiatement conformes aux règles générales de +% présentation matérielle de la Faculté des études supérieures et +% postdoctorales (FESP) de l'Université Laval. Ces règles définissent +% principalement la présentation de la page titre des thèses et mémoires +% ainsi que la disposition du texte sur la page. La classe en elle-même +% est donc relativement simple. +% +% Cependant, la classe \textsf{ulthese} est basée sur la classe +% \textsf{memoir}, une extension de la classe standard \textsf{book} +% facilitant à plusieurs égards la préparation de documents d'allure +% professionnelle dans {\LaTeX}. La classe \textsf{memoir} est très +% configurable et incorpore d'office plus de 30 des paquetages +% (\emph{packages}) les plus populaires\footnote{% +% Consulter la section~18.24 de la documentation de \textsf{memoir} +% pour la liste ou encore le fichier journal (\emph{log}) de la +% compilation d'un document utilisant la classe \textsf{ulthese}.}. % +% L'intégralité des fonctionnalités de \textsf{memoir} est disponible +% dans \textsf{ulthese}. +% +% La classe \textsf{memoir} fait maintenant partie des distributions +% {\LaTeX} modernes; elle devrait donc être installée et disponible sur +% votre système. La classe est livrée avec une documentation exhaustive: +% le manuel d'instructions fait près de 600~pages! Il peut être utile de +% s'y référer de temps à autre pour réaliser une mise en page +% particulière. Rechercher sur votre système le fichier |memman.pdf| ou +% le % +% \href{http://mirrors.ctan.org/macros/latex/contrib/memoir/memman.pdf}{% +% consulter en ligne} % +% sur le site % +% \href{http://www.ctan.org/}{% +% \emph{Comprehensive R Archive Network} (CTAN)}. +% +% +% \section{Installation} +% +% L'installation de la classe consiste à créer le fichier +% |ulthese.cls| et plusieurs gabarits |.tex| à partir du code source +% documenté se trouvant dans le fichier |ulthese.dtx|. Il est +% recommandé de simplement créer ces fichiers dans le dossier de +% travail de la thèse ou du mémoire. +% +% Pour procéder à l'installation, décompresser l'archive |ulthese.zip| +% dans le dossier de travail, puis compiler avec {\LaTeX} le fichier +% |ulthese.ins| en exécutant +% \begin{quote} +% |latex ulthese.ins| +% \end{quote} +% depuis une invite de commande. Si l'on est peu familier avec +% l'invite de commande, on peut aussi procéder comme avec tout +% document {\LaTeX}, soit ouvrir le fichier |ulthese.ins| dans son +% éditeur de texte favori et lancer la compilation avec {\LaTeX}, +% pdf{\TeX} ou pdf{\LaTeX} depuis celui-ci. +% +% \section{Utilisation} +% +% La classe est chargée avec la commande +% \begin{quote} +% |\documentclass[|\meta{options}|]{ulthese}| +% \end{quote} +% Les marges, l'interligne et la numérotation des pages sont adaptées +% aux règles de présentation matérielle de la FESP. Les options et les +% commandes définies par la classe sont décrites dans les sections +% suivantes. +% +% \subsection{Options de la classe} +% \label{sec:utilisation:options} +% +% Les \meta{options} que l'on peut spécifier au chargement de la classe +% sont les suivantes: +% \begin{description} +% \item[|10pt|] sélectionne une taille de police de 10~points; +% \item[|11pt|] sélectionne une taille de police de 11~points; +% \item[|12pt|] sélectionne une taille de police de 12~points; +% \item[|nonatbib|] empêche de charger le paquetage \textsf{natbib} (voir +% la section~\ref{sec:bibliographystyle}); +% \item[|english, francais, ...|] langues utilisées dans le document; +% \item[\mdseries\meta{options memoir}] autres options du paquetage +% \textsf{memoir}. +% \end{description} +% +% Si aucune taille de police n'est précisée, la classe utilisera une +% taille par défaut de 11~points. La taille des polices de la page +% titre n'est pas affectée par les options |10pt|, |11pt| et |12pt|. +% +% Le paquetage \textsf{natbib} est normalement chargé par la classe. +% L'option |nonatbib| permet d'empêcher le chargement en cas de +% conflit avec un autre paquetage de mise en forme de la +% bibliographie. +% +% Les langues sont passées au paquetage \textsf{babel}; voir la +% section~\ref{sec:langues}. Leur libellé devrait donc correspondre +% aux options de ce paquetage. La dernière langue spécifiée est la +% langue active par défaut dans le document. +% +% Toute autre option sera passée à la classe \textsf{memoir} dont, +% entre autres, le format du papier. Le format lettre nord-américain +% (option |letterpaper|) est utilisé par défaut. Si la thèse doit être +% imprimée en format international A4, utiliser l'option |a4paper|. La +% classe \textsf{memoir} est toujours chargée avec les options +% |twoside|, et |openright|. +% +% \subsection{Nouvelles commandes} +% +% La classe \textsf{ulthese} définit quelques nouvelles commandes +% servant principalement à créer la page titre et des éléments des pages +% liminaires. +% +% \subsubsection{Commandes de la page titre} +% \label{sec:commandes-pagetitre} +% +% Les commandes ci-dessous servent à définir les divers éléments de la +% page titre et leur disposition sur la page. +% +% \begin{DescribeMacro}{\titre} +% Titre principal de la thèse ou du mémoire. Ne pas utiliser la +% commande |\title| de {\LaTeX} pour ce faire. +% +% Un titre très long devra être coupé manuellement avec |\\| ou +% |\newline|. Dans un tel cas, il est également nécessaire de +% retrancher de l'espacement vertical pour éviter que les autres +% éléments de la page titre ne soient poussés vers le bas. Pour ce +% faire, ajouter la déclaration +% \begin{quote} +% |\vspace*{-|$n$|\baselineskip}| +% \end{quote} +% à la fin de la \emph{dernière} ligne de titre s'il y a $n + 1$ +% lignes de titre au total. +% +% Par exemple, la déclaration d'un titre d'une seule ligne est: +% \begin{quote} +% |\titre{Ceci est un titre d'une seule ligne}| +% \end{quote} +% Pour un titre de deux lignes, on écrira: +% \begin{quote} +% |\titre{Ceci est la première ligne d'un long titre \\| \\ +% | et ceci est la seconde \vspace*{-1\baselineskip}}| +% \end{quote} +% Pour un titre de trois lignes, c'est alors: +% \begin{quote} +% |\titre{Ceci est la première ligne d'un très long titre \\| \\ +% | pour lequel deux lignes ne suffisent pas \\| \\ +% | alors voici la troisième \vspace*{-2\baselineskip}}| +% \end{quote} +% Ainsi de suite. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\soustitre} +% Sous-titre de la thèse ou du mémoire, le cas échéant. Les remarques +% sur un long titre principal s'appliquent également au sous-titre. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\auteur} +% Nom complet de l'auteur de la thèse ou du mémoire, sous la forme +% |Prénom Nom| avec seulement des majuscules initiales. Ne pas +% utiliser la commande |\author| de {\LaTeX} pour le nom de l'auteur. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\annee} +% Année du dépôt final de la thèse ou du mémoire. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\programme} +% Nom du complet officiel du programme d'études comme «Doctorat en +% informatique» ou «Maîtrise en mathématiques». Si le programme +% comporte une majeure, séparer sa mention de celle du programme +% principal par un tiret demi-quadratin (obtenu avec |--|). +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\PhD,} +% \begin{DescribeMacro}{\MSc,} +% \begin{DescribeMacro}{\MA, \dots} +% Déclaration du type de grade auquel donne droit la thèse ou le +% mémoire. Consulter le tableau~\ref{tab:grades} pour la liste +% complète des commandes et des grades correspondants. +% \end{DescribeMacro} +% \end{DescribeMacro} +% \end{DescribeMacro} +% +% \begin{table} +% \centering +% \begin{tabular}{lp{9cm}} +% \toprule +% Commande & Nom du grade (sigle) \\ +% \midrule +% |\LLD|\index{LLD=\verb+\LLD+} +% & Docteur en droit (L.L.D.) \\ +% |\DPsy|\index{DPsy=\verb+\DPsy+} +% & Docteur en psychologie (D.Psy.) \\ +% |\DThP|\index{DThP=\verb+\DThP+} +% & Docteur en théologie pratique (D.Th.P.) \\ +% |\PhD|\index{PhD=\verb+\PhD+} +% & Philosophi{\ae} doctor (Ph.D.) \\ +% \addlinespace[6pt] +% |\LLM|\index{LLM=\verb+\LLM+} +% & Maître en droit (L.L.M.) \\ +% |\MA|\index{MA=\verb+\MA+} +% & Maître ès arts (M.A.) \\ +% |\MMus|\index{MMus=\verb+\MMus+} +% & Maître en musique (M.Mus.) \\ +% |\MSc|\index{MSc=\verb+\MSc+} +% & Maître ès sciences (M.Sc.) \\ +% |\MServSoc|\index{MServSoc=\verb+\MServSoc+} +% & Maître en service social (M.Serv.Soc.) \\ +% |\MScGeogr|\index{MScGeogr=\verb+\MScGeogr+} +% & Maître en sciences géographiques (M.Sc.G\'eogr.) \\ +% |\MATDR|\index{MATDR=\verb+\MATDR+} +% & Maître en aménagement du territoire et développement régional +% (M.ATDR) \\ +% \bottomrule +% \end{tabular} +% \caption{Commandes de déclaration du grade et libellés correspondants} +% \label{tab:grades} +% \end{table} +% +% \begin{DescribeMacro}{\multifacultaire} +% Déclaration d'une thèse ou d'un mémoire multifacultaire. Requiert +% d'utiliser aussi la commande |\faculteUL| ci-dessous. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\cotutelle} +% Déclaration d'une thèse ou d'un mémoire réalisé en cotutelle avec +% une autre université. Requiert d'utiliser aussi les commandes +% |\univcotutelle| et |\gradecotutelle| ci-dessous. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\univcotutelle} +% Nom, ville et pays de l'université de cotutelle, déclarés sous la +% forme +% \begin{quote} +% |\univcotutelle{Nom de l'université \\ Ville, Pays}| +% \end{quote} +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\gradecotutelle} +% Grade conféré par l'université de cotutelle, déclaré sous la forme +% \begin{quote} +% |\gradecotutelle{Nom du grade (sigle)}| +% \end{quote} +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\extensionUdeS} +% Déclaration que la thèse est réalisée en extension à l'Université +% de Sherbrooke. Requiert d'utiliser aussi les commandes +% |\faculteUL| et |\faculteUdeS| ci-dessous. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\extensionUQO} +% Déclaration que la thèse est réalisée en extension à l'Université du +% Québec en Outaouais. Requiert d'utiliser aussi les commandes +% |\faculteUL| et |\faculteUQO| ci-dessous. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\extensionUQAC} +% Déclaration que le mémoire est réalisé en extension à l'Université +% du Québec à Chicoutimi. Requiert d'utiliser aussi les commandes +% |\faculteUL| et |\faculteUQAC| ci-dessous. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\faculteUL} +% Cette macro a deux usages: +% \begin{enumerate} +% \item noms des facultés pour les thèses et mémoires +% multifacultaires, séparés par des commandes |\\|; +% \item nom de la faculté de l'Université Laval où sont réalisés les +% thèses et mémoires en extension aux universités de Sherbrooke, du +% Québec en Outaouais ou du Québec à Chicoutimi. +% \end{enumerate} +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\faculteUdeS} +% Nom de la faculté de l'Université de Sherbrooke hébergeant la thèse +% en extension. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\faculteUQO} +% Nom de la faculté de l'Université du Québec en Outaouais hébergeant +% la thèse en extension. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\faculteUQAC} +% Nom de la faculté de l'Université du Québec à Chicoutimi hébergeant +% le mémoire en extension. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\pagetitre} +% Déclaration de création de la page titre. Ne pas utiliser la +% commande |\pagetitle| de {\LaTeX} pour ce faire. De toutes les +% commandes ci-dessus, c'est la seule qui doit se trouver dans le +% corps du document plutôt que dans le préambule. +% \end{DescribeMacro} +% +% \subsubsection{Commandes des pages liminaires} +% +% La classe définit deux commandes pour créer des pages liminaires +% prévues aux règles de présentation matérielle. +% +% \begin{DescribeMacro}{\dedicace} +% La commande |\dedicace| ajoute une dédicace («À mes parents», «À +% Camille») à la thèse ou au mémoire. La dédicace est disposée seule +% sur une page recto, à une dizaine de lignes de la marge du haut et +% alignée à droite. Par défaut, elle est composée en italique. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\epigraphe} +% La commande |\epigraph| sert à ajouter une épigraphe au début du +% document. Comme la dédicace, l'épigraphe est disposée seule sur une +% page recto, à une dizaine de lignes de la marge du haut et alignée à +% droite. La commande accepte deux arguments, soit le texte de la +% citation et son auteur ou la source, dans l'ordre. +% \end{DescribeMacro} +% +% Pour ajouter une épigraphe au début d'un ou plusieurs chapitres, +% utiliser directement la commande |\epigraph| de \textsf{memoir}, sur +% laquelle |\dedicace| et |\epigraphe| sont d'ailleurs basées. +% +% \subsection{Citations} +% \label{sec:citations} +% +% \begin{DescribeEnv}{quote} +% {\LaTeX} offre deux environnements pour les citations dans le +% texte: |quote| et |quotation|. Le premier sert pour les citations +% «courtes», quelques lignes au plus. Dans la classe, le texte est +% alors placé en retrait des marges normales de 10~mm à gauche et à +% droite. +% \end{DescribeEnv} +% +% \begin{DescribeEnv}{quotation} +% L'environnement |quotation|, quant à lui, doit être utilisé pour +% les citations «longues», celles qui peuvent s'étendre sur plus de +% cinq lignes ou, surtout, plus d'un paragraphe. Dans la classe, le +% texte est alors toujours placé en retrait de 10~mm, mais également +% à interligne simple. De plus, les paragraphes, le cas échéant, +% sont séparés d'un espace vertical afin de bien les distinguer les +% uns des autres. +% \end{DescribeEnv} +% +% \subsection{Interligne} +% +% \begin{DescribeMacro}{\OnehalfSpacing} +% L'espacement d'un interligne et demi utilisé dans la classe est +% obtenu avec la commande |\OnehalfSpacing| de \textsf{memoir}. +% L'interligne simple est automatiquement rétabli pour la page +% titre, la table des matières, la liste des tableaux, la liste +% des figures et les longues citations (section~\ref{sec:citations}). +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\SingleSpacing} +% Si ce devait être nécessaire ailleurs dans le document, la +% commande |\SingleSpacing| permet de passer à l'interligne simple. +% \end{DescribeMacro} +% +% +% \subsection{Autres paquetages chargés} +% \label{sec:paquetages} +% +% Outre \textsf{memoir} et \textsf{babel}, la classe \textsf{ulthese} +% charge pour son usage quelques paquetages qui peuvent aussi être +% utiles pour l'utilisateur de la classe. Il n'est donc pas nécessaire +% de charger de nouveau les paquetages suivants: +% \begin{description} +% \item[\normalfont\textsf{natbib}] gestion de la bibliographie (si +% l'option |nonatbib| de la classe est absente; voir aussi la +% section~\ref{sec:bibliographystyle}); +% \item[\normalfont\textsf{numprint}] permet de composer +% automatiquement des nombres avec un séparateur toutes les trois +% positions (une espace en français). Requis par la commande +% |\nombre| de \textsf{babel}; +% \item[\normalfont\textsf{graphicx}] support pour l'insertion et la +% manipulation de graphiques; +% \item[\normalfont\textsf{xcolor}] extension du paquetage +% \textsf{color} pour gérer les couleurs dans le texte; +% \item[\normalfont\textsf{textcomp}] multitude de symboles spéciaux, +% dont un beau symbole de copyright, \textcopyright. +% \item[\normalfont\textsf{helvet}] remplacement de la police sans +% empattements (\emph{serifs}) standard par la police Helvetica pour +% composer la page titre. Le paquetage est chargé avec l'option +% |scaled=0.92|. Consulter la section~\ref{sec:police} pour des +% informations additionnelles. +% \end{description} +% La section~\ref{sec:meo} sur la mise en {\oe}uvre de la classe +% fournit plus de détails sur la liste des paquetages chargés et les +% raisons pour lesquelles ils sont requis dans la classe. +% +% \section{Français et autres langues} +% \label{sec:langues} +% +% Une complication additionnelle pour les auteurs rédigeant dans une +% langue autre que l'anglais consiste à adapter {\LaTeX} à leur +% langue, qu'il s'agisse des mots clés, de la typographie ou de la +% césure des mots. La solution à ce problème provient du paquetage +% standard \textsf{babel}. Celui-ci permet de combiner plusieurs +% langues dans un même document et de passer de l'une à l'autre +% facilement. Il est chargé par la classe \textsf{ulthese}. +% +% Aucune langue n'est spécifiée dans la classe. La plupart des auteurs +% auront recours à l'anglais et au français, ne serait-ce que pour les +% deux résumés demandés par la FESP. Les langues utilisées dans le +% document doivent être spécifiées comme options à la classe, tel que +% mentionné à la section~\ref{sec:utilisation:options}. La +% \emph{dernière} langue spécifiée devient par défaut la langue active +% du document. +% +% \begin{DescribeMacro}{\selectlanguage} +% La commande |\selectlanguage| de \textsf{babel} permet de passer de +% la langue courante à la langue spécifiée en argument. +% \end{DescribeMacro} +% +% \begin{DescribeEnv}{otherlanguage} +% L'environnement |otherlanguage| de \textsf{babel} permet de faire +% la même chose que la commande |\selectlanguage|, sauf que le +% changement de langue est local à l'environnement --- utile pour +% les brefs changements de langue. +% \end{DescribeEnv} +% +% Si vous n'êtes pas autrement familier avec le paquetage \textsf{babel}, +% consulter sa documentation, ne serait-ce que les chapitres consacrés +% aux langues utilisées dans votre thèse ou mémoire. Rechercher sur +% votre système le fichier |babel.pdf| ou le % +% \href{http://mirrors.ctan.org/macros/latex/required/babel/babel.pdf}{% +% consulter en ligne} % +% sur CTAN. +% +% \section{Police de caractère du document} +% \label{sec:police} +% +% Les documents {\LaTeX} sont facilement reconnaissables par leur +% police de caractère par défaut, Computer Modern. Avec toute +% distribution {\LaTeX} moderne, il est maintenant simple d'utiliser +% l'une ou l'autre des polices Postscript standards. D'ailleurs la +% classe \textsf{ulthese} utilise la police sans empattements +% Helvetica pour composer la page titre. +% +% La FESP permet l'utilisation des polices Times et +% Palatino\footnote{% +% Police utilisée dans le présent document.} % +% dans les thèse et mémoires {\LaTeX}. Pour utiliser ces polices, +% charger les paquetages \textsf{mathptmx} ou \textsf{mathpazo}, +% respectivement. Pour les détails, consulter la documentation de +% l'ensemble de paquetages PSNFSS. Rechercher sur votre système le +% fichier |psnfss2e.pdf| ou le % +% \href{http://mirrors.ctan.org/macros/latex/required/psnfss/psnfss2e.pdf}{% +% consulter en ligne} % +% sur CTAN. +% +% \section{Gabarits} +% +% Il est recommandé de segmenter tout document d'une certaine ampleur +% dans des fichiers |.tex| distincts pour chaque partie --- +% habituellement un fichier par chapitre. Le document complet est +% composé à l'aide d'un fichier maître qui contient le préambule +% {\LaTeX} et un ensemble de commandes +% |\include|\index{include=\verb+\include+} pour réunir les parties +% dans un tout. +% +% La classe \textsf{ulthese} est livrée avec un ensemble de gabarits sur +% lesquels se baser pour: +% \begin{itemize} +% \item les fichiers maîtres de divers types de thèses et mémoires +% (standard, sur mesure, en cotutelle, en extension, etc.); +% \item les fichiers des parties les plus usuelles (résumés français +% et anglais, avant-propos, introduction, chapitres, conclusion, +% etc.). +% \end{itemize} +% Les noms des fichiers devraient permettre de facilement identifier +% leur contenu (une bonne pratique; |rappels.tex| est plus parlant et +% résiste mieux aux changements à l'ordre des chapitres que +% |chapitre1.tex|). +% +% Pour débuter la rédaction, renommer le gabarit de document maître +% approprié d'après votre numéro de dossier. Par exemple, l'étudiante +% dont le numéro de dossier est 900352789 et qui entame la rédaction +% d'une thèse multifacultaire renommera le fichier +% \begin{quote} +% |gabarit-doctorat-multifacultaire.tex| +% \end{quote} +% en +% \begin{quote} +% |900352789.tex|. +% \end{quote} +% Les autres gabarits de documents maîtres peuvent alors être +% supprimés. +% +% Les gabarits comportent des commentaires succincts pour vous guider +% dans la préparation de votre document. Les sections suivantes +% fournissent des détails additionnels, et ce, dans l'ordre où les +% commandes apparaissent dans les gabarits. +% +% \subsection{Encodage des fichiers} +% +% Taper de longs textes en français en {\LaTeX} devient rapidement +% pénible si on utilise les commandes |\'e|, |\`a| ou |\^e| pour entrer +% les lettres accentuées. Afin de pouvoir plutôt entrer directement |é|, +% |à| ou |ê|, {\LaTeX} doit être configuré pour reconnaître les lettres +% accentuées. C'est le rôle du paquetage \textsf{inputenc}. +% +% Cependant, il existe plusieurs manières différentes d'encoder --- ou +% d'enregistrer --- les lettres accentuées et autres caractères +% spéciaux (comme, par exemple, le symbole de l'euro) dans un +% ordinateur. La méthode la plus répandue et celle standard sur les +% versions récentes des systèmes d'exploitation Linux et OS~X est +% l'UTF-8 de la norme +% \href{http://fr.wikipedia.org/wiki/Unicode}{Unicode}. Les gabarits +% sont livrés dans ce type d'encodage. +% +% La déclaration +% \begin{quote} +% |\usepackage[utf8]{inputenc}| +% \end{quote} +% dans le préambule assure que {\LaTeX} traitera correctement des +% fichiers source encodés en UTF-8. +% +% La norme Unicode n'est pas aussi uniformément supportée par Windows. +% Selon l'éditeur de texte employé et la version du système +% d'exploitation, il peut être nécessaire d'utiliser les normes +% d'encodage % +% \href{http://fr.wikipedia.org/wiki/ISO_8859-1}{ISO~8859-1} % +% (ou Latin-1; option |latin1| de \textsf{inputenc}), % +% \href{http://fr.wikipedia.org/wiki/ISO_8859-15}{ISO~8859-15} % +% (ou Latin-9; option |latin9|) ou % +% \href{http://fr.wikipedia.org/wiki/Windows-1252}{Windows-1252} % +% (options |cp1252| ou |ansinew|). +% +% +% \subsection{Paquetages additionnels} +% +% Tel qu'explicité à la section~\ref{sec:paquetages}, la classe +% charge déjà quelques paquetages. Cependant, il est fort probable que +% vous devrez en charger d'autres pour composer votre document. Les +% gabarits prévoient un endroit pour le chargement de paquetages +% additionnels. Il est recommandé d'inscrire vos commandes +% |\usepackage| à cet endroit afin de respecter un certain ordre de +% chargement; voir la section suivante. +% +% Si vous utilisez un paquetage non standard dans les distributions +% courantes (MiK\TeX, \TeX~Live, Mac\TeX), vous devez le fournir avec +% le code source de votre document lors du dépôt final. +% +% \subsection{Hyperliens} +% \label{sec:hyperref} +% +% Le paquetage \textsf{hyperref} permet de transformer toutes les +% références en hyperliens cliquables lorsque le document {\LaTeX} est +% produit avec |pdflatex|. L'interaction de ce paquetage avec les autres +% est parfois (voire souvent) délicate. Pour cette raison, il est +% habituellement nécessaire de charger \textsf{hyperref} en tout +% dernier. C'est pourquoi il n'est pas chargé dans la classe, mais +% plutôt dans les gabarits. Prendre soin de maintenir le dernier rang de +% chargement lors de l'édition d'un gabarit. +% +% La configuration du paquetage dans les gabarits fait en sorte que +% les liens sont simplement signalés par une couleur de texte +% légèrement contrastante. L'utilisation de couleurs dans un document +% requiert le paquetage |xcolor|, chargé par la classe. La couleur de +% lien par défaut, |ULlinkcolor|, est définie dans la classe; voir la +% section~\ref{sec:couleurs}. +% +% \subsection{Police sans empattements} +% +% Tel que mentionné à la section~\ref{sec:police}, la police sans +% empattements Helvetica est chargée par la classe pour composer la page +% titre. Ainsi, toute utilisation des commandes |\textsf| et |\sffamily| +% sélectionnera cette police. +% +% Si vous composez la thèse ou le mémoire dans la police par défaut de +% {\LaTeX}, vous souhaiterez sans doute que les commandes ci-dessus +% sélectionnent plutôt la version sans empattements de Computer +% Modern; comparer +% \begin{quote} +% \sffamily Texte sans empattements en Helvetica +% \end{quote} +% et +% \begin{quote} +% \fontfamily{cmss}\selectfont Texte sans empattements en Computer Modern +% \end{quote} +% Pour ce faire, décommentez la ligne du gabarit afin que la +% commande +% \begin{quote} +% |\renewcommand{\sfdefault}{cmss}| +% \end{quote} +% apparaisse dans le préambule. +% +% \subsection{Options de \textsf{babel}} +% +% \begin{DescribeMacro}{\frenchbsetup} +% La commande |\frenchbsetup|\index{frenchbsetup=\verb+\frenchbsetup+} +% de \textsf{babel} permet de contrôler certains ajustements +% typographiques apportés par le paquetage en mode français. Consulter +% la documentation de \textsf{babel} pour la liste des options de +% configuration disponibles. +% \end{DescribeMacro} +% +% Les concepteurs de la classe \textsf{ulthese} proposent deux +% ajustements dans les gabarits: +% \begin{enumerate} +% \item l'option |CompactItemize=false| évite que le mode français de +% \textsf{babel} ne diminue l'espacement vertical dans les listes; +% \item avec l'option |ThinSpaceInFrenchNumbers=true|, une espace fine +% sera utilisée comme séparateur des milliers dans les nombres plutôt +% qu'une espace pleine. +% \end{enumerate} +% \begin{DescribeMacro}{\nombre} +% D'ailleurs, à ce sujet, le paquetage \textsf{numprint} étant +% chargé dans la classe, on peut utiliser la commande |\nombre| pour +% formater automatiquement les nombres. Par exemple, le résultat de +% |\nombre{123456789}| est \nombre{123456789}. +% \end{DescribeMacro} +% +% \subsection{Style de la bibliographie} +% \label{sec:bibliographystyle} +% +% \begin{DescribeMacro}{\bibliographystyle} +% Il est fortement recommandé d'utiliser BIB{\TeX} pour la préparation +% de la bibliographie. Le formatage de la bibliographie est contrôlé +% par un style choisi par la commande |\bibliographystyle|. Les styles +% standards de {\LaTeX} sont |plain|, |unsrt|, |alpha| et |abbrv|. +% \end{DescribeMacro} +% +% Pour plus de flexibilité, il est recommandé d'utiliser le paquetage +% \textsf{natbib} pour la gestion des références et des styles de la +% bibliographie. Entre autres choses, ce paquetage supporte le style de +% citation auteur-année fréquemment employé en sciences naturelles, +% plusieurs commandes de citation, un grand nombres de styles de +% bibliographie ainsi que des entrées spécifiques pour les numéros ISBN +% et les URL. Le paquetage fournit des styles de bibliographie +% |plainnat|, |unsrtnat| et |abbrvnat| similaires aux styles standards, +% mais plus complets. Il existe des +% \href{http://mirrors.ctan.org/biblio/bibtex/contrib/bib-fr/}{versions +% francisées} de ces styles (et de quelques autres) dans CTAN. +% +% Afin d'assurer le bon fonctionnement avec \textsf{babel}, le +% paquetage \textsf{natbib} est chargé par la classe \textsf{ulthese} +% (à moins que l'option |nonatbib| ne soit spécifiée). Pour consulter +% sa documentation, rechercher sur votre système le fichier +% |natbib.pdf| ou le % +% \href{http://mirrors.ctan.org/macros/latex/contrib/natbib/natbib.pdf}{% +% consulter en ligne} % +% sur CTAN. +% +% De plus, Vincent Goulet a préparé la feuille de style +% |francais.bst| compatible avec \textsf{natbib}; consulter la +% section \href{http://vgoulet.act.ulaval.ca/latex/}{LaTeX en +% français} de son site Web. La feuille de style permet de composer +% des bibliographies auteur-année respectant les normes de typographie +% française telles qu'édictées dans \cite{Malo:1996}. Pour utiliser ce +% style, on spécifera dans le préambule du document LaTeX +% \begin{quote} +% |\bibliographystyle{francais}| +% \end{quote} +% +% Autrement, la FESP n'a pas d'exigences particulières quant à la +% présentation de la bibliographie (présentation du titre, des auteurs +% et autres informations bibliographiques). +% +% \subsection{Déclarations de la page titre} +% +% Les gabarits comportent toutes les déclarations nécessaires pour +% composer la page titre des divers types de thèse ou de mémoires. +% Vous devez remplacer les éléments se trouvant entre crochets <~> en +% respectant la forme indiquée. Assurez-vous de supprimer les +% caractères < et > afin qu'il n'apparaissent pas sur la page titre de +% votre document. +% +% \subsection{Pages liminaires} +% +% \begin{DescribeMacro}{\frontmatter} +% La commande |\frontmatter| déclare que {\LaTeX} doit considérer le +% matériel qui suit comme des pages liminaires. En pratique, cela +% résulte essentiellement en une numérotation des pages en chiffres +% romains. +% \end{DescribeMacro} +% +% Les normes de présentation de la FESP édictent que les thèses et +% mémoires devraient comporter les pages liminaires suivantes, dans +% l'ordre: +% \begin{enumerate} +% \item la page titre (obligatoire); +% \item un résumé en français (obligatoire); +% \item un résumé en anglais (recommandé mais non obligatoire); +% \item une table des matières (obligatoire); +% \item une liste des tableaux; +% \item une liste des figures; +% \item une liste des abbréviations et des sigles; +% \item une dédicace; +% \item une épigraphe; +% \item des remerciements; +% \item un avant-propos (obligatoire dans le cas d'un mémoire ou d'une +% thèse avec insertion d'articles). +% \end{enumerate} +% +% Les commandes +% \begin{quote} +% |\pagetitre| \\ +% |\tableofcontents| \\ +% |\listoftables| \\ +% |\listoffigures| \\ +% |\dedicace{|\meta{texte}|}| \\ +% |\epigraphe{|\meta{texte}|}{|\meta{auteur}|}| +% \end{quote} +% permettent de générer les pages correspondantes. Seules les deux +% dernières commandes admettent des arguments. +% +% \begin{DescribeMacro}{\chapter*} +% Les résumés, la liste des abbréviations et des sigles, les +% remerciements et l'avant-propos sont composés comme des chapitres +% normaux, mais sans être numérotés. Il faut donc définir ces éléments +% avec la commande |\chapter*|. +% \end{DescribeMacro} +% +% \begin{DescribeMacro}{\phantomsection} +% \begin{DescribeMacro}{\addcontentsline} +% Les sections declarées avec la commande |\chapter*| +% n'apparaissent pas dans la table des matières. Comme les normes +% de présentation de la FESP exigent que toutes les pages +% liminaires y figurent, on fait suivre les commandes +% |\chapter*{|\meta{Titre}|}| des commandes +% \end{DescribeMacro} +% \end{DescribeMacro} +% \begin{quote} +% |\phantomsection\addcontentsline{toc}{chapter}{|\meta{Titre}|}| +% \end{quote} +% Celles-ci ajoutent à la table des matières (|toc|) une section de +% niveau |chapter| dont le titre est \meta{Titre}. La commande +% |\phantomsection| est rendue nécessaire (ou recommandée) par le +% paquetage \textsf{hyperref}. +% +% \subsection{Corps du document} +% +% \begin{DescribeMacro}{\mainmatter} +% La commande |\mainmatter| délimite le début du corps du document. La +% numérotation des pages passe en chiffres arabes. +% \end{DescribeMacro} +% +% Le corps du document devrait normalement compter une introduction (non +% numérotée), un développement divisé en chapitres (numérotés) et une +% conclusion (non numérotée). +% +% \subsection{Annexes} +% +% \begin{DescribeMacro}{\appendix} +% Si la thèse ou le mémoire comporte une ou plusieurs annexes, +% composer celles-ci comme des chapitres normaux insérés dans le +% document maître après la commande |\appendix|. Cette commande a +% pour effet de passer d'un mode de numération numérique (1, 2, 3, +% \dots) à un mode alphabétique (A, B, C, \dots). +% \end{DescribeMacro} +% +% \subsection{Bibliographie} +% +% \begin{DescribeMacro}{\bibliography} +% Si vous utilisez BIB{\TeX}, la bibliographie est insérée dans le +% document à l'endroit où apparait la commande |\bibliography| dans le +% code source. +% \end{DescribeMacro} +% +% Consulter la section~\ref{sec:bibliographystyle} pour des +% informations additionnelles sur la préparation de la bibliographie. +% +% \section{Aide additionnelle} +% +% Pour obtenir de l'aide additionnelle sur l'utilisation de la classe +% \textsf{ulthese} (et sur celle de {\LaTeX} en général), prière de +% consulter d'abord +% \begin{enumerate} +% \item le \href{http://www.theses.ulaval.ca/wiki/}{WikiThèse} de +% l'Université Laval; +% \item les +% \href{http://listes.ulaval.ca/listserv/archives/ulthese-aide.html}{archives} +% de la liste de distribution \texttt{ulthese-aide}. +% \end{enumerate} +% Si la réponse à votre question ne se trouve ni dans le wiki, ni dans +% les archives, alors écrire à l'adresse +% \href{mailto:ulthese-aide@bibl.ulaval.ca}{\url{ulthese-aide@bibl.ulaval.ca}}. +% +% \StopEventually{\PrintChanges\PrintIndex} +% +% ^^A Début du code de la classe +% +% \appendix +% \section{Mise en {\oe}uvre} +% \label{sec:meo} +% +% Cette section passe en revue le code {\TeX} et {\LaTeX} de la +% classe. Elle ne risque d'intéresser que les personnes qui souhaitent +% explorer comment la classe est programmée. +% +% \subsection{Options de la classe} +% +% Il y a deux options propres à la classe: la taille de la police de +% caractères en points et la possibilité d'empêcher le chargement du +% paquetage \textsf{natbib}. +% +% \begin{macro}{nonatbib} +% Pour des raisons de programmation, on traite l'option |nonatbib| +% en premier. Celle-ci permet d'empêcher la classe de charger le +% paquetage \textsf{natbib} en cas d'incompatibilité avec d'autres +% paquetages spécialisés de mise en forme de la bibliographie. Le +% paquetage \textsf{ifthen} est nécessaire pour définir cette option +% booléenne et pour plusieurs autres usages internes à la classe. +% \begin{macrocode} +%<*class> +\RequirePackage{ifthen} +\newboolean{UL@natbib} +\setboolean{UL@natbib}{true} +\DeclareOption{nonatbib}{\setboolean{UL@natbib}{false}} +% \end{macrocode} +% \end{macro} +% +% Les valeurs possibles pour la taille de la police de caractères sont +% |10pt|, |11pt| et |12pt|. Cette option est gérée au niveau de la +% classe afin de s'assurer que les divers éléments sur la page titre +% sont toujours de la même taille. La taille de la police par défaut +% permet de déterminer si, par exemple, le titre du document doit être +% dans la taille |\Huge|, |\huge| ou |\LARGE| de \textsf{memoir}. +% +% \begin{macro}{\UL@ptsize} +% La taille de la police est passée à \textsf{memoir} et la macro +% |\UL@ptsize| stocke la taille des caractères pour usage futur. +% \begin{macrocode} +\newcommand*{\UL@ptsize}{} +\DeclareOption{10pt}{% + \PassOptionsToClass{10pt}{memoir} + \renewcommand*{\UL@ptsize}{10}} +\DeclareOption{11pt}{% + \PassOptionsToClass{11pt}{memoir} + \renewcommand*{\UL@ptsize}{11}} +\DeclareOption{12pt}{% + \PassOptionsToClass{12pt}{memoir} + \renewcommand*{\UL@ptsize}{12}} +% \end{macrocode} +% \end{macro} +% +% \subsection{Chargement de la classe \textsf{memoir}} +% +% Toutes les options de la classe sont passées à \textsf{memoir}. Le +% format de papier et la taille de police par défaut sont, dans +% l'ordre, |letterpaper| et |11pt|. Les options de \textsf{memoir} +% |twoside| et |openright| sont explicitement déclarées afin d'éviter +% toute tentative de passer outre à ces exigences de la FESP. +% \begin{macrocode} +\DeclareOption*{\PassOptionsToClass{\CurrentOption}{memoir}} +\ExecuteOptions{11pt,letterpaper} +\ProcessOptions +\LoadClass[twoside,openright]{memoir} +% \end{macrocode} +% +% \subsection{Paquetages requis} +% +% La classe s'efforce de charger un minimum de paquetages afin d'éviter +% les conflits potentiels. +% +% Il est aujourd'hui préférable d'utiliser les polices T1. +% \begin{macrocode} +\RequirePackage[T1]{fontenc} +% \end{macrocode} +% +% Le paquetage \textsf{natbib} doit être chargé avant \textsf{babel} +% pour bien fonctionner. C'est pourquoi il est chargé dans la classe, +% à moins que l'option |nonatbib| n'ait été spécifiée au chargement de +% la classe. +% \begin{macrocode} +\ifthenelse{\boolean{UL@natbib}}{\RequirePackage{natbib}}{} +% \end{macrocode} +% +% Le support pour les langues autres que l'anglais est offert par le +% paquetage \textsf{babel}. Les langues sont passées en option de la +% classe, et non du paquetage. Le paquetage \textsf{numprint} est +% requis par \textsf{babel} pour la définition de la commande de mise +% en forme des nombres |\nombre|. +% \begin{macrocode} +\RequirePackage{babel} +\RequirePackage[autolanguage]{numprint} +% \end{macrocode} +% +% L'insertion du logo de l'Université sur la page titre requiert +% \textsf{graphicx}. On définit également une couleur pour les +% hyperliens dans le document (mais \textsf{hyperref} est chargé dans +% les gabarits afin de demeurer le dernier paquetage chargé; voir la +% section~\ref{sec:hyperref}). +% \begin{macrocode} +\RequirePackage{graphicx} +\RequirePackage{xcolor} +% \end{macrocode} +% +% La commande |\textcopyright| utilisée sur la page titre requiert le +% paquetage \textsf{textcomp} pour obtenir un beau signe de copyright. +% \begin{macrocode} +\RequirePackage{textcomp} +% \end{macrocode} +% +% \subsection{Police pour la page titre} +% +% La page titre est composée dans la police Helvetica. Le chargement +% du paquetage \textsf{helvet} a pour effet de redéfinir la commande +% |\sfdefault|. +% \begin{macrocode} +\RequirePackage[scaled=0.92]{helvet} +% \end{macrocode} +% +% \subsection{Couleur des hyperliens} +% \label{sec:couleurs} +% +% La classe définit une couleur standard pour les hyperliens, une +% teinte de bleu assez foncée pour être à fois visible en couleur et +% peu contrastante si le document est imprimé en noir et blanc. +% \begin{macrocode} +\definecolor{ULlinkcolor}{rgb}{0,0,0.3} +% \end{macrocode} +% +% \subsection{Marges} +% +% Les marges exigées par les normes de présentation de la FESP sont de +% 25~mm partout, sauf 35~mm pour la marge de reliure (gauche pour les +% pages impaires, droite pour les pages paires). Le pied de page est +% placé de sorte que le folio de page se retrouve à 10~mm du bas de la +% page. +% \begin{macrocode} +\setlrmarginsandblock{35mm}{25mm}{*} +\setulmarginsandblock{25mm}{*}{1} +\setheadfoot{\baselineskip}{20mm} +\checkandfixthelayout[lines] +% \end{macrocode} +% +% \subsection{Interligne} +% +% L'espacement entre les lignes est d'un interligne et demi. +% L'espacement «double» entre les paragraphes est fixé à +% |0.5\baselineskip| afin d'en arriver à une disposition agréable à +% l'{\oe}il. Le retrait de première ligne est supprimé puisque plus +% nécessaire suite à l'ajout de l'espacement entre les paragraphes. +% \begin{macrocode} +\OnehalfSpacing +\setlength{\parskip}{0.5\baselineskip} +\setlength{\parindent}{0em} +% \end{macrocode} +% +% La table des matières, la liste des tableaux et la liste des figures +% sont composées à interligne simple. +% \begin{macrocode} +\renewcommand{\tocheadstart}{\SingleSpacing\chapterheadstart} +\renewcommand{\lotheadstart}{\SingleSpacing\chapterheadstart} +\renewcommand{\lofheadstart}{\SingleSpacing\chapterheadstart} +% \end{macrocode} +% +% \subsection{Entêtes et pieds de page} +% +% Les règles pour les entêtes et pieds de page sont uniformes pour +% tout le document: aucun entête et folio sur le bord extérieur du +% pied de page. On définit un style de page pour ce faire ainsi qu'un +% alias entre le nouveau style |ul| et le style standard |plain|. +% Raison: les premières pages de chapitres utilisent par défaut le +% style |plain|; avec l'alias c'est le style |ul| qui sera activé. +% \begin{macrocode} +\makepagestyle{ul} + \makeevenfoot{ul}{\thepage}{}{} + \makeoddfoot{ul}{}{}{\thepage} +\aliaspagestyle{plain}{ul} +\pagestyle{ul} +% \end{macrocode} +% +% \subsection{Page titre} +% +% La page titre est composée avec les polices présentées au +% tableau~\ref{tab:polices}. Les commandes sélectionnant ces polices +% sont adaptées selon la taille de police choisie pour le document +% afin d'être toujours identiques. +% +% \begin{table} +% \centering +% \begin{tabular}{ll} +% \toprule +% Élément & Police \\ +% \midrule +% Titre & 20~points gras \\ +% Sous-titre & 17~points gras \\ +% Auteur & 14~points gras \\ +% Nom du programme & 14~points gras \\ +% Autres éléments & 14~points normal \\ +% \bottomrule +% \end{tabular} +% \caption{Taille et type de police des éléments de la page titre} +% \label{tab:polices} +% \end{table} +% +% \begin{macrocode} +\ifthenelse{\UL@ptsize=10}{% + \newcommand*{\UL@fonttitle}{\normalfont\Huge\bfseries\sffamily} + \newcommand*{\UL@fontsubtitle}{\normalfont\huge\bfseries\sffamily} + \newcommand*{\UL@fontauthor}{\normalfont\LARGE\bfseries\sffamily} + \newcommand*{\UL@fontprogram}{\UL@fontauthor} + \newcommand*{\UL@fontbase}{\normalfont\LARGE\sffamily}}{} +\ifthenelse{\UL@ptsize=11}{% + \newcommand*{\UL@fonttitle}{\normalfont\huge\bfseries\sffamily} + \newcommand*{\UL@fontsubtitle}{\normalfont\LARGE\bfseries\sffamily} + \newcommand*{\UL@fontauthor}{\normalfont\Large\bfseries\sffamily} + \newcommand*{\UL@fontprogram}{\UL@fontauthor} + \newcommand*{\UL@fontbase}{\normalfont\Large\sffamily}}{} +\ifthenelse{\UL@ptsize=12}{% + \newcommand*{\UL@fonttitle}{\normalfont\LARGE\bfseries\sffamily} + \newcommand*{\UL@fontsubtitle}{\normalfont\Large\bfseries\sffamily} + \newcommand*{\UL@fontauthor}{\normalfont\large\bfseries\sffamily} + \newcommand*{\UL@fontprogram}{\UL@fontauthor} + \newcommand*{\UL@fontbase}{\normalfont\large\sffamily}}{} +% \end{macrocode} +% +% Définition de deux valeurs booléennes servant à adapter la +% composition de la page titre plus loin: y a-t-il un sous-titre ou +% non et est-ce que le type de programme (doctorat ou maîtrise) est +% masculin ou non? +% \begin{macrocode} +\newboolean{UL@hassubtitle} +\newboolean{UL@isprogmasc} +% \end{macrocode} +% +% Définition des commandes permettant de construire la page titre. +% L'interface utilisateur est basée sur un ensemble de commandes +% internes. On commence par celles-ci. +% \begin{macrocode} +\newcommand{\UL@maintitle}{} +\newcommand{\UL@subtitle}{} +\newcommand*{\UL@author}{} +\newcommand*{\UL@program}{} +\newcommand*{\UL@year}{} +\newcommand*{\UL@typeofdoc}{} +\newcommand*{\UL@degree}{} +\newcommand*{\UL@nameother}{} +\newcommand*{\UL@degreeother}{} +\newcommand*{\UL@facUL}{} +\newcommand*{\UL@facother}{} +\newcommand*{\UL@extensionat}{} +\newcommand*{\UL@extensionloc}{} +% \end{macrocode} +% Puis les commandes visibles pour les utilisateurs, qui redéfinissent +% les commandes internes. Voir la +% section~\ref{sec:commandes-pagetitre} pour leur signification. +% \begin{macrocode} +\newcommand{\titre}[1]{\renewcommand{\UL@maintitle}{#1}} +\newcommand{\soustitre}[1]{% + \setboolean{UL@hassubtitle}{true} + \renewcommand{\UL@subtitle}{#1}} +\newcommand*{\auteur}[1]{\renewcommand*{\UL@author}{#1}} +\newcommand*{\annee}[1]{\renewcommand*{\UL@year}{#1}} +\newcommand*{\programme}[1]{\renewcommand*{\UL@program}{#1}} +\newcommand*{\LLD}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Docteur en droit (L.L.D.)}} +\newcommand*{\DPsy}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Docteur en psychologie (D.Psy.)}} +\newcommand*{\DThP}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Docteur en th\'eologie pratique (D.Th.P.)}} +\newcommand*{\PhD}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{true} + \renewcommand*{\UL@typeofdoc}{Th\`ese} + \renewcommand*{\UL@degree}{Philosophi{\ae} doctor (Ph.D.)}} +\newcommand*{\LLM}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en droit (L.L.M.)}} +\newcommand*{\MA}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre \`es arts (M.A.)}} +\newcommand*{\MMus}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en musique (M.Mus.)}} +\newcommand*{\MSc}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre \`es sciences (M.Sc.)}} +\newcommand*{\MServSoc}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en service social (M.Serv.Soc.)}} +\newcommand*{\MScGeogr}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en sciences géographiques (M.Sc.G\'eogr.)}} +\newcommand*{\MATDR}{% + \newcommand*{\UL@typenum}{0} + \setboolean{UL@isprogmasc}{false} + \renewcommand*{\UL@typeofdoc}{M\'emoire} + \renewcommand*{\UL@degree}{Ma\^itre en am\'enagement du territoire et d\'eveloppement r\'egional (M.ATDR)}} +\newcommand*{\multifacultaire}{\renewcommand*{\UL@typenum}{1}} +\newcommand*{\cotutelle}{% + \renewcommand*{\UL@typenum}{2} + \edef\UL@typeofdoc{\UL@typeofdoc\ en cotutelle}} +\newcommand*{\univcotutelle}[1]{\renewcommand*{\UL@nameother}{#1}} +\newcommand*{\gradecotutelle}[1]{\renewcommand*{\UL@degreeother}{#1}} +\newcommand*{\extensionUdeS}{% + \renewcommand*{\UL@typenum}{3} + \renewcommand*{\UL@extensionat}{Universit\'e de Sherbrooke} + \renewcommand*{\UL@extensionloc}{Sherbrooke, Qu\'ebec}} +\newcommand*{\extensionUQO}{% + \renewcommand*{\UL@typenum}{3} + \renewcommand*{\UL@extensionat}{Universit\'e du Qu\'ebec en Outaouais} + \renewcommand*{\UL@extensionloc}{Gatineau, Qu\'ebec}} +\newcommand*{\extensionUQAC}{% + \renewcommand*{\UL@typenum}{3} + \renewcommand*{\UL@extensionat}{Universit\'e du Qu\'ebec \`a Chicoutimi} + \renewcommand*{\UL@extensionloc}{Chicoutimi, Qu\'ebec}} +\newcommand{\faculteUL}[1]{\renewcommand*{\UL@facUL}{#1}} +\newcommand*{\faculteUdeS}[1]{\renewcommand*{\UL@facother}{#1}} +\newcommand*{\faculteUQO}[1]{\renewcommand*{\UL@facother}{#1}} +\newcommand*{\faculteUQAC}[1]{\renewcommand*{\UL@facother}{#1}} +% \end{macrocode} +% +% Outre le nom de l'auteur et la notice de copyright en bas de page, +% la page titre peut être divisée en trois grands blocs: +% \begin{enumerate} +% \item le titre et le sous-titre, le cas échéant; +% \item la description du type de document (thèse, thèse en cotutelle, +% mémoire, etc.); +% \item les détails sur la ou les facultés, la ou les universités, +% etc. +% \end{enumerate} +% À ces blocs s'ajoutent quatre grandes catégories de disposition des +% éléments sur la page titre selon le type de thèse ou de mémoire: +% standard, multifacultaire, en cotutelle et en extension. Les +% commandes internes ci-dessous prennent en compte toutes les +% combinaisons possibles. +% +% \begin{macro}{\UL@title} +% La commande |\UL@title| prépare le titre. Si le document ne +% comporte pas de sous-titre, un espace est ajouté pour ne pas +% décaler vers le haut les autres éléments de la page. +% \begin{macrocode} +\newcommand{\UL@title}{% + {\UL@fonttitle\UL@maintitle\par} + {\UL@fontsubtitle + \ifthenelse{\boolean{UL@hassubtitle}}{\UL@subtitle}{% + \vspace*{\baselineskip}}\par}} +% \end{macrocode} +% \end{macro} +% \begin{macro}{\Ul@docid} +% La commande |\Ul@docid| prépare la mention du type de document. La +% thèse ou le mémoire en cotutelle requiert un traitement différent +% puisque le programme d'étude apparaît immédiatement sous la mention. +% \begin{macrocode} +\newcommand{\UL@docid}{% + {\UL@fontprogram\UL@typeofdoc\par + \ifnum\UL@typenum=2 \UL@program\par \fi}} +% \end{macrocode} +% \end{macro} +% \begin{macro}{\Ul@details} +% La commande |\Ul@details| est la plus complexe puisque la +% disposition des informations additionnelles sur le document +% varie beaucoup selon le type de thèse ou de mémoire. +% \begin{macrocode} +\newcommand{\UL@details}{% + \ifcase\UL@typenum\relax% 0 standard + \vspace{96pt} + {\UL@fontprogram\UL@program}\par + \UL@degree\par + \vspace{112pt} + Qu\'ebec, Canada\par + \or% 1 multifacultaire + \vspace{96pt} + {\UL@fontprogram\UL@program}\par + \UL@degree\par + \vspace{36pt} + \UL@facUL\par + \vspace{48pt} + Qu\'ebec, Canada\par + \or% 2 cotutelle + \vspace{72pt} + Universit\'e Laval\par Qu\'ebec, Canada\par + \UL@degree\par + \vspace{\baselineskip} et\par \vspace{\baselineskip} + \UL@nameother\par + \UL@degreeother\par + \or% 3 extension + \vspace{48pt} + {\UL@fontprogram\UL@program\ de l'Universit\'e Laval\par + \ifthenelse{\boolean{UL@isprogmasc}}{offert}{offerte} + en extension \`a l'\UL@extensionat}\par + \vspace{36pt} + \UL@degree\par + \vspace{36pt} + \UL@facother\par \UL@extensionat\par \UL@extensionloc\par + \vspace{\baselineskip} + \UL@facUL\par Universit\'e Laval\par Qu\'ebec, Canada\par + \fi} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\pagetitre} +% Construction de la page titre. Le nom de l'auteur et la notice de +% copyright sont insérées directement dans le code de la commande +% |\pagetitre|. On doit rétablir sur la page titre l'interligne +% simple et l'espacement nul entre les paragraphes (|\parskip|). La +% page titre des thèses et mémoires en cotutelle ou en extension ne +% comporte pas de logo de l'Université Laval. +% \begin{macrocode} +\newcommand{\pagetitre}{{% + \clearpage + \thispagestyle{empty} + \SingleSpacing\setlength{\parskip}{0pt} + \centering + \UL@fontbase + \ifthenelse{\UL@typenum > 1}{\vspace*{0pt}\par}{% + \includegraphics[height=40px,keepaspectratio=true]{ul_p}\par} + \vspace{72pt} + \UL@title + \vspace{48pt} + \UL@docid + \vspace{72pt} + {\UL@fontauthor\UL@author}\par + \UL@details + \vfill + {\textcopyright} \UL@author, \UL@year\par + \cleardoublepage + }} +% \end{macrocode} +% \end{macro} +% +% \subsection{Listes des figures et des tableaux} +% +% Le paquetage \textsf{babel} définit comme titre pour la liste des +% figures «Table des figures», alors que la liste des tableaux est +% «Liste des tableaux». Pour une plus grande symétrie, la classe +% redéfinit le titre correspondant à |\listoffigures|. La commande +% |\addto| est nécessaire pour éviter que \textsf{babel} redéfinisse +% le titre à |\begin{document}|. +% \begin{macrocode} +\addto\captionsfrench{\renewcommand{\listfigurename}{Liste des figures}} +% \end{macrocode} +% +% \subsection{Dédicace et épigraphe} +% +% La dédicace et l'épigraphe sont mises en forme avec la commande +% |\epigraph| de \textsf{memoir}. +% \begin{macro}{\dedicace} +% La dédicace est une épigraphe simplifiée placée seule sur une +% page, alignée à droite à une dizaine de lignes de la marge +% supérieure, sans auteur ou source et sans ligne de démarcation. +% \begin{macrocode} +\newcommand{\dedicace}[1]{{% + \clearpage + \pagestyle{empty} + \setlength{\beforeepigraphskip}{10\baselineskip} + \setlength{\epigraphrule}{0pt} + \epigraphtextposition{flushright} + \mbox{}\epigraph{\itshape #1}{} + }} +% \end{macrocode} +% \end{macro} +% \begin{macro}{\epigraphe} +% L'épigraphe de début de document est placée seule sur une page à +% une dizaine de lignes de la marge supérieure. Pour le reste, on +% s'en remet à la commande |\epigraph| de \textsf{memoir}. +% \begin{macrocode} +\newcommand{\epigraphe}[2]{{% + \clearpage + \pagestyle{empty} + \setlength{\beforeepigraphskip}{10\baselineskip} + \mbox{}\epigraph{#1}{#2} + }} +% \end{macrocode} +% \end{macro} +% +% \subsection{Citations} +% +% \begin{environment}{quote} +% La classe redéfinit l'environnement |quote| de \textsf{memoir} +% afin que le texte des citations se trouve en retrait de 10~mm à +% gauche et à droite, conformément aux règles de présentation de la +% FESP. +% \begin{macrocode} +\renewenvironment{quote}{% + \list{}{\rightmargin 10mm \leftmargin 10mm}% + \item[]}{\endlist} +% \end{macrocode} +% \end{environment} +% +% \begin{environment}{quotation} +% Il en va de même de l'environnement |quotation|. Cependant, cet +% environnement passe également à l'interligne simple et la classe +% ajuste l'espacement vertical entre les paragraphes afin que +% ceux-ci soient bien distincts les uns des autres tout en demeurant +% raisonnablement compacts. Cet espacement est ici fixé à 6~points. +% \begin{macrocode} +\renewenvironment{quotation}{% + \list{}{% + \SingleSpacing + \listparindent 0em + \itemindent \listparindent + \leftmargin 10mm + \rightmargin \leftmargin + \parsep 6\p@ \@plus\p@}% + \item[]}{\endlist} +% \end{macrocode} +% \end{environment} +% +% \subsection{Numérotation des divisions du document} +% +% Par défaut, \textsf{memoir} numérote les divisions du document +% seulement jusqu'au niveau des sections. La classe étend la +% numérotation aux sous-sections. +% \begin{macrocode} +\setsecnumdepth{subsection} +% +% \end{macrocode} +% ^^A Fin du code de la classe +% +% \begin{thebibliography}{1} +% \providecommand{\bibnamefont}[1]{#1} +% \providecommand{\selectlanguage}[1]{\relax} +% \bibitem[{Malo(1996)}]{Malo:1996} +% \bibnamefont{Malo}, M. 1996, {\selectlanguage{francais}\emph{Guide de la +% communication écrite au cégep, à l'université et en entreprise}}, Québec +% Amérique. +% \end{thebibliography} +% +% \Finale +% +% \iffalse +% ^^A Gabarits du document maître +%<*gabarit> +%%% GABARIT POUR THÈSE STANDARD +%%% GABARIT POUR THÈSE SUR MESURE +%%% GABARIT POUR THÈSE MULTIFACULTAIRE +%%% GABARIT POUR THÈSE EN COTUTELLE +%%% GABARIT POUR THÈSE EN EXTENSION À L'UNIVERSITÉ DE SHERBROOKE +%%% GABARIT POUR THÈSE EN EXTENSION À L'UQO +%%% GABARIT POUR MÉMOIRE STANDARD +%%% GABARIT POUR MÉMOIRE SUR MESURE +%%% GABARIT POUR MÉMOIRE EN EXTENSION À L'UQAC +%% +%% Consulter la documentation de la classe ulthese pour une +%% description détaillée de la classe, de ce gabarit et des options +%% disponibles. +%% +%% [Ne pas hésiter à supprimer les commentaires après les avoir lus.] +%% +%% Déclaration de la classe avec les langues les plus courantes. Le +%% français sera la langue par défaut du document. +\documentclass[11pt,english,francais]{ulthese} + %% Encodage utilisé pour les caractères accentués dans les fichiers + %% source du document. Les gabarits sont encodés en UTF-8. + \usepackage[utf8]{inputenc} + + %% Charger ici les autres paquetages nécessaires pour le document. + %% Quelques exemples; décommenter au besoin. + %\usepackage{amsmath} % recommandé pour les mathématiques + %\usepackage{mathpazo} % texte en Palatino + %\usepackage{mathptmx} % texte en Times + + %% Gestion des hyperliens dans le document. S'assurer que hyperref + %% est le dernier paquetage chargé. + \usepackage{hyperref} + \hypersetup{colorlinks,allcolors=ULlinkcolor} + + %% Pour utiliser Computer Modern comme police sans empattements + %% (serif) plutôt que Helvetica, décommenter la ligne ci-dessous. + %\renewcommand{\sfdefault}{cmss} + + %% Options de mise en forme du mode français de babel. Consulter la + %% documentation du paquetage babel pour les options disponibles. + \frenchbsetup{% + CompactItemize=false, % ne pas compacter les listes + ThinSpaceInFrenchNumbers=true % espace fine dans les nombres + } + + %% Style de la bibliographie. + \bibliographystyle{} + + %% Déclarations de la page titre. Remplacer les éléments entre < >. + %% Supprimer les caractères < >. Couper un long titre ou un long + %% sous-titre manuellement avec \\. Terminer une éventuelle deuxième + %% ligne de titre ou de sous-titre avec \vspace*{-\baselineskip}. + \titre{} + % \titre{Ceci est un exemple de long titre \\ + % avec saut de ligne manuel \vspace*{-\baselineskip}} + % \soustitre{Sous-titre le cas échéant} + % \soustitre{Ceci est un exemple de long-titre \\ + % avec saut de ligne manuel \vspace*{-\baselineskip}} + \auteur{} +% \programme{Doctorat en -- } +% \programme{Doctorat sur mesure en -- } +% \programme{Doctorat en } +% \programme{Maîtrise en -- } +% \programme{Maîtrise sur mesure en -- } +% \programme{Maîtrise en } + \annee{<20xx>} +% \PhD % ou l'un de \LLD, \DPsy, \DThP +% \MSc % ou l'un de \LLM, \MA, \MMus, \MServSoc, \MScGeogr, \MATDR +% \cotutelle +% \univcotutelle{ \\ , } +% \gradecotutelle{ ()} +% \multifacultaire +% \faculteUL{ \\ } +% \extensionUdeS +% \extensionUQO +% \extensionUQAC +% \faculteUL{} +% \faculteUdeS{} +% \faculteUQO{} +% \faculteUQAC{} + +\begin{document} + +\frontmatter % pages liminaires + +\pagetitre % production de la page titre + +\include{resume} % résumé français +\include{abstract} % résumé anglais +\cleardoublepage + +\tableofcontents % production de la TdM +\cleardoublepage + +\listoftables % production de la liste des tableaux +\cleardoublepage + +\listoffigures % production de la liste des figures +\cleardoublepage + +\dedicace{Dédicace si désiré} +\cleardoublepage + +\epigraphe{Texte de l'épigraphe}{Source ou auteur} +\cleardoublepage + +\include{remerciements} % remerciements +\include{avantpropos} % avant-propos + +\mainmatter % corps du document + +\include{introduction} % introduction +\include{chapitre1} % chapitre 1 +\include{chapitre2} % chapitre 2, etc. +\include{conclusion} % conclusion + +\appendix % annexes le cas échéant + +\include{annexe} % annexe A + +\bibliography{} % production de la bibliographie + +\end{document} +% +% +% ^^A Gabarits des parties du document +%<*resume> +\chapter*{Résumé} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Résumé} % inclure dans TdM + +\begin{otherlanguage*}{francais} + Texte du résumé en français. +\end{otherlanguage*} +% +% +%<*abstract> +\chapter*{Abstract} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Abstract} % inclure dans TdM + +\begin{otherlanguage*}{english} + Text of English abstract. +\end{otherlanguage*} +% +% +%<*remerciements> +\chapter*{Remerciements} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Remerciements} % inclure dans TdM + +Texte des remerciements en prose. +% +% +%<*avantpropos> +\chapter*{Avant-propos} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Avant-propos} % inclure dans TdM + +L'avant-propos est surtout nécessaire pour une thèse par article. +% +% +%<*introduction> +\chapter*{Introduction} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Introduction} % inclure dans TdM + +Une thèse ou un mémoire devrait normalement débuter par une +introduction. Celle-ci est traitée comme un chapitre normal, sauf +qu'elle n'est pas numérotée. +% +% +%<*chapitre> +\chapter{Titre du chapitre} % numéroté + +Texte du chapitre. +% +% +%<*conclusion> +\chapter*{Conclusion} % ne pas numéroter +\phantomsection\addcontentsline{toc}{chapter}{Conclusion} % dans TdM + +Une thèse ou un mémoire devrait normalement se terminer par une +conclusion, placée avant les annexes, le cas échéant. Celle-ci est +traitée comme un chapitre normal, sauf qu'elle n'est pas numérotée. +% +% +%<*annexe> +\chapter{Titre de l'annexe} % numérotée + +Texte de l'annexe. +% +% Local Variables: +% mode: doctex +% coding: utf-8 +% TeX-master: t +% End: +% \fi diff --git a/depotfinal/908144032/ulthese.ins b/depotfinal/908144032/ulthese.ins new file mode 100644 index 0000000..5599434 --- /dev/null +++ b/depotfinal/908144032/ulthese.ins @@ -0,0 +1,102 @@ +%% +%% Copyright (C) 2012 Universite Laval +%% +%% This file may be distributed and/or modified under the conditions +%% of the LaTeX Project Public License, either version 1.3c of this +%% license or (at your option) any later version. The latest version +%% of this license is in: +%% +%% http://www.latex-project.org/lppl.txt +%% +%% and version 1.3c or later is part of all distributions of LaTeX +%% version 2006/05/20 or later. +%% + +\input docstrip.tex +\keepsilent +\askforoverwritefalse + +\preamble + +This is a generated file. + +Copyright (C) 2012 Universite Laval + +This file may be distributed and/or modified under the conditions +of the LaTeX Project Public License, either version 1.3c of this +license or (at your option) any later version. The latest version +of this license is in: + + http://www.latex-project.org/lppl.txt + +and version 1.3c or later is part of all distributions of LaTeX +version 2006/05/20 or later. + +This work has the LPPL maintenance status `maintained'. + +The Current Maintainer of this work is Universite Laval +. + +This work consists of the files ulthese.dtx and ulthese.ins and the +derived files listed in the README file. + +\endpreamble + +\generate{ + \file{ulthese.cls}{\from{ulthese.dtx}{class}} + \file{gabarit-doctorat.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,standard}} + \file{gabarit-doctorat-mesure.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,mesure}} + \file{gabarit-doctorat-multifacultaire.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,multifac}} + \file{gabarit-doctorat-cotutelle.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,cotutelle}} + \file{gabarit-doctorat-extension-UdeS.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,UdeS}} + \file{gabarit-doctorat-extension-UQO.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,phd,UQO}} + \file{gabarit-maitrise.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,m,standard}} + \file{gabarit-maitrise-mesure.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,m,mesure}} + \file{gabarit-maitrise-extension-UQAC.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{gabarit,m,UQAC}} + \file{resume.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{resume}} + \file{abstract.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{abstract}} + \file{remerciements.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{remerciements}} + \file{avantpropos.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{avantpropos}} + \file{introduction.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{introduction}} + \file{chapitre1.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{chapitre}} + \file{chapitre2.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{chapitre}} + \file{conclusion.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{conclusion}} + \file{annexe.tex}{\nopreamble\nopostamble + \from{ulthese.dtx}{annexe}} +} + +\obeyspaces +\Msg{*************************************************************} +\Msg{* *} +\Msg{* La classe ulthese est maintenant prete a l'emploi. *} +\Msg{* *} +\Msg{* Utiliser un des gabarits comme base pour votre fichier *} +\Msg{* maitre. Vous pouvez effacer les gabarits inutiles. *} +\Msg{* Taper le contenu du document dans des fichiers .tex *} +\Msg{* distincts; quelques gabarits sont fournis avec la classe. *} +\Msg{* *} +\Msg{* Pour produire la documentation de la classe, passer le *} +\Msg{* fichier ulthese.dtx dans LaTeX. *} +\Msg{* *} +\Msg{* Bonne redaction! *} +\Msg{* *} +\Msg{*************************************************************} + +\endbatchfile \ No newline at end of file diff --git a/memoire/.Rhistory b/memoire/.Rhistory new file mode 100644 index 0000000..e69de29 diff --git a/memoire/gabarit-maitrise.aux b/memoire/908144032.aux similarity index 93% rename from memoire/gabarit-maitrise.aux rename to memoire/908144032.aux index ce8ea44..fb01afd 100644 --- a/memoire/gabarit-maitrise.aux +++ b/memoire/908144032.aux @@ -21,7 +21,7 @@ \gdef\HyperFirstAtBeginDocument#1{#1} \providecommand\HyField@AuxAddToFields[1]{} \providecommand\HyField@AuxAddToCoFields[2]{} -\bibstyle{plainnatmod} +\bibstyle{plainnat-fr} \select@language{french} \@writefile{toc}{\select@language{french}} \@writefile{lof}{\select@language{french}} @@ -62,8 +62,8 @@ \bibcite{black1973pricing}{{7}{1973}{{Black et Scholes}}{{}}} \bibcite{buckle1995bayesian}{{8}{1995}{{Buckle}}{{}}} \bibcite{butler2007saddlepoint}{{9}{2007}{{Butler}}{{}}} -\bibcite{carr1999option}{{10}{1999}{{Carr et Madan}}{{}}} -\bibcite{madan1998variance}{{11}{1998}{{Carr et~al.}}{{Carr, Chang, et Madan}}} +\bibcite{madan1998variance}{{10}{1998}{{Carr \emph {et~al.}}}{{Carr, Chang, et Madan}}} +\bibcite{carr1999option}{{11}{1999}{{Carr et Madan}}{{}}} \bibcite{crowder1986consistency}{{12}{1986}{{Crowder}}{{}}} \bibcite{crowder1987linear}{{13}{1987}{{Crowder}}{{}}} \@writefile{toc}{\contentsline {chapter}{Bibliographie}{131}{section*.62}} @@ -87,7 +87,7 @@ \bibcite{hogg1978introduction}{{31}{1978}{{Hogg et Craig}}{{}}} \bibcite{itkin2005pricing}{{32}{2005}{{Itkin}}{{}}} \bibcite{wang2003evaluating}{{33}{2003}{{{Jingbo Wang} et Marsaglia}}{{}}} -\bibcite{kotz2001laplace}{{34}{2001}{{Kotz et~al.}}{{Kotz, Kozubowski, et Podg{\'o}rski}}} +\bibcite{kotz2001laplace}{{34}{2001}{{Kotz \emph {et~al.}}}{{Kotz, Kozubowski, et Podg{\'o}rski}}} \bibcite{KOUTROUVELIS01011980}{{35}{1980}{{Koutrouvelis}}{{}}} \bibcite{kozubowski1999class}{{36}{1999}{{Kozubowski et Podg{\'o}rski}}{{}}} \bibcite{kozubowski2001asymmetric}{{37}{2001}{{Kozubowski et Podg{\'o}rski}}{{}}} @@ -98,7 +98,7 @@ \bibcite{madan1990variance}{{42}{1990}{{Madan et Seneta}}{{}}} \bibcite{mandelbrot1963variation}{{43}{1963}{{Mandelbrot}}{{}}} \bibcite{merton1976option}{{44}{1976}{{Merton}}{{}}} -\bibcite{barndorff2001levy}{{45}{2001}{{Mikosch et~al.}}{{Mikosch, Resnick, et Barndorff-Nielsen}}} +\bibcite{barndorff2001levy}{{45}{2001}{{Mikosch \emph {et~al.}}}{{Mikosch, Resnick, et Barndorff-Nielsen}}} \bibcite{mitchell1916critique}{{46}{1916}{{Mitchell}}{{}}} \bibcite{newey1994large}{{47}{1994}{{Newey et McFadden}}{{}}} \bibcite{newey1987hypothesis}{{48}{1987}{{Newey et West}}{{}}} diff --git a/memoire/908144032.bbl b/memoire/908144032.bbl new file mode 100644 index 0000000..00a4532 --- /dev/null +++ b/memoire/908144032.bbl @@ -0,0 +1,412 @@ +\begin{thebibliography}{63} +\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi +\expandafter\ifx\csname fonteauteurs\endcsname\relax +\def\fonteauteurs{\scshape}\fi +\expandafter\ifx\csname url\endcsname\relax + \def\url#1{{\tt #1}}% + \message{You should include the url package}\fi + +\bibitem[Abramowitz et Stegun(1965)]{abramowitz1965handbook} +Milton \bgroup\fonteauteurs\bgroup Abramowitz\egroup\egroup{} et Irene~A. + \bgroup\fonteauteurs\bgroup Stegun\egroup\egroup{} : +\newblock {\em {H}andbook of {M}athematical {F}unctions: with {F}ormulas, + {G}raphs, and {M}athematical {T}ables}, volume~55. +\newblock Dover Publications, 1965. + +\bibitem[Applebaum(2004)]{applebaum2004levy} +David \bgroup\fonteauteurs\bgroup Applebaum\egroup\egroup{} : +\newblock {L}{\'e}vy {P}rocesses: {F}rom {P}robability to {F}inance and + {Q}uantum {G}roups. +\newblock {\em Notices of the American Mathematical Society}, 51\penalty0 + (11)\string:\penalty500\relax 1336--1347, 2004. + +\bibitem[Bachelier(1900)]{bachelier1900theorie} +Louis \bgroup\fonteauteurs\bgroup Bachelier\egroup\egroup{} : +\newblock {\em {T}h{\'e}orie de la sp{\'e}culation}. +\newblock Gauthier-Villars, 1900. + +\bibitem[Berkson(1980)]{berkson1980minimum} +Joseph \bgroup\fonteauteurs\bgroup Berkson\egroup\egroup{} : +\newblock {M}inimum {C}hi-square, not {M}aximum {L}ikelihood! +\newblock {\em The Annals of Statistics}, pages 457--487, 1980. + +\bibitem[Bingham et Kiesel(2004)]{bingham2004risk} +Nicholas~H. \bgroup\fonteauteurs\bgroup Bingham\egroup\egroup{} et R{\"u}diger + \bgroup\fonteauteurs\bgroup Kiesel\egroup\egroup{} : +\newblock {\em {R}isk-Neutral {V}aluation: {P}ricing and {H}edging of + {F}inancial {D}erivatives}. +\newblock Springer, 2004. + +\bibitem[Black(1976)]{black1976pricing} +Fischer \bgroup\fonteauteurs\bgroup Black\egroup\egroup{} : +\newblock {T}he {P}ricing of {C}ommodity {C}ontracts. +\newblock {\em Journal of Financial Economics}, 3\penalty0 + (1)\string:\penalty500\relax 167--179, 1976. + +\bibitem[Black et Scholes(1973)]{black1973pricing} +Fischer \bgroup\fonteauteurs\bgroup Black\egroup\egroup{} et Myron + \bgroup\fonteauteurs\bgroup Scholes\egroup\egroup{} : +\newblock {T}he {P}ricing of {O}ptions and {C}orporate {L}iabilities. +\newblock {\em The Journal of Political Economy}, pages 637--654, 1973. + +\bibitem[Buckle(1995)]{buckle1995bayesian} +D.~J. \bgroup\fonteauteurs\bgroup Buckle\egroup\egroup{} : +\newblock {B}ayesian {I}nference for {S}table {D}istributions. +\newblock {\em Journal of the American Statistical Association}, 90\penalty0 + (430)\string:\penalty500\relax 605--613, 1995. + +\bibitem[Butler(2007)]{butler2007saddlepoint} +Ronald~W. \bgroup\fonteauteurs\bgroup Butler\egroup\egroup{} : +\newblock {\em {S}addlepoint {A}pproximations with {A}pplications}, volume~22. +\newblock Cambridge University Press, 2007. + +\bibitem[Carr \emph{et~al.}(1998)Carr, Chang, et Madan]{madan1998variance} +Peter~P. \bgroup\fonteauteurs\bgroup Carr\egroup\egroup{}, Eric~C. + \bgroup\fonteauteurs\bgroup Chang\egroup\egroup{} et Dilip~B. + \bgroup\fonteauteurs\bgroup Madan\egroup\egroup{} : +\newblock {T}he {V}ariance {G}amma {P}rocess and {O}ption {P}ricing. +\newblock {\em European Finance Review}, 2\penalty0 + (1)\string:\penalty500\relax 79--105, 1998. + +\bibitem[Carr et Madan(1999)]{carr1999option} +Peter~P. \bgroup\fonteauteurs\bgroup Carr\egroup\egroup{} et Dilip~B. + \bgroup\fonteauteurs\bgroup Madan\egroup\egroup{} : +\newblock {O}ption {V}aluation {U}sing the {F}ast {F}ourier {T}ransform. +\newblock {\em Journal of Computational Finance}, 2\penalty0 + (4)\string:\penalty500\relax 61--73, 1999. + +\bibitem[Crowder(1986)]{crowder1986consistency} +Martin \bgroup\fonteauteurs\bgroup Crowder\egroup\egroup{} : +\newblock {O}n {C}onsistency and {I}nconsistency of {E}stimating {E}quations. +\newblock {\em Econometric Theory}, pages 305--330, 1986. + +\bibitem[Crowder(1987)]{crowder1987linear} +Martin \bgroup\fonteauteurs\bgroup Crowder\egroup\egroup{} : +\newblock {O}n {L}inear and {Q}uadratic {E}stimating {F}unctions. +\newblock {\em Biometrika}, 74\penalty0 (3)\string:\penalty500\relax 591--597, + 1987. + +\bibitem[Daniels(1954)]{daniels1954saddlepoint} +Henry~E. \bgroup\fonteauteurs\bgroup Daniels\egroup\egroup{} : +\newblock {S}addlepoint {A}pproximations in {S}tatistics. +\newblock {\em The Annals of Mathematical Statistics}, pages 631--650, 1954. + +\bibitem[Derman(1996)]{derman1996modelrisk} +Emanuel \bgroup\fonteauteurs\bgroup Derman\egroup\egroup{} : +\newblock {M}odel {R}isk. +\newblock Rapport technique, Goldman Sachs, 1996. + +\bibitem[Dodge(2004)]{dodge2004statistique} +Yadolah \bgroup\fonteauteurs\bgroup Dodge\egroup\egroup{} : +\newblock {\em {S}tatistique: {D}ictionnaire Encyclop{\'e}dique}. +\newblock Springer Verlag France, 2004. + +\bibitem[Epps(2007)]{epps2007pricing} +Thomas~W. \bgroup\fonteauteurs\bgroup Epps\egroup\egroup{} : +\newblock {\em {P}ricing {D}erivative {S}ecurities}. +\newblock World Scientific Publishing Company incorporated, 2007. + +\bibitem[Epps et Pulley(1983)]{epps1983test} +Thomas~W. \bgroup\fonteauteurs\bgroup Epps\egroup\egroup{} et Lawrence~B. + \bgroup\fonteauteurs\bgroup Pulley\egroup\egroup{} : +\newblock {A} {T}est for {N}ormality {B}ased on the {E}mpirical + {C}haracteristic {F}unction. +\newblock {\em Biometrika}, 70\penalty0 (3)\string:\penalty500\relax 723--726, + 1983. + +\bibitem[Everitt et Skrondal(2006)]{everitt2006cambridge} +Brian \bgroup\fonteauteurs\bgroup Everitt\egroup\egroup{} et Anders + \bgroup\fonteauteurs\bgroup Skrondal\egroup\egroup{} : +\newblock {\em {T}he {C}ambridge {D}ictionary of {S}tatistics}, volume~4. +\newblock Cambridge University Press, 2006. + +\bibitem[Fama et French(1993)]{fama1993common} +Eugene~F. \bgroup\fonteauteurs\bgroup Fama\egroup\egroup{} et Kenneth~R. + \bgroup\fonteauteurs\bgroup French\egroup\egroup{} : +\newblock {C}ommon {R}isk {F}actors in the {R}eturns on {S}tocks and {B}onds. +\newblock {\em Journal of Financial Economics}, 33\penalty0 + (1)\string:\penalty500\relax 3--56, 1993. + +\bibitem[Feuerverger et McDunnough(1981)]{feuerverger1981efficiency} +Andrey \bgroup\fonteauteurs\bgroup Feuerverger\egroup\egroup{} et Philip + \bgroup\fonteauteurs\bgroup McDunnough\egroup\egroup{} : +\newblock {O}n the {E}fficiency of {E}mpirical {C}haracteristic {F}unction + {P}rocedures. +\newblock {\em Journal of the Royal Statistical Society. Series B + (methodological)}, pages 20--27, 1981. + +\bibitem[Fox et Taqqu(1986)]{fox1986large} +Robert \bgroup\fonteauteurs\bgroup Fox\egroup\egroup{} et Murad~S. + \bgroup\fonteauteurs\bgroup Taqqu\egroup\egroup{} : +\newblock {L}arge-sample {P}roperties of {P}arameter {E}stimates for {S}trongly + {D}ependent {S}tationary {G}aussian {T}ime {S}eries. +\newblock {\em The Annals of Statistics}, 14\penalty0 + (2)\string:\penalty500\relax 517--532, 1986. + +\bibitem[Gil-Pelaez(1951)]{gil1951note} +J.~\bgroup\fonteauteurs\bgroup Gil-Pelaez\egroup\egroup{} : +\newblock {N}ote on the {I}nversion {T}heorem. +\newblock {\em Biometrika}, 38\penalty0 (3-4)\string:\penalty500\relax + 481--482, 1951. + +\bibitem[Gourieroux et Monfort(1989)]{gourieroux1989statistique} +Christian \bgroup\fonteauteurs\bgroup Gourieroux\egroup\egroup{} et Alain + \bgroup\fonteauteurs\bgroup Monfort\egroup\egroup{} : +\newblock {\em {S}tatistique et Mod{\`e}les {\'E}conom{\'e}triques: Notions + G{\'e}n{\'e}rales, Estimation, Pr{\'e}vision, Algorithmes}, volume~1. +\newblock Economica, 1989. + +\bibitem[Hall(2005)]{hall2005generalized} +Alastair~R. \bgroup\fonteauteurs\bgroup Hall\egroup\egroup{} : +\newblock {\em {G}eneralized {M}ethod of {M}oments}. +\newblock Oxford University Press, 2005. + +\bibitem[Hamilton(1994)]{hamilton1994time} +James~Douglas \bgroup\fonteauteurs\bgroup Hamilton\egroup\egroup{} : +\newblock {\em {T}ime {S}eries {A}nalysis}, volume~2. +\newblock Cambridge University Press, 1994. + +\bibitem[Hansen(1982)]{hansen1982large} +Lars~Peter \bgroup\fonteauteurs\bgroup Hansen\egroup\egroup{} : +\newblock {L}arge {S}ample {P}roperties of {G}eneralized {M}ethod of {M}oments + {E}stimators. +\newblock {\em Econometrica}, pages 1029--1054, 1982. + +\bibitem[Henze(1990)]{henze1990approximation} +Norbert \bgroup\fonteauteurs\bgroup Henze\egroup\egroup{} : +\newblock {A}n {A}pproximation to the {L}imit {D}istribution of the + {E}pps-{P}ulley {T}est {S}tatistic for {N}ormality. +\newblock {\em Metrika}, 37\penalty0 (1)\string:\penalty500\relax 7--18, 1990. + +\bibitem[Heston(1993)]{heston1993closed} +Steven~L. \bgroup\fonteauteurs\bgroup Heston\egroup\egroup{} : +\newblock {A} {C}losed-{F}orm {S}olution for {O}ptions with {S}tochastic + {V}olatility with {A}pplications to {B}ond and {C}urrency {O}ptions. +\newblock {\em Review of Financial Studies}, 6\penalty0 + (2)\string:\penalty500\relax 327--343, 1993. + +\bibitem[Hinkley et Revankar(1977)]{hinkley1977estimation} +David~V. \bgroup\fonteauteurs\bgroup Hinkley\egroup\egroup{} et Nagesh~S. + \bgroup\fonteauteurs\bgroup Revankar\egroup\egroup{} : +\newblock {E}stimation of the {P}areto {L}aw from {U}nderreported {D}ata: {A} + {F}urther {A}nalysis. +\newblock {\em Journal of Econometrics}, 5\penalty0 + (1)\string:\penalty500\relax 1--11, 1977. + +\bibitem[Hogg et Craig(1978)]{hogg1978introduction} +Robert~V. \bgroup\fonteauteurs\bgroup Hogg\egroup\egroup{} et Allen + \bgroup\fonteauteurs\bgroup Craig\egroup\egroup{} : +\newblock {\em {I}ntroduction to {M}athematical {S}tatistics}. +\newblock Macmillan, 1978. + +\bibitem[Itkin(2005)]{itkin2005pricing} +Andrey \bgroup\fonteauteurs\bgroup Itkin\egroup\egroup{} : +\newblock {P}ricing {O}ptions with {VG} {M}odel {U}sing {FFT}. +\newblock {\em arXiv preprint physics/0503137}, 2005. + +\bibitem[{Jingbo Wang} et Marsaglia(2003)]{wang2003evaluating} +Wai Wan~Tsang \bgroup\fonteauteurs\bgroup {Jingbo Wang}\egroup\egroup{} et + George \bgroup\fonteauteurs\bgroup Marsaglia\egroup\egroup{} : +\newblock {E}valuating {K}olmogorov's {D}istribution. +\newblock {\em Journal of Statistical Software}, 8\penalty0 + (18)\string:\penalty500\relax 1--4, 2003. + +\bibitem[Kotz \emph{et~al.}(2001)Kotz, Kozubowski, et + Podg{\'o}rski]{kotz2001laplace} +Samuel \bgroup\fonteauteurs\bgroup Kotz\egroup\egroup{}, Tomasz~J. + \bgroup\fonteauteurs\bgroup Kozubowski\egroup\egroup{} et Krzystof + \bgroup\fonteauteurs\bgroup Podg{\'o}rski\egroup\egroup{} : +\newblock {\em {T}he {L}aplace {D}istribution and {G}eneralizations: {A} + {R}evisit with {A}pplications to {C}ommunications, {E}conomics, + {E}ngineering, and {F}inance}. +\newblock Progress in Mathematics Series. Birkh{\"a}user, 2001. + +\bibitem[Koutrouvelis(1980)]{KOUTROUVELIS01011980} +Ioannis~A. \bgroup\fonteauteurs\bgroup Koutrouvelis\egroup\egroup{} : +\newblock {{A} {G}oodness-{O}f-{F}it {T}est of {S}imple {H}ypotheses {B}ased on + the {E}mpirical {C}haracteristic {F}unction}. +\newblock {\em {B}iometrika}, 67\penalty0 (1)\string:\penalty500\relax + 238--240, 1980. + +\bibitem[Kozubowski et Podg{\'o}rski(1999)]{kozubowski1999class} +Tomasz~J. \bgroup\fonteauteurs\bgroup Kozubowski\egroup\egroup{} et Krzysztof + \bgroup\fonteauteurs\bgroup Podg{\'o}rski\egroup\egroup{} : +\newblock {A} {C}lass of {A}symmetric {D}istributions. +\newblock {\em Actuarial Research Clearing House}, 1\string:\penalty500\relax + 113--134, 1999. + +\bibitem[Kozubowski et Podg{\'o}rski(2001)]{kozubowski2001asymmetric} +Tomasz~J. \bgroup\fonteauteurs\bgroup Kozubowski\egroup\egroup{} et Krzysztof + \bgroup\fonteauteurs\bgroup Podg{\'o}rski\egroup\egroup{} : +\newblock {A}symmetric {L}aplace {L}aws and {M}odeling {F}inancial {D}ata. +\newblock {\em {M}athematical and {C}omputer {M}odeling}, 34\penalty0 + (9)\string:\penalty500\relax 1003--1021, 2001. + +\bibitem[Kyprianou(2007)]{kyprianou2007introductory} +Andreas~E. \bgroup\fonteauteurs\bgroup Kyprianou\egroup\egroup{} : +\newblock {\em {I}ntroductory {L}ectures on {F}luctuations of {L}{\'e}vy + {P}rocesses with {A}pplications}. +\newblock Springer, 2007. + +\bibitem[Lugannani et Rice(1980)]{lugannani1980saddle} +Robert \bgroup\fonteauteurs\bgroup Lugannani\egroup\egroup{} et Stephen + \bgroup\fonteauteurs\bgroup Rice\egroup\egroup{} : +\newblock {S}addle {P}oint {A}pproximation for the {D}istribution of the {S}um + of {I}ndependent {R}andom {V}ariables. +\newblock {\em Advances in Applied Probability}, pages 475--490, 1980. + +\bibitem[Lukacs(1960)]{lukacs1960characteristic} +Eugene \bgroup\fonteauteurs\bgroup Lukacs\egroup\egroup{} : +\newblock {\em {C}haracteristic {F}unctions}, volume~4. +\newblock Griffin London, 1960. + +\bibitem[Luong et Thompson(1987)]{luong1987minimum} +Andrew \bgroup\fonteauteurs\bgroup Luong\egroup\egroup{} et Mary~E. + \bgroup\fonteauteurs\bgroup Thompson\egroup\egroup{} : +\newblock {M}inimum-{D}istance {M}ethods {B}ased on {Q}uadratic {D}istances for + {T}ransforms. +\newblock {\em Canadian Journal of Statistics}, 15\penalty0 + (3)\string:\penalty500\relax 239--251, 1987. + +\bibitem[Madan et Seneta(1990)]{madan1990variance} +Dilip~B. \bgroup\fonteauteurs\bgroup Madan\egroup\egroup{} et Eugene + \bgroup\fonteauteurs\bgroup Seneta\egroup\egroup{} : +\newblock {T}he {V}ariance {G}amma {M}odel for {S}hare {M}arket {R}eturns. +\newblock {\em Journal of Business}, pages 511--524, 1990. + +\bibitem[Mandelbrot(1963)]{mandelbrot1963variation} +Benoit \bgroup\fonteauteurs\bgroup Mandelbrot\egroup\egroup{} : +\newblock {T}he {V}ariation of {C}ertain {S}peculative {P}rices. +\newblock {\em Journal of Business}, pages 394--419, 1963. + +\bibitem[Merton(1976)]{merton1976option} +Robert~C. \bgroup\fonteauteurs\bgroup Merton\egroup\egroup{} : +\newblock {O}ption {P}ricing when {U}nderlying {S}tock {R}eturns {A}re + {D}iscontinuous. +\newblock {\em Journal of Financial Economics}, 3\penalty0 + (1)\string:\penalty500\relax 125--144, 1976. + +\bibitem[Mikosch \emph{et~al.}(2001)Mikosch, Resnick, et + Barndorff-Nielsen]{barndorff2001levy} +Thomas \bgroup\fonteauteurs\bgroup Mikosch\egroup\egroup{}, Sidney~I. + \bgroup\fonteauteurs\bgroup Resnick\egroup\egroup{} et Ole~E. + \bgroup\fonteauteurs\bgroup Barndorff-Nielsen\egroup\egroup{} : +\newblock {\em {L}{\'e}vy {P}rocesses: {T}heory and {A}pplications}. +\newblock Birkh{\"a}user, 2001. + +\bibitem[Mitchell(1916)]{mitchell1916critique} +Wesley~C. \bgroup\fonteauteurs\bgroup Mitchell\egroup\egroup{} : +\newblock {A} {C}ritique of {I}ndex {N}umbers of the {P}rices of {S}tocks. +\newblock {\em The Journal of Political Economy}, pages 625--693, 1916. + +\bibitem[Newey et McFadden(1994)]{newey1994large} +Whitney~K. \bgroup\fonteauteurs\bgroup Newey\egroup\egroup{} et Daniel + \bgroup\fonteauteurs\bgroup McFadden\egroup\egroup{} : +\newblock {L}arge {S}ample {E}stimation and {H}ypothesis {T}esting. +\newblock {\em Handbook of Econometrics}, 4\string:\penalty500\relax + 2111--2245, 1994. + +\bibitem[Newey et West(1987)]{newey1987hypothesis} +Whitney~K. \bgroup\fonteauteurs\bgroup Newey\egroup\egroup{} et Kenneth~D. + \bgroup\fonteauteurs\bgroup West\egroup\egroup{} : +\newblock {H}ypothesis {T}esting with {E}fficient {M}ethod of {M}oments + {E}stimation. +\newblock {\em International Economic Review}, 28\penalty0 + (3)\string:\penalty500\relax 777--787, 1987. + +\bibitem[Praetz(1972)]{praetz1972distribution} +Peter~D. \bgroup\fonteauteurs\bgroup Praetz\egroup\egroup{} : +\newblock {T}he {D}istribution of {S}hare {P}rice {C}hanges. +\newblock {\em Journal of Business}, pages 49--55, 1972. + +\bibitem[Press(1967)]{press1967compound} +S.~James \bgroup\fonteauteurs\bgroup Press\egroup\egroup{} : +\newblock {A} {C}ompound {E}vents {M}odel for {S}ecurity {P}rices. +\newblock {\em Journal of Business}, pages 317--335, 1967. + +\bibitem[{Sato}(1999)]{sato1999levy} +{Ken-Iti} \bgroup\fonteauteurs\bgroup {Sato}\egroup\egroup{} : +\newblock {\em {L}{\'e}vy {P}rocesses and {I}nfinitely {D}ivisible + {D}istributions}. +\newblock Cambridge Studies in Advanced Mathematics. Cambridge University + Press, 1999. + +\bibitem[Schoutens(2003)]{schoutens2003levy} +Wim \bgroup\fonteauteurs\bgroup Schoutens\egroup\egroup{} : +\newblock {\em {L}{\'e}vy {P}rocesses in {F}inance}. +\newblock Wiley, 2003. + +\bibitem[Scott et Dong(2012)]{RpackageVarianceGamma} +David \bgroup\fonteauteurs\bgroup Scott\egroup\egroup{} et Christine~Yang + \bgroup\fonteauteurs\bgroup Dong\egroup\egroup{} : +\newblock {\em {{v}ariancegamma: {T}he {V}ariance {G}amma {D}istribution}}, + 2012. +\newblock URL \url{http://CRAN.R-project.org/package=VarianceGamma}. +\newblock Consulté le 24 janvier 2014. + +\bibitem[Seneta(2004)]{seneta2004fitting} +Eugene \bgroup\fonteauteurs\bgroup Seneta\egroup\egroup{} : +\newblock {F}itting the {V}ariance {G}amma {M}odel to {F}inancial {D}ata. +\newblock {\em Journal of Applied Probability}, pages 177--187, 2004. + +\bibitem[Shapiro et Wilk(1965)]{shapiro1965analysis} +Samuel~Sanford \bgroup\fonteauteurs\bgroup Shapiro\egroup\egroup{} et Martin~B. + \bgroup\fonteauteurs\bgroup Wilk\egroup\egroup{} : +\newblock {A}n {A}nalysis of {V}ariance {T}est for {N}ormality ({C}omplete + {S}amples). +\newblock {\em Biometrika}, 52\penalty0 (3/4)\string:\penalty500\relax + 591--611, 1965. + +\bibitem[Shephard(1991)]{shephard1991characteristic} +Neil~G. \bgroup\fonteauteurs\bgroup Shephard\egroup\egroup{} : +\newblock {F}rom {C}haracteristic {F}unction to {D}istribution {F}unction: {A} + {S}imple {F}ramework for the {T}heory. +\newblock {\em Econometric Theory}, 7\penalty0 (04)\string:\penalty500\relax + 519--529, 1991. + +\bibitem[Spiegel et Liu(1999)]{spiegel1999schaum} +Murray~R. \bgroup\fonteauteurs\bgroup Spiegel\egroup\egroup{} et John + \bgroup\fonteauteurs\bgroup Liu\egroup\egroup{} : +\newblock {\em {S}chaum's {M}athematical {H}andbook of {F}ormulas and + {T}ables}, volume 1000. +\newblock McGraw-Hill, 1999. + +\bibitem[Stuart et Ord(1987)]{stuart1987kendall} +Alan \bgroup\fonteauteurs\bgroup Stuart\egroup\egroup{} et J.~Keith + \bgroup\fonteauteurs\bgroup Ord\egroup\egroup{} : +\newblock {\em {K}endall{\rq}s {A}dvanced {T}heory of {S}tatistics, {V}ol. 1}. +\newblock Oxford University Press, 1987. + +\bibitem[Teschl(2004)]{teschl2004topics} +Gerald \bgroup\fonteauteurs\bgroup Teschl\egroup\egroup{} : +\newblock {\em {T}opics in {R}eal and {F}unctional {A}nalysis}, 2004. +\newblock URL \url{http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html}. +\newblock Consulté le 24 janvier 2014. + +\bibitem[Walter(1995)]{walterlevy} +Christian \bgroup\fonteauteurs\bgroup Walter\egroup\egroup{} : +\newblock {L}{\'e}vy-{S}tability {U}nder {A}ddition and {F}ractal {S}tructure + of {M}arkets: {I}mplications for the {A}ctuaries and {E}mphasized + {E}xamination of {MATIF} {N}ational {C}ontract. +\newblock \emph{In} {\em Proceedings of the 5th AFIR colloquium}. International + Actuarial Association, 1995. + +\bibitem[Wendel(1961)]{wendel1961non} +J.~G. \bgroup\fonteauteurs\bgroup Wendel\egroup\egroup{} : +\newblock {T}he {N}on-{A}bsolute {C}onvergence of {G}il-{P}elaez' {I}nversion + {I}ntegral. +\newblock {\em The Annals of Mathematical Statistics}, 32\penalty0 + (1)\string:\penalty500\relax 338--339, 1961. + +\bibitem[Wolfowitz(1957)]{wolfowitz1957minimum} +Jacob \bgroup\fonteauteurs\bgroup Wolfowitz\egroup\egroup{} : +\newblock {T}he {M}inimum {D}istance {M}ethod. +\newblock {\em The Annals of Mathematical Statistics}, pages 75--88, 1957. + +\bibitem[Wooldridge(2001)]{wooldridge2001econometric} +Jeffrey~M. \bgroup\fonteauteurs\bgroup Wooldridge\egroup\egroup{} : +\newblock {\em {E}conometric {A}nalysis of {C}ross {S}ection and {P}anel + {D}ata}. +\newblock MIT press, 2001. + +\end{thebibliography} diff --git a/memoire/gabarit-maitrise.blg b/memoire/908144032.blg similarity index 73% rename from memoire/gabarit-maitrise.blg rename to memoire/908144032.blg index 5221feb..d9ed9aa 100644 --- a/memoire/gabarit-maitrise.blg +++ b/memoire/908144032.blg @@ -1,7 +1,7 @@ This is BibTeX, Version 0.99d (TeX Live 2013/Debian) Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 -The top-level auxiliary file: gabarit-maitrise.aux -The style file: plainnatmod.bst +The top-level auxiliary file: 908144032.aux +The style file: plainnat-fr.bst A level-1 auxiliary file: resume.aux A level-1 auxiliary file: abstract.aux A level-1 auxiliary file: remerciements.aux @@ -22,37 +22,37 @@ A level-1 auxiliary file: annexe3.aux A level-1 auxiliary file: deed.aux Database file #1: memoire.bib You've used 63 entries, - 2773 wiz_defined-function locations, - 966 strings with 14138 characters, -and the built_in function-call counts, 25331 in all, are: -= -- 2355 -> -- 954 + 2774 wiz_defined-function locations, + 985 strings with 14543 characters, +and the built_in function-call counts, 28653 in all, are: += -- 3018 +> -- 897 < -- 81 + -- 342 - -- 279 -* -- 1694 -:= -- 3810 -add.period$ -- 194 +* -- 1747 +:= -- 5134 +add.period$ -- 131 call.type$ -- 63 -change.case$ -- 261 +change.case$ -- 324 chr.to.int$ -- 63 -cite$ -- 126 +cite$ -- 63 duplicate$ -- 1376 -empty$ -- 2359 +empty$ -- 2323 format.name$ -- 390 -if$ -- 5435 +if$ -- 6133 int.to.chr$ -- 1 int.to.str$ -- 1 missing$ -- 80 -newline$ -- 325 +newline$ -- 326 num.names$ -- 252 pop$ -- 595 preamble$ -- 1 -purify$ -- 219 +purify$ -- 282 quote$ -- 0 -skip$ -- 1044 +skip$ -- 1089 stack$ -- 0 -substring$ -- 1128 +substring$ -- 1758 swap$ -- 194 text.length$ -- 21 text.prefix$ -- 0 @@ -61,4 +61,4 @@ type$ -- 593 warning$ -- 0 while$ -- 279 width$ -- 0 -write$ -- 816 +write$ -- 817 diff --git a/memoire/gabarit-maitrise.lof b/memoire/908144032.lof similarity index 100% rename from memoire/gabarit-maitrise.lof rename to memoire/908144032.lof diff --git a/memoire/gabarit-maitrise.log b/memoire/908144032.log similarity index 93% rename from memoire/gabarit-maitrise.log rename to memoire/908144032.log index 332be00..12c34ba 100644 --- a/memoire/gabarit-maitrise.log +++ b/memoire/908144032.log @@ -1,9 +1,9 @@ -This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2013.11.30) 29 JAN 2014 23:01 +This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2013.11.30) 3 FEB 2014 18:19 entering extended mode restricted \write18 enabled. %&-line parsing enabled. -**\input gabarit-maitrise.tex -(./gabarit-maitrise.tex (./ulthese.cls +**\input 908144032.tex +(./908144032.tex (./ulthese.cls Document Class: ulthese 2012/09/30 v1.0 Classe pour les theses et memoires de l 'Universite Laval (/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty @@ -1511,12 +1511,12 @@ Package: environ 2013/04/01 v0.3 A new environment syntax (/usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty Package: trimspaces 2009/09/17 v1.1 Trim spaces around a token list )))) -(./gabarit-maitrise.aux (./resume.aux) (./abstract.aux) (./remerciements.aux) +(./908144032.aux (./resume.aux) (./abstract.aux) (./remerciements.aux) (./introduction.aux) (./chapitre1.aux) (./chapitre2.aux) (./chapitre3.aux) (./chapitre4.aux) (./chapitre5.aux) (./chapitre6.aux) (./chapitre7.aux) (./chapitre8.aux) (./chapitre9.aux) (./conclusion.aux) (./annexe1.aux) (./annexe2.aux) (./annexe3.aux) (./deed.aux)) -\openout1 = `gabarit-maitrise.aux'. +\openout1 = `908144032.aux'. 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Couper un long titre ou un long +%% sous-titre manuellement avec \\. Terminer une éventuelle deuxième +%% ligne de titre ou de sous-titre avec \vspace*{-\baselineskip}. +\titre{Modélisation des rendements financiers \\ à l'aide de la + distribution de Laplace \\ asymétrique généralisée + \vspace*{-\baselineskip}} +% \titre{Ceci est un exemple de long titre \\ +% avec saut de ligne manuel \vspace*{-\baselineskip}} +% \soustitre{Sous-titre le cas échéant} +% \soustitre{Ceci est un exemple de long-titre \\ +% avec saut de ligne manuel \vspace*{-\baselineskip}} +\auteur{François Pelletier} \programme{Maîtrise en actuariat} +\annee{2014} +\MSc % ou l'un de \LLM, \MA, \MMus, \MServSoc, \MScGeogr, \MATDR +\setcounter{tocdepth}{4} +\begin{document} +\newtheorem{theo}{Théorème}[subsection] +\newtheorem{hypothese}{Hypothèse}[chapter] + +\frontmatter % pages liminaires + +\pagetitre % production de la page titre + +\include{resume} % résumé français +\include{abstract} % résumé anglais +\cleardoublepage + +\tableofcontents % production de la TdM +\cleardoublepage + +\listoftables % production de la liste des tableaux +\cleardoublepage + +\listoffigures % production de la liste des figures +\cleardoublepage + +%\dedicace{Dédicace si désiré} +%\cleardoublepage + +\epigraphe{Dieu ne se soucie pas de nos difficultés mathématiques. Il intègre empiriquement.}{Albert Einstein} +\cleardoublepage + +\include{remerciements} % remerciements + +\mainmatter % corps du document + +\include{introduction} % introduction +\include{chapitre1} % chapitre 1 +\include{chapitre2} % chapitre 2, etc. +\include{chapitre3} +\include{chapitre4} +\include{chapitre5} +\include{chapitre6} +\include{chapitre7} +\include{chapitre8} +\include{chapitre9} +\include{conclusion} % conclusion + +\appendix % annexes le cas échéant + +\include{annexe1} % Annexe 1 +\include{annexe2} % Annexe 2 +\include{annexe3} % Annexe 3 + +\bibliography{memoire} % production de la bibliographie + +\include{deed} + +\end{document} diff --git a/memoire/gabarit-maitrise.toc b/memoire/908144032.toc similarity index 100% rename from memoire/gabarit-maitrise.toc rename to memoire/908144032.toc diff --git a/memoire/MD5SUM b/memoire/MD5SUM deleted file mode 100644 index 402fee4..0000000 --- a/memoire/MD5SUM +++ /dev/null @@ -1 +0,0 @@ -cb86f43ece0fa685e93c596c18426965 908144032.pdf diff --git a/memoire/chapitre1.tex b/memoire/chapitre1.tex index 10db086..a60d6d6 100644 --- a/memoire/chapitre1.tex +++ b/memoire/chapitre1.tex @@ -385,7 +385,7 @@ titres boursiers, entre 1890 et 1915 (figure \ref{fig:mitchell1}). Une asymétrie négative des rendements sera alors présente. \begin{figure}[!ht] \centering - \includegraphics[scale=0.75]{../graphiques/mitchell1.pdf} + \includegraphics[scale=0.75]{./graphiques/mitchell1.pdf} \caption{Distribution des rendements annuels de 40 titres boursiers, de 1890 à 1915, Table XVIII de \cite{mitchell1916critique}} \label{fig:mitchell1} diff --git a/memoire/chapitre2.tex b/memoire/chapitre2.tex index a6fa87e..c9d440e 100644 --- a/memoire/chapitre2.tex +++ b/memoire/chapitre2.tex @@ -220,7 +220,7 @@ hauteur de saut $X_1$, de distribution normale. On obtient ainsi le premier incrément de la trajectoire, tel qu'illustré à la figure \ref{fig:increment1}. \begin{figure}[!ht] - \centering \input{"../graphiques/increment1.tex"} + \centering \input{"./graphiques/increment1.tex"} \caption{Premier incrément d'un processus subordonné} \label{fig:increment1} \end{figure} @@ -230,7 +230,7 @@ trouve à la figure \ref{fig:simgammagauss}. \begin{figure}[!ht] \centering - \includegraphics[scale=.8]{"../graphiques/CH3-SIMGAMMAGAUSS"} + \includegraphics[scale=.8]{"./graphiques/CH3-SIMGAMMAGAUSS"} \caption{Simulation d'un processus de Wiener subordonné par un processus gamma} \label{fig:simgammagauss} @@ -548,7 +548,7 @@ avec différents paramètres d'asymétrie et d'aplatissement à la figure \ref{fig:densiteGAL}. \begin{figure}[!ht] \centering - \includegraphics[scale=0.8]{../graphiques/dGAL-exemples.pdf} + \includegraphics[scale=0.8]{./graphiques/dGAL-exemples.pdf} \caption{Fonction de densité de la distribution Laplace asymétrique généralisée avec différents paramètres: $GAL(y;\theta,\sigma,\kappa,\tau)$} @@ -732,7 +732,7 @@ valeur supérieure à 1. \begin{figure}[!ht] \centering - \includegraphics[scale=0.8]{"../graphiques/CH3-SIMULGAL0121"} + \includegraphics[scale=0.8]{"./graphiques/CH3-SIMULGAL0121"} \caption{Histogramme et estimateur de densité par noyau de 2500 réalisations de la variable aléatoire $Y\sim GAL(\theta=0,\sigma=1,\kappa=2,\tau=1)$} diff --git a/memoire/chapitre3.tex b/memoire/chapitre3.tex index 4d79fbd..d132663 100644 --- a/memoire/chapitre3.tex +++ b/memoire/chapitre3.tex @@ -19,7 +19,7 @@ celui-ci a comme désavantage d'être très peu précis dans l'évaluation des probabilités aux extrémités du support de la densité (figure \ref{fig:probdroite}). \begin{figure}[!ht] - \centering \input{../graphiques/probdroite.tex} + \centering \input{./graphiques/probdroite.tex} \caption{Probabilité à l'extrémité du support} \label{fig:probdroite} \end{figure} @@ -96,7 +96,7 @@ variable aléatoire $Z_{w_0}$ de distribution normale standard tronquée de telle sorte qu'elle ne puisse prendre des valeurs supérieures au point $w_0$. \begin{figure}[!ht] - \centering \input{../graphiques/normaletronque.tex} + \centering \input{./graphiques/normaletronque.tex} \caption{Distribution normale tronquée à $w_0$} \label{fig:normaletronque} \end{figure} diff --git a/memoire/chapitre9.tex b/memoire/chapitre9.tex index 67559dc..4f30cf2 100644 --- a/memoire/chapitre9.tex +++ b/memoire/chapitre9.tex @@ -18,7 +18,7 @@ la figure \ref{fig:seriechronoR1} sous forme de série chronologique. \begin{figure}[!ht] \centering \includegraphics[height=4in, - width=4in]{../graphiques/ABBEYN-chronologie.pdf} + width=4in]{./graphiques/ABBEYN-chronologie.pdf} \caption{Représentation en série chronologique de l'échantillon $R_1$} \label{fig:seriechronoR1} @@ -69,7 +69,7 @@ d'une courbe de densité à la figure \ref{fig:distributionR1}. \begin{figure}[!ht] \centering - \includegraphics[height=4in,width=4in]{../graphiques/ABBEYN-histogramme.pdf} + \includegraphics[height=4in,width=4in]{./graphiques/ABBEYN-histogramme.pdf} \caption{Distribution de la variable aléatoire $R_1$} \label{fig:distributionR1} \end{figure} @@ -109,7 +109,7 @@ quantiles empiriques avec ceux de la loi normale présenté à la figure \begin{figure}[!ht] \centering \includegraphics[height=4in, - width=4in]{../graphiques/ABBEYN-qq.pdf} + width=4in]{./graphiques/ABBEYN-qq.pdf} \caption{Graphique Quantile-Quantile} \label{fig:qqplotR1} \end{figure} @@ -395,7 +395,7 @@ On résout d'abord l'équation du point de selle $r=0.01$ à la figure \ref{fig:equationptselle0.01R1} : \begin{figure}[!ht] \centering - \includegraphics[scale=0.5]{../graphiques/pointdeselleGMM.png} + \includegraphics[scale=0.5]{./graphiques/pointdeselleGMM.png} \caption{Équation du point de selle pour $r=0.01$} \label{fig:equationptselle0.01R1} \end{figure} @@ -513,7 +513,7 @@ la table \ref{tab:intapproxpointselleR1}. \begin{figure}[!ht] \centering \includegraphics[height=6in, - width=6in]{../graphiques/ABBEYN-densiteGALmu-7.pdf} + width=6in]{./graphiques/ABBEYN-densiteGALmu-7.pdf} \caption{Densité de $R_1^{*}$ selon la méthode des moments généralisée} \label{fig:densite1R1} @@ -522,7 +522,7 @@ la table \ref{tab:intapproxpointselleR1}. \begin{figure}[!ht] \centering \includegraphics[height=6in, - width=6in]{../graphiques/ABBEYN-densiteGALmu-5.pdf} + width=6in]{./graphiques/ABBEYN-densiteGALmu-5.pdf} \caption{Densité de $R_1^{*}$ selon la méthode de l'équation d'estimation optimale} \label{fig:densite3R1} @@ -681,7 +681,7 @@ d'ordre 1 est très précise dans ce contexte. \begin{figure}[!ht] \centering \includegraphics[height=6in, - width=6in]{../graphiques/ABBEYN-callGAL-7.pdf} + width=6in]{./graphiques/ABBEYN-callGAL-7.pdf} \caption{Prix de l'option selon les paramètres estimés avec la méthode des moments généralisée} \label{fig:prix1R1-1} @@ -690,7 +690,7 @@ d'ordre 1 est très précise dans ce contexte. \begin{figure}[!ht] \centering \includegraphics[height=6in, - width=6in]{../graphiques/ABBEYN-callGAL-5.pdf} + width=6in]{./graphiques/ABBEYN-callGAL-5.pdf} \caption{Prix de l'option selon les paramètres estimés avec la méthode de l'équation d'estimation optimale} \label{fig:prix1R1-3} diff --git a/memoire/gabarit-maitrise.bbl b/memoire/gabarit-maitrise.bbl deleted file mode 100644 index 2a01fc7..0000000 --- a/memoire/gabarit-maitrise.bbl +++ /dev/null @@ -1,371 +0,0 @@ -\begin{thebibliography}{63} -\providecommand{\natexlab}[1]{#1} -\providecommand{\url}[1]{\texttt{#1}} -\expandafter\ifx\csname urlstyle\endcsname\relax - \providecommand{\doi}[1]{doi: #1}\else - \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi - 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+\pgfsetlinewidth{0.100000\du} +\pgfsetdash{}{0pt} +\pgfsetdash{}{0pt} +\pgfsetbuttcap +{ +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetfillcolor{dialinecolor} +% was here!!! +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (20.900000\du,-0.200000\du)--(20.900000\du,0.000000\du); +} +\end{tikzpicture} diff --git a/memoire/plainnat-fr.bst b/memoire/plainnat-fr.bst new file mode 100644 index 0000000..76e3707 --- /dev/null +++ b/memoire/plainnat-fr.bst @@ -0,0 +1,1503 @@ +%% +%% Bib. style "plainnat-fr" (version francisee de plainnat.bst) +%% +%% NM, 2005/10/16 +%% markey@lsv.ens-cachan.fr +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Original Copyright (Patrick Daly): +%% +%% Copyright 1993-1999 Patrick W Daly +%% Max-Planck-Institut f\"ur Aeronomie +%% Max-Planck-Str. 2 +%% D-37191 Katlenburg-Lindau +%% Germany +%% E-mail: daly@linmpi.mpg.de +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN +%% archives in directory macros/latex/base/lppl.txt; either +%% version 1 of the License, or any later version. +%% + % Version and source file information: + % \ProvidesFile{natbst.mbs}[1999/05/11 1.6 (PWD)] + % + % BibTeX `plainnat' family + % version 0.99b for BibTeX versions 0.99a or later, + % for LaTeX versions 2.09 and 2e. + % + % For use with the `natbib.sty' package; emulates the corresponding + % member of the `plain' family, but with author-year citations. + % + % With version 6.0 of `natbib.sty', it may also be used for numerical + % citations, while retaining the commands \citeauthor, \citefullauthor, + % and \citeyear to print the corresponding information. + % + % For version 7.0 of `natbib.sty', the KEY field replaces missing + % authors/editors, and the date is left blank in \bibitem. + % + % Includes fields ISBN and ISSN. + % + % Includes field URL for Internet addresses (best used with + % with the url.sty package of Donald Arseneau + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +ENTRY + { address + author + booktitle + chapter + edition + editor + howpublished + institution + isbn + issn + journal + key + month + note + number + organization + pages + publisher + school + series + title + type + url + volume + year + } + {} + { label extra.label sort.label short.list } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% +%% C'est ici que je definis les "traductions". Normalement, y a que ca a changer +%% pour franciser le style... +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +FUNCTION{fr.and}{ %% le "et" entre les deux derniers auteurs +" et " +} + +FUNCTION{fr.editeur}{ %% ", {\'e}diteur" au singulier, ou " ({\'e}diteur)"... +", {\'e}diteur" +} + +FUNCTION{fr.editeurs}{ %% ", {\'e}diteurs" au pluriel, ou " ({\'e}diteurs)"... +", {\'e}diteurs" +} + +FUNCTION{fr.et.al}{ %% " et~al." +" \emph{et~al.}" +} + +FUNCTION{fr.in}{ %% "Dans " +"\emph{In} " +} + +FUNCTION{fr.in.spc}{ %% " dans " +" \emph{in} " +} + +FUNCTION{fr.of}{ %% " de " +" de " +} + +FUNCTION{fr.number}{ %% "num{\'e}ro " +"num{\'e}ro " +} + +FUNCTION{fr.number.maj}{ %% "Num{\'e}ro " +"Num{\'e}ro " +} + +FUNCTION{fr.volume}{ %% "volume" +"volume" +} + +FUNCTION{fr.volume.maj}{ %% "Volume" +"Volume" +} + +FUNCTION{fr.edition}{ %% " {\'e}dition" +" {\'e}dition" +} + +FUNCTION{fr.pages}{ %% "pages" +"pages" +} + +FUNCTION{fr.page}{ %% "page" +"page" +} + +FUNCTION{fr.chapter}{ %% "chapitre" +"chapitre" +} + +FUNCTION{fr.tech.rep}{ %% "Rapport Technique" +"Rapport technique" +} + +FUNCTION{fr.master}{ %% "M{\'e}moire de D.E.A." +"M{\'e}moire de D.E.A." +} + +FUNCTION{fr.phd}{ %% "Th{\`e}se de doctorat" +"Th{\`e}se de doctorat" +} + +FUNCTION{fr.auteurs.style}{ + %% si on veut des small caps sur le LASTNAME de l'auteur + %% Cette fonction est utilisee dans la definition d'une + %% fonction LaTeX. +"\scshape" +} + +FUNCTION{fr.ponctuation.apres.auteurs}{ %% Comme son nom l'indique... +" : " +} + +FUNCTION{fr.deuxpoints}{ %% Pour eviter l'espace avant les ":". +"\string:\penalty500\relax " +} + +MACRO {jan} {"janvier"} +MACRO {feb} {"f\'evrier"} +MACRO {mar} {"mars"} +MACRO {apr} {"avril"} +MACRO {may} {"mai"} +MACRO {jun} {"juin"} +MACRO {jul} {"juillet"} +MACRO {aug} {"ao\^ut"} +MACRO {sep} {"septembre"} +MACRO {oct} {"octobre"} +MACRO {nov} {"novembre"} +MACRO {dec} {"d\'ecembre"} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% +%% La suite, normalement, y a rien a changer... +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +INTEGERS { output.state before.all mid.sentence after.sentence after.block after.authors } + +FUNCTION {init.state.consts} +{ #0 'before.all := + #1 'mid.sentence := + #2 'after.sentence := + #3 'after.block := + #0 'after.authors := +} + +STRINGS { s t } + +FUNCTION{ fr.add.period }{ +after.authors + { fr.ponctuation.apres.auteurs empty$ + { add.period$ } + { fr.ponctuation.apres.auteurs * #0 'after.authors := } + if$} + { add.period$ } +if$ +} + +FUNCTION {output.nonnull} +{ 's := + output.state mid.sentence = + { ", " * write$ } + { output.state after.block = + { fr.add.period write$ + newline$ + "\newblock " write$ + } + { output.state before.all = + 'write$ + { add.period$ " " * write$ } + if$ + } + if$ + mid.sentence 'output.state := + } + if$ + s +} + +FUNCTION {output} +{ duplicate$ empty$ + 'pop$ + 'output.nonnull + if$ +} + +FUNCTION {output.check} +{ 't := + duplicate$ empty$ + { pop$ "empty " t * " in " * cite$ * warning$ } + 'output.nonnull + if$ +} + +FUNCTION {fin.entry} +{ add.period$ + write$ + newline$ +} + +FUNCTION {new.block} +{ output.state before.all = + 'skip$ + { after.block 'output.state := } + if$ +} + +FUNCTION {new.sentence} +{ output.state after.block = + 'skip$ + { output.state before.all = + 'skip$ + { after.sentence 'output.state := } + if$ + } + if$ +} + +FUNCTION {not} +{ { #0 } + { #1 } + if$ +} + +FUNCTION {and} +{ 'skip$ + { pop$ #0 } + if$ +} + +FUNCTION {or} +{ { pop$ #1 } + 'skip$ + if$ +} + +FUNCTION {new.block.checka} +{ empty$ + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.block.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.sentence.checka} +{ empty$ + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {new.sentence.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {field.or.null} +{ duplicate$ empty$ + { pop$ "" } + 'skip$ + if$ +} + +FUNCTION {emphasize} +{ duplicate$ empty$ + { pop$ "" } + { "{\em " swap$ * "}" * } + if$ +} + +INTEGERS { nameptr namesleft numnames } + +FUNCTION {format.names} +{ 's := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr "{ff~}{vv~}\bgroup\fonteauteurs\bgroup {ll}\egroup\egroup{{}}{, jj}" format.name$ 't := + nameptr #1 > + { namesleft #1 > + { ", " * t * } + { %numnames #2 > + % { "," * } + % 'skip$ + %if$ + t "\bgroup\fonteauteurs\bgroup others\egroup\egroup{}" = + { fr.et.al * } + { fr.and * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {format.key} +{ empty$ + { key field.or.null } + { "" } + if$ +} + +FUNCTION {format.authors} +{ author empty$ + { "" } + { author format.names #1 'after.authors := } + if$ +} + +FUNCTION {format.editors} +{ editor empty$ + { "" } + { editor format.names + editor num.names$ #1 > + { fr.editeurs * } + { fr.editeur * } + if$ + } + if$ +} + +FUNCTION {format.isbn} +{ isbn empty$ + { "" } + { new.block "ISBN " isbn * } + if$ +} + +FUNCTION {format.issn} +{ issn empty$ + { "" } + { new.block "ISSN " issn * } + if$ +} + +FUNCTION {format.url} +{ url empty$ + { "" } + { new.block "URL \url{" url * "}" * } + if$ +} + +FUNCTION {format.title} +{ title empty$ + { "" } + { title "t" change.case$ } + if$ +} + +FUNCTION {format.full.names} +{% + % Ici, on laisse les noms en minuscules, on ne les mets pas en small caps. + % + 's := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr + "{vv~}{ll}" format.name$ 't := + nameptr #1 > + { + namesleft #1 > + { ", " * t * } + { + numnames #2 > + { "," * } + 'skip$ + if$ + t "\bgroup\fonteauteurs\bgroup others\egroup\egroup{}" = + { fr.et.al * } + { fr.and * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {author.editor.full} +{ author empty$ + { editor empty$ + { "" } + { editor format.full.names } + if$ + } + { author format.full.names } + if$ +} + +FUNCTION {author.full} +{ author empty$ + { "" } + { author format.full.names } + if$ +} + +FUNCTION {editor.full} +{ editor empty$ + { "" } + { editor format.full.names } + if$ +} + +FUNCTION {make.full.names} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.full + { type$ "proceedings" = + 'editor.full + 'author.full + if$ + } + if$ +} + +FUNCTION {output.bibitem} +{ newline$ + "\bibitem[" write$ + label write$ + ")" make.full.names duplicate$ short.list = + { pop$ } + { * } + if$ + "]{" * write$ + cite$ write$ + "}" write$ + newline$ + "" + before.all 'output.state := +} + +FUNCTION {n.dashify} +{ 't := + "" + { t empty$ not } + { t #1 #1 substring$ "-" = + { t #1 #2 substring$ "--" = not + { "--" * + t #2 global.max$ substring$ 't := + } + { { t #1 #1 substring$ "-" = } + { "-" * + t #2 global.max$ substring$ 't := + } + while$ + } + if$ + } + { t #1 #1 substring$ * + t #2 global.max$ substring$ 't := + } + if$ + } + while$ +} + +FUNCTION {format.date} +{ year duplicate$ empty$ + { "empty year in " cite$ * warning$ + pop$ "" } + 'skip$ + if$ + month empty$ + 'skip$ + { month + " " * swap$ * + } + if$ + extra.label * +} + +FUNCTION {format.btitle} +{ title emphasize +} + +FUNCTION {tie.or.space.connect} +{ duplicate$ text.length$ #3 < + { "~" } + { " " } + if$ + swap$ * * +} + +FUNCTION {either.or.check} +{ empty$ + 'pop$ + { "can't use both " swap$ * " fields in " * cite$ * warning$ } + if$ +} + +FUNCTION {format.bvolume} +{ volume empty$ + { "" } + { fr.volume volume tie.or.space.connect + series empty$ + 'skip$ + { fr.of * series emphasize * } + if$ + "volume and number" number either.or.check + } + if$ +} + +FUNCTION {format.number.series} +{ volume empty$ + { number empty$ + { series field.or.null } + { output.state mid.sentence = + { fr.number } + { fr.number.maj } + if$ + number tie.or.space.connect + series empty$ + { "there's a number but no series in " cite$ * warning$ } + { fr.in.spc * series * } + if$ + } + if$ + } + { "" } + if$ +} + +FUNCTION {format.edition} +{ edition empty$ + { "" } + { output.state mid.sentence = + { edition "l" change.case$ fr.edition * } + { edition "t" change.case$ fr.edition * } + if$ + } + if$ +} + +INTEGERS { multiresult } + +FUNCTION {multi.page.check} +{ 't := + #0 'multiresult := + { multiresult not + t empty$ not + and + } + { t #1 #1 substring$ + duplicate$ "-" = + swap$ duplicate$ "," = + swap$ "+" = + or or + { #1 'multiresult := } + { t #2 global.max$ substring$ 't := } + if$ + } + while$ + multiresult +} + +FUNCTION {format.pages} +{ pages empty$ + { "" } + { pages multi.page.check + { fr.pages pages n.dashify tie.or.space.connect } + { fr.page pages tie.or.space.connect } + if$ + } + if$ +} + +FUNCTION {format.vol.num.pages} +{ volume field.or.null + number empty$ + 'skip$ + { "\penalty0 (" number * ")" * * + volume empty$ + { "there's a number but no volume in " cite$ * warning$ } + 'skip$ + if$ + } + if$ + pages empty$ + 'skip$ + { duplicate$ empty$ + { pop$ format.pages } + { fr.deuxpoints * pages n.dashify * } + if$ + } + if$ +} + +FUNCTION {format.chapter.pages} +{ chapter empty$ + 'format.pages + { type empty$ + { fr.chapter } + { type "l" change.case$ } + if$ + chapter tie.or.space.connect + pages empty$ + 'skip$ + { ", " * format.pages * } + if$ + } + if$ +} + +FUNCTION {format.in.ed.booktitle} +{ booktitle empty$ + { "" } + { editor empty$ + { fr.in booktitle emphasize * } + { fr.in format.editors #1 'after.authors := fr.add.period " " * * booktitle emphasize * } + if$ + } + if$ +} + +FUNCTION {empty.misc.check} +{ author empty$ title empty$ howpublished empty$ + month empty$ year empty$ note empty$ + and and and and and + key empty$ not and + { "all relevant fields are empty in " cite$ * warning$ } + 'skip$ + if$ +} + +FUNCTION {format.thesis.type} +{ type empty$ + 'skip$ + { pop$ + type "t" change.case$ + } + if$ +} + +FUNCTION {format.tr.number} +{ type empty$ + { fr.tech.rep } + 'type + if$ + number empty$ + { "t" change.case$ } + { number tie.or.space.connect } + if$ +} + +FUNCTION {format.article.crossref} +{ key empty$ + { journal empty$ + { "need key or journal for " cite$ * " to crossref " * crossref * + warning$ + "" + } + { fr.in "{\em " * journal * "\/}" * } + if$ + } + { fr.in key * } + if$ + " \citep{" * crossref * "}" * +} + +FUNCTION {format.book.crossref} +{ volume empty$ + { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ + fr.in + } + { fr.volume.maj volume tie.or.space.connect + fr.of * + } + if$ + editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { series empty$ + { "need editor, key, or series for " cite$ * " to crossref " * + crossref * warning$ + "" * + } + { "{\em " * series * "\/}" * } + if$ + } + { key * } + if$ + } + 'skip$ + if$ + ", \citet{" * crossref * "}" * +} + +FUNCTION {format.incoll.inproc.crossref} +{ editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { booktitle empty$ + { "need editor, key, or booktitle for " cite$ * " to crossref " * + crossref * warning$ + "" + } + { fr.in "{\em " * booktitle * "\/}" * } + if$ + } + { fr.in key * } + if$ + } + { fr.in } + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {article} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { journal emphasize "journal" output.check + format.vol.num.pages output + format.date "year" output.check + } + { format.article.crossref output.nonnull + format.pages output + } + if$ + format.issn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {book} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {booklet} +{ output.bibitem + format.authors output + author format.key output + new.block + format.title "title" output.check + howpublished address new.block.checkb + howpublished output + address output + format.date output + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inbook} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + format.chapter.pages "chapter and pages" output.check + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { format.chapter.pages "chapter and pages" output.check + new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {incollection} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.chapter.pages output + new.sentence + publisher "publisher" output.check + address output + format.edition output + format.date "year" output.check + } + { format.incoll.inproc.crossref output.nonnull + format.chapter.pages output + } + if$ + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inproceedings} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.pages output + address empty$ + { organization publisher new.sentence.checkb + organization output + publisher output + format.date "year" output.check + } + { address output.nonnull + format.date "year" output.check + new.sentence + organization output + publisher output + } + if$ + } + { format.incoll.inproc.crossref output.nonnull + format.pages output + } + if$ + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {conference} { inproceedings } + +FUNCTION {manual} +{ output.bibitem + format.authors output + author format.key output + new.block + format.btitle "title" output.check + organization address new.block.checkb + organization output + address output + format.edition output + format.date output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {mastersthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + fr.master format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {misc} +{ output.bibitem + format.authors output + author format.key output + title howpublished new.block.checkb + format.title output + howpublished new.block.checka + howpublished output + format.date output + format.issn output + format.url output + new.block + note output + fin.entry + empty.misc.check +} + +FUNCTION {phdthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.btitle "title" output.check + new.block + fr.phd format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {proceedings} +{ output.bibitem + format.editors output + editor format.key output + new.block + format.btitle "title" output.check + format.bvolume output + format.number.series output + address output + format.date "year" output.check + new.sentence + organization output + publisher output + format.isbn output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {techreport} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + format.tr.number output.nonnull + institution "institution" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {unpublished} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + format.url output + new.block + note "note" output.check + format.date output + fin.entry +} + +FUNCTION {default.type} { misc } + +MACRO {acmcs} {"ACM Computing Surveys"} +MACRO {acta} {"Acta Informatica"} +MACRO {cacm} {"Communications of the ACM"} +MACRO {ibmjrd} {"IBM Journal of Research and Development"} +MACRO {ibmsj} {"IBM Systems Journal"} +MACRO {ieeese} {"IEEE Transactions on Software Engineering"} +MACRO {ieeetc} {"IEEE Transactions on Computers"} +MACRO {ieeetcad} {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"} +MACRO {ipl} {"Information Processing Letters"} +MACRO {jacm} {"Journal of the ACM"} +MACRO {jcss} {"Journal of Computer and System Sciences"} +MACRO {scp} {"Science of Computer Programming"} +MACRO {sicomp} {"SIAM Journal on Computing"} +MACRO {tocs} {"ACM Transactions on Computer Systems"} +MACRO {tods} {"ACM Transactions on Database Systems"} +MACRO {tog} {"ACM Transactions on Graphics"} +MACRO {toms} {"ACM Transactions on Mathematical Software"} +MACRO {toois} {"ACM Transactions on Office Information Systems"} +MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"} +MACRO {tcs} {"Theoretical Computer Science"} + + +READ + +FUNCTION {sortify} +{ purify$ + "l" change.case$ +} + +INTEGERS { len } + +FUNCTION {chop.word} +{ 's := + 'len := + s #1 len substring$ = + { s len #1 + global.max$ substring$ } + 's + if$ +} + +FUNCTION {format.lab.names} +% +% Ici, on laisse les noms en minuscules, on ne les mets pas +% en small caps. +% +{ 's := + s #1 "{vv~}{ll}" format.name$ + s num.names$ duplicate$ + #2 > + { pop$ fr.et.al * } + { #2 < + 'skip$ + { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = + { fr.et.al * } + { fr.and * s #2 "{vv~}{ll}" format.name$ * } + if$ + } + if$ + } + if$ +} + +FUNCTION {author.key.label} +{ author empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.editor.key.label} +{ author empty$ + { editor empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { editor format.lab.names } + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.key.organization.label} +{ author empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "Le " #3 + "La " #3 + "Les " #4 + "The " #4 organization chop.word + chop.word + chop.word + chop.word + #3 text.prefix$ } + if$ + } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {editor.key.organization.label} +{ editor empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "Le " #3 + "La " #3 + "Les " #4 + "The " #4 organization chop.word + chop.word + chop.word + chop.word + #3 text.prefix$ } + if$ + } + 'key + if$ + } + { editor format.lab.names } + if$ +} + +FUNCTION {calc.short.authors} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.key.label + { type$ "proceedings" = + 'editor.key.organization.label + { type$ "manual" = + 'author.key.organization.label + 'author.key.label + if$ + } + if$ + } + if$ + 'short.list := +} + +FUNCTION {calc.label} +{ calc.short.authors + short.list + "(" + * + year duplicate$ empty$ + short.list key field.or.null = or + { pop$ "" } + 'skip$ + if$ + * + 'label := +} + +FUNCTION {sort.format.names} +{ 's := + #1 'nameptr := + "" + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { nameptr #1 > + { " " * } + 'skip$ + if$ + s nameptr "{vv{ } }{ll{ }}{ ff{ }}{ jj{ }}" format.name$ 't := + nameptr numnames = t "others" = and + { fr.et.al * } %% sera "purify"e ensuite... + { t sortify * } + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {sort.format.title} +{ 't := + "Un " #3 + "Une " #4 + "Des " #4 + "Le " #3 + "La " #3 + "Les " #4 + "A " #2 + "An " #3 + "The " #4 t chop.word + chop.word + chop.word + chop.word + chop.word + chop.word + chop.word + chop.word + chop.word + sortify + #1 global.max$ substring$ +} + +FUNCTION {author.sort} +{ author empty$ + { key empty$ + { "to sort, need author or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.editor.sort} +{ author empty$ + { editor empty$ + { key empty$ + { "to sort, need author, editor, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { editor sort.format.names } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.organization.sort} +{ author empty$ + { organization empty$ + { key empty$ + { "to sort, need author, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "Le " #3 + "La " #3 + "Les " #4 + "The " #4 organization chop.word + chop.word + chop.word + chop.word + sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {editor.organization.sort} +{ editor empty$ + { organization empty$ + { key empty$ + { "to sort, need editor, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "Le " #3 + "La " #3 + "Les " #4 + "The " #4 organization chop.word + chop.word + chop.word + chop.word + sortify } + if$ + } + { editor sort.format.names } + if$ +} + +FUNCTION {presort} +{ calc.label + label sortify + " " + * + type$ "book" = + type$ "inbook" = + or + 'author.editor.sort + { type$ "proceedings" = + 'editor.organization.sort + { type$ "manual" = + 'author.organization.sort + 'author.sort + if$ + } + if$ + } + if$ + " " + * + year field.or.null sortify + * + " " + * + title field.or.null + sort.format.title + * + #1 entry.max$ substring$ + 'sort.label := + sort.label * + #1 entry.max$ substring$ + 'sort.key$ := +} + +ITERATE {presort} + +SORT + +STRINGS { longest.label last.label next.extra } + +INTEGERS { longest.label.width last.extra.num number.label } + +FUNCTION {initialize.longest.label} +{ "" 'longest.label := + #0 int.to.chr$ 'last.label := + "" 'next.extra := + #0 'longest.label.width := + #0 'last.extra.num := + #0 'number.label := +} + +FUNCTION {forward.pass} +{ last.label label = + { last.extra.num #1 + 'last.extra.num := + last.extra.num int.to.chr$ 'extra.label := + } + { "a" chr.to.int$ 'last.extra.num := + "" 'extra.label := + label 'last.label := + } + if$ + number.label #1 + 'number.label := +} + +FUNCTION {reverse.pass} +{ next.extra "b" = + { "a" 'extra.label := } + 'skip$ + if$ + extra.label 'next.extra := + extra.label + duplicate$ empty$ + 'skip$ + { "{\natexlab{" swap$ * "}}" * } + if$ + 'extra.label := + label extra.label * 'label := +} + +EXECUTE {initialize.longest.label} + +ITERATE {forward.pass} + +REVERSE {reverse.pass} + +FUNCTION {bib.sort.order} +{ sort.label 'sort.key$ := +} + +ITERATE {bib.sort.order} + +SORT + +FUNCTION {begin.bib} +{ preamble$ empty$ + 'skip$ + { preamble$ write$ newline$ } + if$ + "\begin{thebibliography}{" number.label int.to.str$ * "}" * + write$ newline$ + "\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi" + write$ newline$ + "\expandafter\ifx\csname fonteauteurs\endcsname\relax" + write$ newline$ + "\def\fonteauteurs{" fr.auteurs.style * "}\fi" * write$ newline$ + "\expandafter\ifx\csname url\endcsname\relax" write$ newline$ + " \def\url#1{{\tt #1}}% " write$ newline$ + " \message{You should include the url package}\fi" write$ newline$ +} + +EXECUTE {begin.bib} + +EXECUTE {init.state.consts} + +ITERATE {call.type$} + +FUNCTION {end.bib} +{ newline$ + "\end{thebibliography}" write$ newline$ +} + +EXECUTE {end.bib} diff --git a/memoire/ulthese.pdf b/memoire/ulthese.pdf deleted file mode 100644 index 72b5432fcdc207c8b8f3c07830bc1a7633ea26bb..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 239896 zcmb4}Q;;aix}@8-wc56A+qP}nwr$(CZQJhEwr9_cxiL3l_QRQasHo?Pip;O_PZD`y zQ5ptX7ATVW<&kwL1~v|SdVD)WOMGr_C^~5qTQg^Kdf>qUTeFOJ|4%P6-oEHt~OB8MUs!{c}nA4Tai-84CiW z1Vg7xNx=A8q9f$7dXC>u8_(E%i<^ymNm$F#{?4`Lj%&@MM2wAVy6K zF@cKIaZr-PVwm6p6{3_;`6dAgXAa215YQ;dSmox42>5d*#e*1w2%Z9 z9Tk%qy;eunef_NB@TQ=A#EGj~YU#avdLX12ChSDHkYJbt=HXDwfWmf%D!5!jwkO&_HfGGj)Z?Wf$ZEsBHx=fMm1r(dLA>$vla7`7FWgRA8Rq=X};aAQ( zVHUzy>}34(RF&)}vYt1WM~t(OsS28IVUX^~?4+Hv!Ke!PKb!*(%m3I++h{wij<7Q% zfo?x{XpHEii?!UgT8>$mthKT#Ic7*Cd{04DmXWt}`c6<|{Z_b;21jCohm1Hufd2vp zx}zC+1j8}PZK%Gq@-}4;YhzR- zAnDy>y)(21#Gyi8m&A{Hcf<`efn1n`+G`9IN&;?}xSfh;(b=v`HdGxM9h~syGAZbq6`vFC1oO{%LNoqEv`=ABw`?utdPlr#NCy1&0a?1-lBh 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