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\begin{thebibliography}{62}
\providecommand{\natexlab}[1]{#1}
\providecommand{\url}[1]{\texttt{#1}}
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\providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi
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\bibitem[Kozubowski et Podg{\'o}rski(1999)]{kozubowski1999class}
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\contentsline {figure}{\numberline {2.1}{\ignorespaces Premier incr\IeC {\'e}ment d'un processus subordonn\IeC {\'e}\relax }}{23}{figure.caption.16}
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\contentsline {table}{\numberline {1.1}{\ignorespaces Domaine et r\IeC {\^o}le des param\IeC {\`e}tres de la distribution L stable de Mandelbrot\relax }}{12}{table.caption.14}
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\contentsline {table}{\numberline {9.14}{\ignorespaces Caract\IeC {\'e}ristiques de l'option\relax }}{109}{table.caption.47}
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\BOOKMARK [0][-]{section*.2}{R\351sum\351}{}% 1
\BOOKMARK [0][-]{section*.4}{Abstract}{}% 2
\BOOKMARK [0][-]{section*.5}{Table des mati\350res}{}% 3
\BOOKMARK [0][-]{section*.6}{Liste des tableaux}{}% 4
\BOOKMARK [0][-]{section*.7}{Liste des figures}{}% 5
\BOOKMARK [0][-]{section*.9}{Remerciements}{}% 6
\BOOKMARK [0][-]{section*.11}{Introduction}{}% 7
\BOOKMARK [0][-]{chapter.1}{Les mod\350les de rendements financiers}{}% 8
\BOOKMARK [1][-]{section.1.1}{L'utilisation de mod\350les en finance}{chapter.1}% 9
\BOOKMARK [2][-]{subsection.1.1.1}{Diff\351rents types de mod\350les}{section.1.1}% 10
\BOOKMARK [2][-]{subsection.1.1.2}{Le risque de mod\351lisation}{section.1.1}% 11
\BOOKMARK [1][-]{section.1.2}{Les rendements financiers}{chapter.1}% 12
\BOOKMARK [2][-]{subsection.1.2.1}{D\351finitions et notation}{section.1.2}% 13
\BOOKMARK [2][-]{subsection.1.2.2}{Rendements cumul\351s}{section.1.2}% 14
\BOOKMARK [2][-]{subsection.1.2.3}{Donn\351es disponibles}{section.1.2}% 15
\BOOKMARK [1][-]{section.1.3}{Les premiers mod\350les}{chapter.1}% 16
\BOOKMARK [2][-]{subsection.1.3.1}{Le mod\350le de Bachelier}{section.1.3}% 17
\BOOKMARK [2][-]{subsection.1.3.2}{Proposition de Mandelbrot}{section.1.3}% 18
\BOOKMARK [2][-]{subsection.1.3.3}{Le mod\350le de Press}{section.1.3}% 19
\BOOKMARK [2][-]{subsection.1.3.4}{Le mod\350le de Praetz}{section.1.3}% 20
\BOOKMARK [1][-]{section.1.4}{Conditions essentielles de Madan et Seneta}{chapter.1}% 21
\BOOKMARK [0][-]{chapter.2}{La distribution de Laplace asym\351trique g\351n\351ralis\351e}{}% 22
\BOOKMARK [1][-]{section.2.1}{Le processus de Laplace}{chapter.2}% 23
\BOOKMARK [2][-]{subsection.2.1.1}{Le processus gamma}{section.2.1}% 24
\BOOKMARK [2][-]{subsection.2.1.2}{Le processus de Wiener}{section.2.1}% 25
\BOOKMARK [2][-]{subsection.2.1.3}{Le processus de Laplace est un processus subordonn\351}{section.2.1}% 26
\BOOKMARK [1][-]{section.2.2}{Distribution de Laplace asym\351trique g\351n\351ralis\351e}{chapter.2}% 27
\BOOKMARK [2][-]{subsection.2.2.1}{Fonction caract\351ristique}{section.2.2}% 28
\BOOKMARK [2][-]{subsection.2.2.2}{Invariance d'\351chelle}{section.2.2}% 29
\BOOKMARK [2][-]{subsection.2.2.3}{Fonctions g\351n\351ratrices}{section.2.2}% 30
\BOOKMARK [2][-]{subsection.2.2.4}{Moments et r\364le des param\350tres}{section.2.2}% 31
\BOOKMARK [2][-]{subsection.2.2.5}{Changement d'\351chelle et de localisation}{section.2.2}% 32
\BOOKMARK [2][-]{subsection.2.2.6}{Repr\351sentation alternative et simulation}{section.2.2}% 33
\BOOKMARK [2][-]{subsection.2.2.7}{Fonction de Bessel et densit\351}{section.2.2}% 34
\BOOKMARK [1][-]{section.2.3}{Cas particuliers}{chapter.2}% 35
\BOOKMARK [2][-]{subsection.2.3.1}{Distribution de Laplace asym\351trique}{section.2.3}% 36
\BOOKMARK [1][-]{section.2.4}{Relation avec le mod\350le de madan1990variance}{chapter.2}% 37
\BOOKMARK [0][-]{chapter.3}{Approximation de la densit\351 et de la fonction de r\351partition}{}% 38
\BOOKMARK [1][-]{section.3.1}{L'approximation de Laplace}{chapter.3}% 39
\BOOKMARK [1][-]{section.3.2}{L'approximation de Temme}{chapter.3}% 40
\BOOKMARK [1][-]{section.3.3}{La m\351thode du point de selle}{chapter.3}% 41
\BOOKMARK [2][-]{subsection.3.3.1}{Approximation de la densit\351}{section.3.3}% 42
\BOOKMARK [2][-]{subsection.3.3.2}{Unicit\351 du point de selle}{section.3.3}% 43
\BOOKMARK [2][-]{subsection.3.3.3}{Approximation de la fonction de r\351partition}{section.3.3}% 44
\BOOKMARK [2][-]{subsection.3.3.4}{Quelques propri\351t\351s des approximations}{section.3.3}% 45
\BOOKMARK [1][-]{section.3.4}{Application de la m\351thode du point de selle}{chapter.3}% 46
\BOOKMARK [2][-]{subsection.3.4.1}{Approximation de la densit\351}{section.3.4}% 47
\BOOKMARK [2][-]{subsection.3.4.2}{Approximation de la fonction de r\351partition}{section.3.4}% 48
\BOOKMARK [0][-]{chapter.4}{M\351thode des moments g\351n\351ralis\351e}{}% 49
\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4}% 50
\BOOKMARK [2][-]{subsection.4.1.1}{M\351thode classique des moments}{section.4.1}% 51
\BOOKMARK [1][-]{section.4.2}{M\351thode des moments g\351n\351ralis\351e}{chapter.4}% 52
\BOOKMARK [2][-]{subsection.4.2.1}{D\351finition}{section.4.2}% 53
\BOOKMARK [2][-]{subsection.4.2.2}{Convergence}{section.4.2}% 54
\BOOKMARK [2][-]{subsection.4.2.3}{Matrice de pond\351ration optimale}{section.4.2}% 55
\BOOKMARK [2][-]{subsection.4.2.4}{M\351thode des moments g\351n\351ralis\351e it\351rative}{section.4.2}% 56
\BOOKMARK [2][-]{subsection.4.2.5}{Distribution asymptotique des estimateurs}{section.4.2}% 57
\BOOKMARK [1][-]{section.4.3}{Estimation sous contrainte}{chapter.4}% 58
\BOOKMARK [2][-]{subsection.4.3.1}{Distribution asymptotique des estimateurs contraints}{section.4.3}% 59
\BOOKMARK [1][-]{section.4.4}{Tests d'hypoth\350ses param\351triques}{chapter.4}% 60
\BOOKMARK [2][-]{subsection.4.4.1}{Test de Wald}{section.4.4}% 61
\BOOKMARK [2][-]{subsection.4.4.2}{Test du multiplicateur de Lagrange}{section.4.4}% 62
\BOOKMARK [2][-]{subsection.4.4.3}{Test bas\351 sur la statistique de m\351trique de distance}{section.4.4}% 63
\BOOKMARK [2][-]{subsection.4.4.4}{En r\351sum\351}{section.4.4}% 64
\BOOKMARK [0][-]{chapter.5}{M\351thode de l'\351quation d'estimation optimale}{}% 65
\BOOKMARK [1][-]{section.5.1}{\311quations quadratiques optimales}{chapter.5}% 66
\BOOKMARK [1][-]{section.5.2}{\311quation d'estimation optimale modifi\351e}{chapter.5}% 67
\BOOKMARK [0][-]{chapter.6}{Estimation des param\350tres de la distribution de Laplace asym\351trique g\351n\351ralis\351e}{}% 68
\BOOKMARK [1][-]{section.6.1}{Vecteur de param\350tres initiaux}{chapter.6}% 69
\BOOKMARK [1][-]{section.6.2}{M\351thode des moments g\351n\351ralis\351e}{chapter.6}% 70
\BOOKMARK [2][-]{subsection.6.2.1}{Matrice de pond\351ration optimale}{section.6.2}% 71
\BOOKMARK [2][-]{subsection.6.2.2}{Variance-covariance des param\350tres}{section.6.2}% 72
\BOOKMARK [2][-]{subsection.6.2.3}{Contraintes lin\351aires}{section.6.2}% 73
\BOOKMARK [1][-]{section.6.3}{M\351thode des \351quations d'estimation optimales}{chapter.6}% 74
\BOOKMARK [0][-]{chapter.7}{Tests statistiques}{}% 75
\BOOKMARK [1][-]{section.7.1}{Test de normalit\351}{chapter.7}% 76
\BOOKMARK [2][-]{subsection.7.1.1}{Test de Shapiro-Wilk}{section.7.1}% 77
\BOOKMARK [2][-]{subsection.7.1.2}{Test d\220Epps-Pulley}{section.7.1}% 78
\BOOKMARK [1][-]{section.7.2}{Tests d'ad\351quation}{chapter.7}% 79
\BOOKMARK [2][-]{subsection.7.2.1}{Test 2 de Pearson}{section.7.2}% 80
\BOOKMARK [2][-]{subsection.7.2.2}{Test de Kolmogorov-Smirnov}{section.7.2}% 81
\BOOKMARK [2][-]{subsection.7.2.3}{Test de distance minimale bas\351 sur la fonction g\351n\351ratrice des moments}{section.7.2}% 82
\BOOKMARK [0][-]{chapter.8}{\311valuation d'options}{}% 83
\BOOKMARK [1][-]{section.8.1}{D\351finitions}{chapter.8}% 84
\BOOKMARK [2][-]{subsection.8.1.1}{\311quation martingale}{section.8.1}% 85
\BOOKMARK [2][-]{subsection.8.1.2}{Param\350tres neutres au risque}{section.8.1}% 86
\BOOKMARK [1][-]{section.8.2}{Aper\347u du mod\350le de Black-Scholes}{chapter.8}% 87
\BOOKMARK [1][-]{section.8.3}{M\351thodes d'\351valuation pour options europ\351ennes}{chapter.8}% 88
\BOOKMARK [2][-]{subsection.8.3.1}{M\351thode de Heston}{section.8.3}% 89
\BOOKMARK [2][-]{subsection.8.3.2}{M\351thode de Carr et Madan}{section.8.3}% 90
\BOOKMARK [2][-]{subsection.8.3.3}{Prix d'exercice hors de la monnaie}{section.8.3}% 91
\BOOKMARK [2][-]{subsection.8.3.4}{Critique de la m\351thode de Carr-Madan}{section.8.3}% 92
\BOOKMARK [2][-]{subsection.8.3.5}{M\351thode d\220Epps}{section.8.3}% 93
\BOOKMARK [1][-]{section.8.4}{Particularit\351s}{chapter.8}% 94
\BOOKMARK [2][-]{subsection.8.4.1}{Option sur actions avec dividendes}{section.8.4}% 95
\BOOKMARK [2][-]{subsection.8.4.2}{Options sur contrats \340 terme et taux de change}{section.8.4}% 96
\BOOKMARK [0][-]{chapter.9}{Exemple d'application}{}% 97
\BOOKMARK [1][-]{section.9.1}{Description des donn\351es}{chapter.9}% 98
\BOOKMARK [1][-]{section.9.2}{Estimation}{chapter.9}% 99
\BOOKMARK [1][-]{section.9.3}{Approximation}{chapter.9}% 100
\BOOKMARK [1][-]{section.9.4}{Graphiques}{chapter.9}% 101
\BOOKMARK [1][-]{section.9.5}{Tests statistiques}{chapter.9}% 102
\BOOKMARK [1][-]{section.9.6}{\311valuation d'options}{chapter.9}% 103
\BOOKMARK [0][-]{section*.52}{Conclusion}{}% 104
\BOOKMARK [0][-]{appendix.A}{\311l\351ments de th\351orie des probabilit\351s}{}% 105
\BOOKMARK [1][-]{section.A.1}{D\351finitions de base}{appendix.A}% 106
\BOOKMARK [1][-]{section.A.2}{Transform\351es d'une variable al\351atoire}{appendix.A}% 107
\BOOKMARK [2][-]{subsection.A.2.1}{La fonction caract\351ristique}{section.A.2}% 108
\BOOKMARK [3][-]{section*.53}{Transform\351e de Fourier}{subsection.A.2.1}% 109
\BOOKMARK [3][-]{section*.54}{D\351finition}{subsection.A.2.1}% 110
\BOOKMARK [3][-]{section*.55}{Les moments}{subsection.A.2.1}% 111
\BOOKMARK [2][-]{subsection.A.2.2}{Inversion de la fonction caract\351ristique}{section.A.2}% 112
\BOOKMARK [3][-]{section*.56}{La densit\351}{subsection.A.2.2}% 113
\BOOKMARK [3][-]{section*.57}{La fonction de r\351partition}{subsection.A.2.2}% 114
\BOOKMARK [2][-]{subsection.A.2.3}{La fonction g\351n\351ratrice des moments}{section.A.2}% 115
\BOOKMARK [2][-]{subsection.A.2.4}{La fonction g\351n\351ratrice des cumulants}{section.A.2}% 116
\BOOKMARK [2][-]{subsection.A.2.5}{La transform\351e d'Esscher}{section.A.2}% 117
\BOOKMARK [1][-]{section.A.3}{La transform\351e de Fourier rapide}{appendix.A}% 118
\BOOKMARK [1][-]{section.A.4}{Processus de L\351vy}{appendix.A}% 119
\BOOKMARK [2][-]{subsection.A.4.1}{D\351finition et propri\351t\351s}{section.A.4}% 120
\BOOKMARK [3][-]{section*.58}{Repr\351sentation de L\351vy-Khintchine}{subsection.A.4.1}% 121
\BOOKMARK [3][-]{section*.59}{Repr\351sentation de L\351vy-It\364}{subsection.A.4.1}% 122
\BOOKMARK [2][-]{subsection.A.4.2}{Processus subordonn\351}{section.A.4}% 123
\BOOKMARK [1][-]{section.A.5}{Th\351or\350mes d'int\351gration}{appendix.A}% 124
\BOOKMARK [2][-]{subsection.A.5.1}{Th\351or\350me de convergence domin\351e de Lebesgue}{section.A.5}% 125
\BOOKMARK [2][-]{subsection.A.5.2}{Th\351or\350me de Fubini}{section.A.5}% 126
\BOOKMARK [0][-]{appendix.B}{\311l\351ments de statistique math\351matique}{}% 127
\BOOKMARK [1][-]{section.B.1}{Loi faible des grands nombres}{appendix.B}% 128
\BOOKMARK [1][-]{section.B.2}{Th\351or\350me central limite}{appendix.B}% 129
\BOOKMARK [2][-]{subsection.B.2.1}{Cas univari\351}{section.B.2}% 130
\BOOKMARK [2][-]{subsection.B.2.2}{Cas multivari\351}{section.B.2}% 131
\BOOKMARK [1][-]{section.B.3}{M\351thode delta multivari\351e}{appendix.B}% 132
\BOOKMARK [0][-]{appendix.C}{Donn\351es}{}% 133
\BOOKMARK [0][-]{section*.62}{Bibliographie}{}% 134
\BOOKMARK [0][-]{section*.64}{Contrat de partage}{}% 135

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%% 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{plainnatmod}
%% 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{2013}
\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}

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@ -1,137 +0,0 @@
\select@language {french}
\select@language {french}
\contentsline {chapter}{R\IeC {\'e}sum\IeC {\'e}}{iii}{section*.2}
\contentsline {chapter}{Abstract}{v}{section*.4}
\contentsline {chapter}{Table des mati{\`e}res}{vii}{section*.5}
\contentsline {chapter}{Liste des tableaux}{xi}{section*.6}
\contentsline {chapter}{Liste des figures}{xiii}{section*.7}
\contentsline {chapter}{Remerciements}{xvii}{section*.9}
\contentsline {chapter}{Introduction}{1}{section*.11}
\contentsline {chapter}{\chapternumberline {1}Les mod\IeC {\`e}les de rendements financiers}{3}{chapter.1}
\contentsline {section}{\numberline {1.1}L'utilisation de mod\IeC {\`e}les en finance}{3}{section.1.1}
\contentsline {subsection}{\numberline {1.1.1}Diff\IeC {\'e}rents types de mod\IeC {\`e}les}{3}{subsection.1.1.1}
\contentsline {subsection}{\numberline {1.1.2}Le risque de mod\IeC {\'e}lisation}{4}{subsection.1.1.2}
\contentsline {section}{\numberline {1.2}Les rendements financiers}{5}{section.1.2}
\contentsline {subsection}{\numberline {1.2.1}D\IeC {\'e}finitions et notation}{5}{subsection.1.2.1}
\contentsline {subsection}{\numberline {1.2.2}Rendements cumul\IeC {\'e}s}{7}{subsection.1.2.2}
\contentsline {subsection}{\numberline {1.2.3}Donn\IeC {\'e}es disponibles}{8}{subsection.1.2.3}
\contentsline {section}{\numberline {1.3}Les premiers mod\IeC {\`e}les}{8}{section.1.3}
\contentsline {subsection}{\numberline {1.3.1}Le mod\IeC {\`e}le de Bachelier}{8}{subsection.1.3.1}
\contentsline {subsection}{\numberline {1.3.2}Proposition de Mandelbrot}{11}{subsection.1.3.2}
\contentsline {subsection}{\numberline {1.3.3}Le mod\IeC {\`e}le de Press}{12}{subsection.1.3.3}
\contentsline {subsection}{\numberline {1.3.4}Le mod\IeC {\`e}le de Praetz}{15}{subsection.1.3.4}
\contentsline {section}{\numberline {1.4}Conditions essentielles de Madan et Seneta}{16}{section.1.4}
\contentsline {chapter}{\chapternumberline {2}La distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{19}{chapter.2}
\contentsline {section}{\numberline {2.1}Le processus de Laplace}{19}{section.2.1}
\contentsline {subsection}{\numberline {2.1.1}Le processus gamma}{20}{subsection.2.1.1}
\contentsline {subsection}{\numberline {2.1.2}Le processus de Wiener}{21}{subsection.2.1.2}
\contentsline {subsection}{\numberline {2.1.3}Le processus de Laplace est un processus subordonn\IeC {\'e}}{22}{subsection.2.1.3}
\contentsline {section}{\numberline {2.2}Distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{25}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}Fonction caract\IeC {\'e}ristique}{25}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}Invariance d'\IeC {\'e}chelle}{26}{subsection.2.2.2}
\contentsline {subsection}{\numberline {2.2.3}Fonctions g\IeC {\'e}n\IeC {\'e}ratrices}{28}{subsection.2.2.3}
\contentsline {subsection}{\numberline {2.2.4}Moments et r\IeC {\^o}le des param\IeC {\`e}tres}{28}{subsection.2.2.4}
\contentsline {subsection}{\numberline {2.2.5}Changement d'\IeC {\'e}chelle et de localisation}{31}{subsection.2.2.5}
\contentsline {subsection}{\numberline {2.2.6}Repr\IeC {\'e}sentation alternative et simulation}{32}{subsection.2.2.6}
\contentsline {subsection}{\numberline {2.2.7}Fonction de Bessel et densit\IeC {\'e}}{33}{subsection.2.2.7}
\contentsline {section}{\numberline {2.3}Cas particuliers}{35}{section.2.3}
\contentsline {subsection}{\numberline {2.3.1}Distribution de Laplace asym\IeC {\'e}trique}{36}{subsection.2.3.1}
\contentsline {section}{\numberline {2.4}Relation avec le mod\IeC {\`e}le de \cite {madan1990variance}}{37}{section.2.4}
\contentsline {chapter}{\chapternumberline {3}Approximation de la densit\IeC {\'e} et de la fonction de r\IeC {\'e}partition}{39}{chapter.3}
\contentsline {section}{\numberline {3.1}L'approximation de Laplace}{39}{section.3.1}
\contentsline {section}{\numberline {3.2}L'approximation de Temme}{41}{section.3.2}
\contentsline {section}{\numberline {3.3}La m\IeC {\'e}thode du point de selle}{42}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}Approximation de la densit\IeC {\'e}}{42}{subsection.3.3.1}
\contentsline {subsection}{\numberline {3.3.2}Unicit\IeC {\'e} du point de selle}{44}{subsection.3.3.2}
\contentsline {subsection}{\numberline {3.3.3}Approximation de la fonction de r\IeC {\'e}partition}{44}{subsection.3.3.3}
\contentsline {subsection}{\numberline {3.3.4}Quelques propri\IeC {\'e}t\IeC {\'e}s des approximations}{45}{subsection.3.3.4}
\contentsline {section}{\numberline {3.4}Application de la m\IeC {\'e}thode du point de selle}{46}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}Approximation de la densit\IeC {\'e}}{46}{subsection.3.4.1}
\contentsline {subsection}{\numberline {3.4.2}Approximation de la fonction de r\IeC {\'e}partition}{47}{subsection.3.4.2}
\contentsline {chapter}{\chapternumberline {4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{49}{chapter.4}
\contentsline {section}{\numberline {4.1}Introduction}{49}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}M\IeC {\'e}thode classique des moments}{50}{subsection.4.1.1}
\contentsline {section}{\numberline {4.2}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{50}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}D\IeC {\'e}finition}{51}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}Convergence}{52}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}Matrice de pond\IeC {\'e}ration optimale}{53}{subsection.4.2.3}
\contentsline {subsection}{\numberline {4.2.4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e it\IeC {\'e}rative}{55}{subsection.4.2.4}
\contentsline {subsection}{\numberline {4.2.5}Distribution asymptotique des estimateurs}{55}{subsection.4.2.5}
\contentsline {section}{\numberline {4.3}Estimation sous contrainte}{58}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}Distribution asymptotique des estimateurs contraints}{59}{subsection.4.3.1}
\contentsline {section}{\numberline {4.4}Tests d'hypoth\IeC {\`e}ses param\IeC {\'e}triques}{61}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}Test de Wald}{61}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}Test du multiplicateur de Lagrange}{62}{subsection.4.4.2}
\contentsline {subsection}{\numberline {4.4.3}Test bas\IeC {\'e} sur la statistique de m\IeC {\'e}trique de distance}{63}{subsection.4.4.3}
\contentsline {subsection}{\numberline {4.4.4}En r\IeC {\'e}sum\IeC {\'e}}{63}{subsection.4.4.4}
\contentsline {chapter}{\chapternumberline {5}M\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale}{65}{chapter.5}
\contentsline {section}{\numberline {5.1}\IeC {\'E}quations quadratiques optimales}{67}{section.5.1}
\contentsline {section}{\numberline {5.2}\IeC {\'E}quation d'estimation optimale modifi\IeC {\'e}e}{70}{section.5.2}
\contentsline {chapter}{\chapternumberline {6}Estimation des param\IeC {\`e}tres de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{73}{chapter.6}
\contentsline {section}{\numberline {6.1}Vecteur de param\IeC {\`e}tres initiaux}{73}{section.6.1}
\contentsline {section}{\numberline {6.2}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{74}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}Matrice de pond\IeC {\'e}ration optimale}{75}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}Variance-covariance des param\IeC {\`e}tres}{76}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}Contraintes lin\IeC {\'e}aires}{77}{subsection.6.2.3}
\contentsline {section}{\numberline {6.3}M\IeC {\'e}thode des \IeC {\'e}quations d'estimation optimales}{78}{section.6.3}
\contentsline {chapter}{\chapternumberline {7}Tests statistiques}{81}{chapter.7}
\contentsline {section}{\numberline {7.1}Test de normalit\IeC {\'e}}{81}{section.7.1}
\contentsline {subsection}{\numberline {7.1.1}Test de Shapiro-Wilk}{81}{subsection.7.1.1}
\contentsline {subsection}{\numberline {7.1.2}Test d\IeC {\textquoteright }Epps-Pulley}{82}{subsection.7.1.2}
\contentsline {section}{\numberline {7.2}Tests d'ad\IeC {\'e}quation}{82}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}Test $\chi ^2$ de Pearson}{83}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}Test de Kolmogorov-Smirnov}{84}{subsection.7.2.2}
\contentsline {subsection}{\numberline {7.2.3}Test de distance minimale bas\IeC {\'e} sur la fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments}{85}{subsection.7.2.3}
\contentsline {chapter}{\chapternumberline {8}\IeC {\'E}valuation d'options}{87}{chapter.8}
\contentsline {section}{\numberline {8.1}D\IeC {\'e}finitions}{87}{section.8.1}
\contentsline {subsection}{\numberline {8.1.1}\IeC {\'E}quation martingale}{88}{subsection.8.1.1}
\contentsline {subsection}{\numberline {8.1.2}Param\IeC {\`e}tres neutres au risque}{89}{subsection.8.1.2}
\contentsline {section}{\numberline {8.2}Aper\IeC {\c c}u du mod\IeC {\`e}le de Black-Scholes}{90}{section.8.2}
\contentsline {section}{\numberline {8.3}M\IeC {\'e}thodes d'\IeC {\'e}valuation pour options europ\IeC {\'e}ennes}{90}{section.8.3}
\contentsline {subsection}{\numberline {8.3.1}M\IeC {\'e}thode de Heston}{91}{subsection.8.3.1}
\contentsline {subsection}{\numberline {8.3.2}M\IeC {\'e}thode de Carr et Madan}{91}{subsection.8.3.2}
\contentsline {subsection}{\numberline {8.3.3}Prix d'exercice hors de la monnaie}{93}{subsection.8.3.3}
\contentsline {subsection}{\numberline {8.3.4}Critique de la m\IeC {\'e}thode de Carr-Madan}{93}{subsection.8.3.4}
\contentsline {subsection}{\numberline {8.3.5}M\IeC {\'e}thode d\IeC {\textquoteright }Epps}{94}{subsection.8.3.5}
\contentsline {section}{\numberline {8.4}Particularit\IeC {\'e}s}{95}{section.8.4}
\contentsline {subsection}{\numberline {8.4.1}Option sur actions avec dividendes}{95}{subsection.8.4.1}
\contentsline {subsection}{\numberline {8.4.2}Options sur contrats \IeC {\`a} terme et taux de change}{95}{subsection.8.4.2}
\contentsline {chapter}{\chapternumberline {9}Exemple d'application}{97}{chapter.9}
\contentsline {section}{\numberline {9.1}Description des donn\IeC {\'e}es}{97}{section.9.1}
\contentsline {section}{\numberline {9.2}Estimation}{99}{section.9.2}
\contentsline {section}{\numberline {9.3}Approximation}{104}{section.9.3}
\contentsline {section}{\numberline {9.4}Graphiques}{105}{section.9.4}
\contentsline {section}{\numberline {9.5}Tests statistiques}{108}{section.9.5}
\contentsline {section}{\numberline {9.6}\IeC {\'E}valuation d'options}{109}{section.9.6}
\contentsline {chapter}{Conclusion}{113}{section*.52}
\contentsline {appendix}{\chapternumberline {A}\IeC {\'E}l\IeC {\'e}ments de th\IeC {\'e}orie des probabilit\IeC {\'e}s}{115}{appendix.A}
\contentsline {section}{\numberline {A.1}D\IeC {\'e}finitions de base}{115}{section.A.1}
\contentsline {section}{\numberline {A.2}Transform\IeC {\'e}es d'une variable al\IeC {\'e}atoire}{116}{section.A.2}
\contentsline {subsection}{\numberline {A.2.1}La fonction caract\IeC {\'e}ristique}{116}{subsection.A.2.1}
\contentsline {subsubsection}{Transform\IeC {\'e}e de Fourier}{116}{section*.53}
\contentsline {subsubsection}{D\IeC {\'e}finition}{116}{section*.54}
\contentsline {subsubsection}{Les moments}{117}{section*.55}
\contentsline {subsection}{\numberline {A.2.2}Inversion de la fonction caract\IeC {\'e}ristique}{117}{subsection.A.2.2}
\contentsline {subsubsection}{La densit\IeC {\'e}}{117}{section*.56}
\contentsline {subsubsection}{La fonction de r\IeC {\'e}partition}{117}{section*.57}
\contentsline {subsection}{\numberline {A.2.3}La fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments}{118}{subsection.A.2.3}
\contentsline {subsection}{\numberline {A.2.4}La fonction g\IeC {\'e}n\IeC {\'e}ratrice des cumulants}{119}{subsection.A.2.4}
\contentsline {subsection}{\numberline {A.2.5}La transform\IeC {\'e}e d'Esscher}{120}{subsection.A.2.5}
\contentsline {section}{\numberline {A.3}La transform\IeC {\'e}e de Fourier rapide}{120}{section.A.3}
\contentsline {section}{\numberline {A.4}Processus de L\IeC {\'e}vy}{121}{section.A.4}
\contentsline {subsection}{\numberline {A.4.1}D\IeC {\'e}finition et propri\IeC {\'e}t\IeC {\'e}s}{121}{subsection.A.4.1}
\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-Khintchine}{122}{section*.58}
\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-It\IeC {\^o}}{122}{section*.59}
\contentsline {subsection}{\numberline {A.4.2}Processus subordonn\IeC {\'e}}{123}{subsection.A.4.2}
\contentsline {section}{\numberline {A.5}Th\IeC {\'e}or\IeC {\`e}mes d'int\IeC {\'e}gration}{123}{section.A.5}
\contentsline {subsection}{\numberline {A.5.1}Th\IeC {\'e}or\IeC {\`e}me de convergence domin\IeC {\'e}e de Lebesgue}{124}{subsection.A.5.1}
\contentsline {subsection}{\numberline {A.5.2}Th\IeC {\'e}or\IeC {\`e}me de Fubini}{124}{subsection.A.5.2}
\contentsline {appendix}{\chapternumberline {B}\IeC {\'E}l\IeC {\'e}ments de statistique math\IeC {\'e}matique}{125}{appendix.B}
\contentsline {section}{\numberline {B.1}Loi faible des grands nombres}{125}{section.B.1}
\contentsline {section}{\numberline {B.2}Th\IeC {\'e}or\IeC {\`e}me central limite}{125}{section.B.2}
\contentsline {subsection}{\numberline {B.2.1}Cas univari\IeC {\'e}}{126}{subsection.B.2.1}
\contentsline {subsection}{\numberline {B.2.2}Cas multivari\IeC {\'e}}{126}{subsection.B.2.2}
\contentsline {section}{\numberline {B.3}M\IeC {\'e}thode delta multivari\IeC {\'e}e}{127}{section.B.3}
\contentsline {appendix}{\chapternumberline {C}Donn\IeC {\'e}es}{129}{appendix.C}
\contentsline {chapter}{Bibliographie}{131}{section*.62}
\contentsline {chapter}{Contrat de partage}{137}{section*.64}

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This package contains a LaTeX class and templates for thesis and
memoirs at Université Laval, Québec, Canada.
LICENCE
LaTeX Project Public License, version 1.3c or (at your option) any
later version.
*** The rest of this file is in French ***
DESCRIPTION
La classe LaTeX ulthese permet de composer 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.
CONTENU DE L'ARCHIVE
- ulthese.ins: fichier d'installation de la classe; voir ci-dessous;
- ulthese.dtx: fichier source documenté de la classe;
- ulthese.pdf: documentation de la classe;
- ul_p.eps: logo de l'Université Laval en format EPS;
- ul_p.pdf: logo de l'Université Laval en format PDF;
- README: le présent fichier.
INSTALLATION
Pour installer la classe ulthese, compiler avec LaTeX le fichier
ulthese.ins, en exécutant
latex ulthese.ins
depuis une invite de commande. On peut aussi ouvrir le fichier
ulthese.ins dans son éditeur LaTeX et lancer la compilation avec
LaTeX, pdfTeX ou pdfLaTeX depuis celui-ci.
La compilation du fichier ulthese.ins générera plusieurs fichiers:
1. la classe elle-même:
- ulthese.cls
2. des gabarits pour les documents maîtres de différents types de
thèses et de mémoires:
- gabarit-doctorat.tex
- gabarit-doctorat-mesure.tex
- gabarit-doctorat-multifacultaire.tex
- gabarit-doctorat-cotutelle.tex
- gabarit-doctorat-extension-UdeS.tex
- gabarit-doctorat-extension-UQO.tex
- gabarit-maitrise.tex
- gabarit-maitrise-mesure.tex
- gabarit-maitrise-extension-UQAC.tex
3. des gabarits pour quelques parties d'un document:
- resume.tex
- abstract.tex
- remerciements.tex
- avantpropos.tex
- introduction.tex
- chapitre1.tex
- chapitre2.tex
- conclusion.tex
- annexe.tex
Les gabarits inutiles peuvent être supprimés.
DOCUMENTATION
Le fichier ulthese.pdf contient la documentation complète de la classe
et des gabarits. Le document peut être recréé à partir du code source
avec les commandes
pdflatex ulthese.dtx
makeindex -s gind.ist ulthese
makeindex -s gglo.ist -o ulthese.gls ulthese.glo
AIDE ADDITIONNELLE
Faire appel aux ressources suivantes, dans l'ordre:
1. consulter le WikiThèse de l'Université Laval
http://www.theses.ulaval.ca/wiki/
2. consulter les archives de la liste de distribution ulthese-aide
http://listes.ulaval.ca/listserv/archives/ulthese-aide.html
3. écrire à la liste de distribution ulthese-aide
ulthese-aide@listes.ulaval.ca

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6f9a85e6a4f4334ba59be60c2a3030db781143bb 908144032.pdf

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\begin{thebibliography}{63}
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\contentsline {table}{\numberline {6.1}{\ignorespaces Contraintes lin\IeC {\'e}aires pour les cas particuliers de la distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e $GAL(\theta ,\sigma ,\mu ,\tau )$\relax }}{77}{table.caption.27}
\addvspace {10pt}
\addvspace {10pt}
\addvspace {10pt}
\contentsline {table}{\numberline {9.1}{\ignorespaces Statistiques d'ordre de l'\IeC {\'e}chantillon $R_1$\relax }}{97}{table.caption.29}
\contentsline {table}{\numberline {9.2}{\ignorespaces Valeurs relatives aux premiers moments de l'\IeC {\'e}chantillon $R_1$\relax }}{98}{table.caption.30}
\contentsline {table}{\numberline {9.3}{\ignorespaces Test de normalit\IeC {\'e} d'Epps-Pulley pour $R_1$\relax }}{98}{table.caption.32}
\contentsline {table}{\numberline {9.4}{\ignorespaces Param\IeC {\`e}tres $\theta _1$ de la premi\IeC {\`e}re optimisation\relax }}{100}{table.caption.34}
\contentsline {table}{\numberline {9.5}{\ignorespaces Param\IeC {\`e}tres $\theta _1$ de la premi\IeC {\`e}re optimisation\relax }}{102}{table.caption.35}
\contentsline {table}{\numberline {9.6}{\ignorespaces Param\IeC {\`e}tres des donn\IeC {\'e}es $R_1$\relax }}{103}{table.caption.36}
\contentsline {table}{\numberline {9.7}{\ignorespaces Approximation de la densit\IeC {\'e} de $R_1$\relax }}{104}{table.caption.38}
\contentsline {table}{\numberline {9.8}{\ignorespaces Approximation de la fonction de r\IeC {\'e}partition de $R_1$\relax }}{105}{table.caption.39}
\contentsline {table}{\numberline {9.9}{\ignorespaces Courbes de densit\IeC {\'e}\relax }}{105}{table.caption.40}
\contentsline {table}{\numberline {9.10}{\ignorespaces Valeur de l'int\IeC {\'e}grale de l'approximation de la densit\IeC {\'e} par la m\IeC {\'e}thode du point de selle \relax }}{105}{table.caption.41}
\contentsline {table}{\numberline {9.11}{\ignorespaces Test du $\chi ^2$\relax }}{108}{table.caption.44}
\contentsline {table}{\numberline {9.12}{\ignorespaces Test de Kolmogorov-Smirnov\relax }}{108}{table.caption.45}
\contentsline {table}{\numberline {9.13}{\ignorespaces Test de distance minimale bas\IeC {\'e} sur la fonction g\IeC {\'e}n\IeC {\'e}ratrice des moments\relax }}{109}{table.caption.46}
\contentsline {table}{\numberline {9.14}{\ignorespaces Caract\IeC {\'e}ristiques de l'option\relax }}{109}{table.caption.47}
\contentsline {table}{\numberline {9.15}{\ignorespaces Param\IeC {\`e}tres neutres au risque\relax }}{109}{table.caption.48}
\addvspace {10pt}
\addvspace {10pt}
\addvspace {10pt}
\contentsline {table}{\numberline {C.1}{\ignorespaces Prix du titre Abbey National (penny sterling) du 31 juillet au 8 octobre 1991\relax }}{129}{table.caption.60}

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\BOOKMARK [0][-]{section*.2}{R\351sum\351}{}% 1
\BOOKMARK [0][-]{section*.4}{Abstract}{}% 2
\BOOKMARK [0][-]{section*.5}{Table des mati\350res}{}% 3
\BOOKMARK [0][-]{section*.6}{Liste des tableaux}{}% 4
\BOOKMARK [0][-]{section*.7}{Liste des figures}{}% 5
\BOOKMARK [0][-]{section*.9}{Remerciements}{}% 6
\BOOKMARK [0][-]{section*.11}{Introduction}{}% 7
\BOOKMARK [0][-]{chapter.1}{Les mod\350les de rendements financiers}{}% 8
\BOOKMARK [1][-]{section.1.1}{L'utilisation de mod\350les en finance}{chapter.1}% 9
\BOOKMARK [2][-]{subsection.1.1.1}{Diff\351rents types de mod\350les}{section.1.1}% 10
\BOOKMARK [2][-]{subsection.1.1.2}{Le risque de mod\351lisation}{section.1.1}% 11
\BOOKMARK [1][-]{section.1.2}{Les rendements financiers}{chapter.1}% 12
\BOOKMARK [2][-]{subsection.1.2.1}{D\351finitions et notations}{section.1.2}% 13
\BOOKMARK [2][-]{subsection.1.2.2}{Rendements cumul\351s}{section.1.2}% 14
\BOOKMARK [2][-]{subsection.1.2.3}{Donn\351es disponibles}{section.1.2}% 15
\BOOKMARK [1][-]{section.1.3}{Les premiers mod\350les}{chapter.1}% 16
\BOOKMARK [2][-]{subsection.1.3.1}{Le mod\350le de Bachelier}{section.1.3}% 17
\BOOKMARK [2][-]{subsection.1.3.2}{Proposition de Mandelbrot}{section.1.3}% 18
\BOOKMARK [2][-]{subsection.1.3.3}{Le mod\350le de Press}{section.1.3}% 19
\BOOKMARK [2][-]{subsection.1.3.4}{Le mod\350le de Praetz}{section.1.3}% 20
\BOOKMARK [1][-]{section.1.4}{Conditions essentielles de Madan et Seneta}{chapter.1}% 21
\BOOKMARK [0][-]{chapter.2}{La distribution de Laplace asym\351trique g\351n\351ralis\351e}{}% 22
\BOOKMARK [1][-]{section.2.1}{Le processus de Laplace}{chapter.2}% 23
\BOOKMARK [2][-]{subsection.2.1.1}{Le processus gamma}{section.2.1}% 24
\BOOKMARK [2][-]{subsection.2.1.2}{Le processus de Wiener}{section.2.1}% 25
\BOOKMARK [2][-]{subsection.2.1.3}{Le processus de Laplace est un processus subordonn\351}{section.2.1}% 26
\BOOKMARK [1][-]{section.2.2}{Distribution de Laplace asym\351trique g\351n\351ralis\351e}{chapter.2}% 27
\BOOKMARK [2][-]{subsection.2.2.1}{Fonction caract\351ristique}{section.2.2}% 28
\BOOKMARK [2][-]{subsection.2.2.2}{Invariance d'\351chelle}{section.2.2}% 29
\BOOKMARK [2][-]{subsection.2.2.3}{Fonctions g\351n\351ratrices}{section.2.2}% 30
\BOOKMARK [2][-]{subsection.2.2.4}{Moments et r\364le des param\350tres}{section.2.2}% 31
\BOOKMARK [2][-]{subsection.2.2.5}{Changement d'\351chelle et de localisation}{section.2.2}% 32
\BOOKMARK [2][-]{subsection.2.2.6}{Repr\351sentation alternative et simulation}{section.2.2}% 33
\BOOKMARK [2][-]{subsection.2.2.7}{Fonction de Bessel et densit\351}{section.2.2}% 34
\BOOKMARK [1][-]{section.2.3}{Cas particuliers}{chapter.2}% 35
\BOOKMARK [2][-]{subsection.2.3.1}{Distribution de Laplace asym\351trique}{section.2.3}% 36
\BOOKMARK [1][-]{section.2.4}{Relation avec le mod\350le de madan1990variance}{chapter.2}% 37
\BOOKMARK [0][-]{chapter.3}{Approximation de la densit\351 et de la fonction de r\351partition}{}% 38
\BOOKMARK [1][-]{section.3.1}{L'approximation de Laplace}{chapter.3}% 39
\BOOKMARK [1][-]{section.3.2}{L'approximation de Temme}{chapter.3}% 40
\BOOKMARK [1][-]{section.3.3}{La m\351thode du point de selle}{chapter.3}% 41
\BOOKMARK [2][-]{subsection.3.3.1}{Approximation de la densit\351}{section.3.3}% 42
\BOOKMARK [2][-]{subsection.3.3.2}{Unicit\351 du point de selle}{section.3.3}% 43
\BOOKMARK [2][-]{subsection.3.3.3}{Approximation de la fonction de r\351partition}{section.3.3}% 44
\BOOKMARK [2][-]{subsection.3.3.4}{Quelques propri\351t\351s des approximations}{section.3.3}% 45
\BOOKMARK [1][-]{section.3.4}{Application de la m\351thode du point de selle}{chapter.3}% 46
\BOOKMARK [2][-]{subsection.3.4.1}{Approximation de la densit\351}{section.3.4}% 47
\BOOKMARK [2][-]{subsection.3.4.2}{Approximation de la fonction de r\351partition}{section.3.4}% 48
\BOOKMARK [0][-]{chapter.4}{M\351thode des moments g\351n\351ralis\351e}{}% 49
\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4}% 50
\BOOKMARK [2][-]{subsection.4.1.1}{M\351thode classique des moments}{section.4.1}% 51
\BOOKMARK [1][-]{section.4.2}{M\351thode des moments g\351n\351ralis\351e}{chapter.4}% 52
\BOOKMARK [2][-]{subsection.4.2.1}{D\351finition}{section.4.2}% 53
\BOOKMARK [2][-]{subsection.4.2.2}{Convergence}{section.4.2}% 54
\BOOKMARK [2][-]{subsection.4.2.3}{Matrice de pond\351ration optimale}{section.4.2}% 55
\BOOKMARK [2][-]{subsection.4.2.4}{M\351thode des moments g\351n\351ralis\351e it\351rative}{section.4.2}% 56
\BOOKMARK [2][-]{subsection.4.2.5}{Distribution asymptotique des estimateurs}{section.4.2}% 57
\BOOKMARK [1][-]{section.4.3}{Estimation sous contraintes}{chapter.4}% 58
\BOOKMARK [2][-]{subsection.4.3.1}{Distribution asymptotique des estimateurs contraints}{section.4.3}% 59
\BOOKMARK [1][-]{section.4.4}{Tests d'hypoth\350ses param\351triques}{chapter.4}% 60
\BOOKMARK [2][-]{subsection.4.4.1}{Test de Wald}{section.4.4}% 61
\BOOKMARK [2][-]{subsection.4.4.2}{Test du multiplicateur de Lagrange}{section.4.4}% 62
\BOOKMARK [2][-]{subsection.4.4.3}{Test bas\351 sur la statistique de m\351trique de distance}{section.4.4}% 63
\BOOKMARK [2][-]{subsection.4.4.4}{En r\351sum\351}{section.4.4}% 64
\BOOKMARK [0][-]{chapter.5}{M\351thode de l'\351quation d'estimation optimale}{}% 65
\BOOKMARK [1][-]{section.5.1}{\311quation d'estimation optimale}{chapter.5}% 66
\BOOKMARK [1][-]{section.5.2}{\311quation d'estimation optimale modifi\351e}{chapter.5}% 67
\BOOKMARK [0][-]{chapter.6}{Estimation des param\350tres de la distribution de Laplace asym\351trique g\351n\351ralis\351e}{}% 68
\BOOKMARK [1][-]{section.6.1}{Vecteur de param\350tres initiaux}{chapter.6}% 69
\BOOKMARK [1][-]{section.6.2}{M\351thode des moments g\351n\351ralis\351e}{chapter.6}% 70
\BOOKMARK [2][-]{subsection.6.2.1}{Matrice de pond\351ration optimale}{section.6.2}% 71
\BOOKMARK [2][-]{subsection.6.2.2}{Variance-covariance des param\350tres}{section.6.2}% 72
\BOOKMARK [2][-]{subsection.6.2.3}{Contraintes lin\351aires}{section.6.2}% 73
\BOOKMARK [1][-]{section.6.3}{M\351thode de l'\351quation d'estimation optimale}{chapter.6}% 74
\BOOKMARK [0][-]{chapter.7}{Tests statistiques}{}% 75
\BOOKMARK [1][-]{section.7.1}{Test de normalit\351}{chapter.7}% 76
\BOOKMARK [2][-]{subsection.7.1.1}{Test de Shapiro-Wilk}{section.7.1}% 77
\BOOKMARK [2][-]{subsection.7.1.2}{Test d\220Epps-Pulley}{section.7.1}% 78
\BOOKMARK [1][-]{section.7.2}{Tests d'ad\351quation}{chapter.7}% 79
\BOOKMARK [2][-]{subsection.7.2.1}{Test 2 de Pearson}{section.7.2}% 80
\BOOKMARK [2][-]{subsection.7.2.2}{Test de Kolmogorov-Smirnov}{section.7.2}% 81
\BOOKMARK [2][-]{subsection.7.2.3}{Test de distance minimale bas\351 sur la fonction g\351n\351ratrice des moments}{section.7.2}% 82
\BOOKMARK [0][-]{chapter.8}{\311valuation d'options}{}% 83
\BOOKMARK [1][-]{section.8.1}{D\351finitions}{chapter.8}% 84
\BOOKMARK [2][-]{subsection.8.1.1}{\311quation martingale}{section.8.1}% 85
\BOOKMARK [2][-]{subsection.8.1.2}{Param\350tres neutres au risque}{section.8.1}% 86
\BOOKMARK [1][-]{section.8.2}{Aper\347u du mod\350le de Black-Scholes}{chapter.8}% 87
\BOOKMARK [1][-]{section.8.3}{M\351thodes d'\351valuation pour options europ\351ennes}{chapter.8}% 88
\BOOKMARK [2][-]{subsection.8.3.1}{M\351thode de Heston}{section.8.3}% 89
\BOOKMARK [2][-]{subsection.8.3.2}{M\351thode de Carr et Madan}{section.8.3}% 90
\BOOKMARK [2][-]{subsection.8.3.3}{Prix d'exercice hors du cours}{section.8.3}% 91
\BOOKMARK [2][-]{subsection.8.3.4}{Critique de la m\351thode de Carr-Madan}{section.8.3}% 92
\BOOKMARK [2][-]{subsection.8.3.5}{M\351thode d\220Epps}{section.8.3}% 93
\BOOKMARK [1][-]{section.8.4}{Particularit\351s}{chapter.8}% 94
\BOOKMARK [2][-]{subsection.8.4.1}{Option sur actions avec dividendes}{section.8.4}% 95
\BOOKMARK [2][-]{subsection.8.4.2}{Options sur contrats \340 terme et taux de change}{section.8.4}% 96
\BOOKMARK [0][-]{chapter.9}{Exemple d'application}{}% 97
\BOOKMARK [1][-]{section.9.1}{Description des donn\351es}{chapter.9}% 98
\BOOKMARK [1][-]{section.9.2}{Estimation}{chapter.9}% 99
\BOOKMARK [1][-]{section.9.3}{Approximation}{chapter.9}% 100
\BOOKMARK [1][-]{section.9.4}{Graphiques}{chapter.9}% 101
\BOOKMARK [1][-]{section.9.5}{Tests statistiques}{chapter.9}% 102
\BOOKMARK [1][-]{section.9.6}{\311valuation d'options}{chapter.9}% 103
\BOOKMARK [0][-]{section*.52}{Conclusion}{}% 104
\BOOKMARK [0][-]{appendix.A}{\311l\351ments de th\351orie des probabilit\351s}{}% 105
\BOOKMARK [1][-]{section.A.1}{D\351finitions de base}{appendix.A}% 106
\BOOKMARK [1][-]{section.A.2}{Transform\351es d'une variable al\351atoire}{appendix.A}% 107
\BOOKMARK [2][-]{subsection.A.2.1}{La fonction caract\351ristique}{section.A.2}% 108
\BOOKMARK [3][-]{section*.53}{Transform\351e de Fourier}{subsection.A.2.1}% 109
\BOOKMARK [3][-]{section*.54}{D\351finition}{subsection.A.2.1}% 110
\BOOKMARK [3][-]{section*.55}{Les moments}{subsection.A.2.1}% 111
\BOOKMARK [2][-]{subsection.A.2.2}{Inversion de la fonction caract\351ristique}{section.A.2}% 112
\BOOKMARK [3][-]{section*.56}{La densit\351}{subsection.A.2.2}% 113
\BOOKMARK [3][-]{section*.57}{La fonction de r\351partition}{subsection.A.2.2}% 114
\BOOKMARK [2][-]{subsection.A.2.3}{La fonction g\351n\351ratrice des moments}{section.A.2}% 115
\BOOKMARK [2][-]{subsection.A.2.4}{La fonction g\351n\351ratrice des cumulants}{section.A.2}% 116
\BOOKMARK [2][-]{subsection.A.2.5}{La transform\351e d'Esscher}{section.A.2}% 117
\BOOKMARK [1][-]{section.A.3}{La transform\351e de Fourier rapide}{appendix.A}% 118
\BOOKMARK [1][-]{section.A.4}{Processus de L\351vy}{appendix.A}% 119
\BOOKMARK [2][-]{subsection.A.4.1}{D\351finition et propri\351t\351s}{section.A.4}% 120
\BOOKMARK [3][-]{section*.58}{Repr\351sentation de L\351vy-Khintchine}{subsection.A.4.1}% 121
\BOOKMARK [3][-]{section*.59}{Repr\351sentation de L\351vy-It\364}{subsection.A.4.1}% 122
\BOOKMARK [2][-]{subsection.A.4.2}{Processus subordonn\351}{section.A.4}% 123
\BOOKMARK [1][-]{section.A.5}{Th\351or\350mes d'int\351gration}{appendix.A}% 124
\BOOKMARK [2][-]{subsection.A.5.1}{Th\351or\350me de convergence domin\351e de Lebesgue}{section.A.5}% 125
\BOOKMARK [2][-]{subsection.A.5.2}{Th\351or\350me de Fubini}{section.A.5}% 126
\BOOKMARK [0][-]{appendix.B}{\311l\351ments de statistique math\351matique}{}% 127
\BOOKMARK [1][-]{section.B.1}{Loi faible des grands nombres}{appendix.B}% 128
\BOOKMARK [1][-]{section.B.2}{Th\351or\350me central limite}{appendix.B}% 129
\BOOKMARK [2][-]{subsection.B.2.1}{Cas univari\351}{section.B.2}% 130
\BOOKMARK [2][-]{subsection.B.2.2}{Cas multivari\351}{section.B.2}% 131
\BOOKMARK [1][-]{section.B.3}{M\351thode delta multivari\351e}{appendix.B}% 132
\BOOKMARK [0][-]{appendix.C}{Donn\351es}{}% 133
\BOOKMARK [0][-]{section*.62}{Bibliographie}{}% 134
\BOOKMARK [0][-]{section*.64}{Contrat de partage}{}% 135

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\select@language {french}
\select@language {french}
\contentsline {chapter}{R\IeC {\'e}sum\IeC {\'e}}{iii}{section*.2}
\contentsline {chapter}{Abstract}{v}{section*.4}
\contentsline {chapter}{Table des mati{\`e}res}{vii}{section*.5}
\contentsline {chapter}{Liste des tableaux}{xi}{section*.6}
\contentsline {chapter}{Liste des figures}{xiii}{section*.7}
\contentsline {chapter}{Remerciements}{xvii}{section*.9}
\contentsline {chapter}{Introduction}{1}{section*.11}
\contentsline {chapter}{\chapternumberline {1}Les mod\IeC {\`e}les de rendements financiers}{3}{chapter.1}
\contentsline {section}{\numberline {1.1}L'utilisation de mod\IeC {\`e}les en finance}{3}{section.1.1}
\contentsline {subsection}{\numberline {1.1.1}Diff\IeC {\'e}rents types de mod\IeC {\`e}les}{3}{subsection.1.1.1}
\contentsline {subsection}{\numberline {1.1.2}Le risque de mod\IeC {\'e}lisation}{4}{subsection.1.1.2}
\contentsline {section}{\numberline {1.2}Les rendements financiers}{5}{section.1.2}
\contentsline {subsection}{\numberline {1.2.1}D\IeC {\'e}finitions et notations}{5}{subsection.1.2.1}
\contentsline {subsection}{\numberline {1.2.2}Rendements cumul\IeC {\'e}s}{7}{subsection.1.2.2}
\contentsline {subsection}{\numberline {1.2.3}Donn\IeC {\'e}es disponibles}{8}{subsection.1.2.3}
\contentsline {section}{\numberline {1.3}Les premiers mod\IeC {\`e}les}{8}{section.1.3}
\contentsline {subsection}{\numberline {1.3.1}Le mod\IeC {\`e}le de Bachelier}{8}{subsection.1.3.1}
\contentsline {subsection}{\numberline {1.3.2}Proposition de Mandelbrot}{11}{subsection.1.3.2}
\contentsline {subsection}{\numberline {1.3.3}Le mod\IeC {\`e}le de Press}{12}{subsection.1.3.3}
\contentsline {subsection}{\numberline {1.3.4}Le mod\IeC {\`e}le de Praetz}{15}{subsection.1.3.4}
\contentsline {section}{\numberline {1.4}Conditions essentielles de Madan et Seneta}{16}{section.1.4}
\contentsline {chapter}{\chapternumberline {2}La distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{19}{chapter.2}
\contentsline {section}{\numberline {2.1}Le processus de Laplace}{19}{section.2.1}
\contentsline {subsection}{\numberline {2.1.1}Le processus gamma}{20}{subsection.2.1.1}
\contentsline {subsection}{\numberline {2.1.2}Le processus de Wiener}{21}{subsection.2.1.2}
\contentsline {subsection}{\numberline {2.1.3}Le processus de Laplace est un processus subordonn\IeC {\'e}}{22}{subsection.2.1.3}
\contentsline {section}{\numberline {2.2}Distribution de Laplace asym\IeC {\'e}trique g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{25}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}Fonction caract\IeC {\'e}ristique}{25}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}Invariance d'\IeC {\'e}chelle}{26}{subsection.2.2.2}
\contentsline {subsection}{\numberline {2.2.3}Fonctions g\IeC {\'e}n\IeC {\'e}ratrices}{28}{subsection.2.2.3}
\contentsline {subsection}{\numberline {2.2.4}Moments et r\IeC {\^o}le des param\IeC {\`e}tres}{28}{subsection.2.2.4}
\contentsline {subsection}{\numberline {2.2.5}Changement d'\IeC {\'e}chelle et de localisation}{30}{subsection.2.2.5}
\contentsline {subsection}{\numberline {2.2.6}Repr\IeC {\'e}sentation alternative et simulation}{32}{subsection.2.2.6}
\contentsline {subsection}{\numberline {2.2.7}Fonction de Bessel et densit\IeC {\'e}}{33}{subsection.2.2.7}
\contentsline {section}{\numberline {2.3}Cas particuliers}{35}{section.2.3}
\contentsline {subsection}{\numberline {2.3.1}Distribution de Laplace asym\IeC {\'e}trique}{35}{subsection.2.3.1}
\contentsline {section}{\numberline {2.4}Relation avec le mod\IeC {\`e}le de \cite {madan1990variance}}{37}{section.2.4}
\contentsline {chapter}{\chapternumberline {3}Approximation de la densit\IeC {\'e} et de la fonction de r\IeC {\'e}partition}{39}{chapter.3}
\contentsline {section}{\numberline {3.1}L'approximation de Laplace}{39}{section.3.1}
\contentsline {section}{\numberline {3.2}L'approximation de Temme}{41}{section.3.2}
\contentsline {section}{\numberline {3.3}La m\IeC {\'e}thode du point de selle}{42}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}Approximation de la densit\IeC {\'e}}{42}{subsection.3.3.1}
\contentsline {subsection}{\numberline {3.3.2}Unicit\IeC {\'e} du point de selle}{44}{subsection.3.3.2}
\contentsline {subsection}{\numberline {3.3.3}Approximation de la fonction de r\IeC {\'e}partition}{44}{subsection.3.3.3}
\contentsline {subsection}{\numberline {3.3.4}Quelques propri\IeC {\'e}t\IeC {\'e}s des approximations}{45}{subsection.3.3.4}
\contentsline {section}{\numberline {3.4}Application de la m\IeC {\'e}thode du point de selle}{46}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}Approximation de la densit\IeC {\'e}}{46}{subsection.3.4.1}
\contentsline {subsection}{\numberline {3.4.2}Approximation de la fonction de r\IeC {\'e}partition}{47}{subsection.3.4.2}
\contentsline {chapter}{\chapternumberline {4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{49}{chapter.4}
\contentsline {section}{\numberline {4.1}Introduction}{49}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}M\IeC {\'e}thode classique des moments}{50}{subsection.4.1.1}
\contentsline {section}{\numberline {4.2}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e}{50}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}D\IeC {\'e}finition}{51}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}Convergence}{52}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}Matrice de pond\IeC {\'e}ration optimale}{53}{subsection.4.2.3}
\contentsline {subsection}{\numberline {4.2.4}M\IeC {\'e}thode des moments g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}e it\IeC {\'e}rative}{55}{subsection.4.2.4}
\contentsline {subsection}{\numberline {4.2.5}Distribution asymptotique des estimateurs}{55}{subsection.4.2.5}
\contentsline {section}{\numberline {4.3}Estimation sous contraintes}{58}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}Distribution asymptotique des estimateurs contraints}{59}{subsection.4.3.1}
\contentsline {section}{\numberline {4.4}Tests d'hypoth\IeC {\`e}ses param\IeC {\'e}triques}{61}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}Test de Wald}{61}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}Test du multiplicateur de Lagrange}{62}{subsection.4.4.2}
\contentsline {subsection}{\numberline {4.4.3}Test bas\IeC {\'e} sur la statistique de m\IeC {\'e}trique de distance}{63}{subsection.4.4.3}
\contentsline {subsection}{\numberline {4.4.4}En r\IeC {\'e}sum\IeC {\'e}}{63}{subsection.4.4.4}
\contentsline {chapter}{\chapternumberline {5}M\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale}{65}{chapter.5}
\contentsline {section}{\numberline {5.1}\IeC {\'E}quation d'estimation optimale}{67}{section.5.1}
\contentsline {section}{\numberline {5.2}\IeC {\'E}quation d'estimation optimale modifi\IeC {\'e}e}{70}{section.5.2}
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\contentsline {subsection}{\numberline {6.2.1}Matrice de pond\IeC {\'e}ration optimale}{75}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}Variance-covariance des param\IeC {\`e}tres}{76}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}Contraintes lin\IeC {\'e}aires}{77}{subsection.6.2.3}
\contentsline {section}{\numberline {6.3}M\IeC {\'e}thode de l'\IeC {\'e}quation d'estimation optimale}{78}{section.6.3}
\contentsline {chapter}{\chapternumberline {7}Tests statistiques}{81}{chapter.7}
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\contentsline {subsection}{\numberline {7.1.1}Test de Shapiro-Wilk}{81}{subsection.7.1.1}
\contentsline {subsection}{\numberline {7.1.2}Test d\IeC {\textquoteright }Epps-Pulley}{82}{subsection.7.1.2}
\contentsline {section}{\numberline {7.2}Tests d'ad\IeC {\'e}quation}{82}{section.7.2}
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\contentsline {subsection}{\numberline {7.2.2}Test de Kolmogorov-Smirnov}{84}{subsection.7.2.2}
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\contentsline {chapter}{\chapternumberline {8}\IeC {\'E}valuation d'options}{87}{chapter.8}
\contentsline {section}{\numberline {8.1}D\IeC {\'e}finitions}{87}{section.8.1}
\contentsline {subsection}{\numberline {8.1.1}\IeC {\'E}quation martingale}{88}{subsection.8.1.1}
\contentsline {subsection}{\numberline {8.1.2}Param\IeC {\`e}tres neutres au risque}{89}{subsection.8.1.2}
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\contentsline {subsection}{\numberline {8.3.2}M\IeC {\'e}thode de Carr et Madan}{91}{subsection.8.3.2}
\contentsline {subsection}{\numberline {8.3.3}Prix d'exercice hors du cours}{93}{subsection.8.3.3}
\contentsline {subsection}{\numberline {8.3.4}Critique de la m\IeC {\'e}thode de Carr-Madan}{93}{subsection.8.3.4}
\contentsline {subsection}{\numberline {8.3.5}M\IeC {\'e}thode d\IeC {\textquoteright }Epps}{93}{subsection.8.3.5}
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\contentsline {subsection}{\numberline {8.4.1}Option sur actions avec dividendes}{95}{subsection.8.4.1}
\contentsline {subsection}{\numberline {8.4.2}Options sur contrats \IeC {\`a} terme et taux de change}{95}{subsection.8.4.2}
\contentsline {chapter}{\chapternumberline {9}Exemple d'application}{97}{chapter.9}
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\contentsline {section}{\numberline {9.2}Estimation}{99}{section.9.2}
\contentsline {section}{\numberline {9.3}Approximation}{104}{section.9.3}
\contentsline {section}{\numberline {9.4}Graphiques}{105}{section.9.4}
\contentsline {section}{\numberline {9.5}Tests statistiques}{108}{section.9.5}
\contentsline {section}{\numberline {9.6}\IeC {\'E}valuation d'options}{109}{section.9.6}
\contentsline {chapter}{Conclusion}{113}{section*.52}
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\contentsline {section}{\numberline {A.1}D\IeC {\'e}finitions de base}{115}{section.A.1}
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\contentsline {subsection}{\numberline {A.2.1}La fonction caract\IeC {\'e}ristique}{116}{subsection.A.2.1}
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\contentsline {subsubsection}{D\IeC {\'e}finition}{116}{section*.54}
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\contentsline {subsubsection}{La densit\IeC {\'e}}{117}{section*.56}
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\contentsline {subsection}{\numberline {A.2.4}La fonction g\IeC {\'e}n\IeC {\'e}ratrice des cumulants}{119}{subsection.A.2.4}
\contentsline {subsection}{\numberline {A.2.5}La transform\IeC {\'e}e d'Esscher}{120}{subsection.A.2.5}
\contentsline {section}{\numberline {A.3}La transform\IeC {\'e}e de Fourier rapide}{120}{section.A.3}
\contentsline {section}{\numberline {A.4}Processus de L\IeC {\'e}vy}{121}{section.A.4}
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\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-Khintchine}{122}{section*.58}
\contentsline {subsubsection}{Repr\IeC {\'e}sentation de L\IeC {\'e}vy-It\IeC {\^o}}{122}{section*.59}
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\contentsline {section}{\numberline {A.5}Th\IeC {\'e}or\IeC {\`e}mes d'int\IeC {\'e}gration}{123}{section.A.5}
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\contentsline {section}{\numberline {B.1}Loi faible des grands nombres}{125}{section.B.1}
\contentsline {section}{\numberline {B.2}Th\IeC {\'e}or\IeC {\`e}me central limite}{125}{section.B.2}
\contentsline {subsection}{\numberline {B.2.1}Cas univari\IeC {\'e}}{126}{subsection.B.2.1}
\contentsline {subsection}{\numberline {B.2.2}Cas multivari\IeC {\'e}}{126}{subsection.B.2.2}
\contentsline {section}{\numberline {B.3}M\IeC {\'e}thode delta multivari\IeC {\'e}e}{127}{section.B.3}
\contentsline {appendix}{\chapternumberline {C}Donn\IeC {\'e}es}{129}{appendix.C}
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A.R. Hall.
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chr.to.int$ -- 0
cite$ -- 4
duplicate$ -- 56
empty$ -- 99
format.name$ -- 14
if$ -- 252
int.to.chr$ -- 0
int.to.str$ -- 4
missing$ -- 7
newline$ -- 23
num.names$ -- 8
pop$ -- 29
preamble$ -- 1
purify$ -- 15
quote$ -- 0
skip$ -- 38
stack$ -- 0
substring$ -- 54
swap$ -- 13
text.length$ -- 1
text.prefix$ -- 0
top$ -- 0
type$ -- 10
warning$ -- 0
while$ -- 11
width$ -- 5
write$ -- 44

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