| Titre : | On the study of some class of self-similar stochastic processes |
| Auteurs : | Idrissi Soumia, Directeur de thèse |
| Type de document : | texte manuscrit |
| Editeur : | [S.l.] : Université de Saida - Dr Moulay Tahar. Faculté des Sciences. Département de Mathématiques., 2017/2018 |
| Format : | 75 ص |
| Accompagnement : | CD |
| Langues: | Français |
| Index. décimale : | BUC-M 008362 |
| Catégories : |
Master Mathématiques:Analyse stochastique, statistique des processus et applications (ASSPA) |
| Mots-clés: | Self-similar processes, Brownian motion, Fractional Brownain motion, Fractional derivatives and integrals, Mittag-Leffler function, Grey noise, Grey Brownian motion. |
| Résumé : |
T his work provides an important step in the construction, definition and the study of
a class of H-sssi stochastic processes (self-similar with stationary increments), which have marginal probability density function that evolves in time according to a differential equation of fractional type. This construction is based on the theory of finite measures on functional spaces. First, we brought the reader through the fundamental notions of stochastic processes and stochastic integration as well. In particular, within the study of H−sssi processes. Then, we focused on fractional Brownian motion (fBm), and introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. We introduced and stud- ied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H−sssi processes. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the un- derline probability space. We also pointed out that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well |
| Note de contenu : |
Contents
Acknowledgments 4 Abstract/Résumé 8 Notations And Symbols 11 Introduction 12 1 Background on stochastic calculus 15 1.1 Basics in stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.1.1 Stochastic processes and filtration . . . . . . . . . . . . . . . . . . . 15 1.1.2 Stationary processes . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.1.3 Self-similar processes . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.1.4 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.1.5 H-sssi processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.1.6 Long-range dependence . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2 Introduction to stochastic integration . . . . . . . . . . . . . . . . . . . . . 27 1.2.1 Itô integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2.2 Extensiens of Itô integral . . . . . . . . . . . . . . . . . . . . . . . . 29 2 Fractional Brownian motion 31 2.1 Fractional Brownian motion definition . . . . . . . . . . . . . . . . . . . . 31 2.2 Fractional Brownian motion characterization. . . . . . . . . . . . . . . . . 32 2.3 Fractional Brownian motion proprieties . . . . . . . . . . . . . . . . . . . 33 2.3.1 Selfsimilarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.2 Markovian property . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.3 Hölder continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5 CONTENTS 6 2.3.4 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.5 The FBm is not a semimartingale . . . . . . . . . . . . . . . . . . . 36 2.3.6 Long-Range Dependence . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Representation of fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.1 Moving averge representation of fBm . . . . . . . . . . . . . . . . . 38 2.4.2 Spectral representation of fBm. . . . . . . . . . . . . . . . . . . . . 40 3 Introduction to Fractional calculus 42 3.1 Fractional integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.1 Fractional integrals definitions . . . . . . . . . . . . . . . . . . . . . 42 3.1.2 Properties of fractional integrals . . . . . . . . . . . . . . . . . . . . 44 3.2 Fractional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.1 Fractional derivatives definitions . . . . . . . . . . . . . . . . . . . . 44 3.2.2 Properties of fractional derivatives . . . . . . . . . . . . . . . . . . 45 3.2.3 Two forms of fractional derivatives . . . . . . . . . . . . . . . . . . 45 3.3 Fractionls integrals and derivatives on the real line . . . . . . . . . . . . . . 46 3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4.1 The Mettag-Leffler function . . . . . . . . . . . . . . . . . . . . . . 47 3.4.2 Fractional representation of fractional Brawnian motion . . . . . . . 48 4 Grey Brawnian motion 50 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 White noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Grey noises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.1 Generalised grey noise space . . . . . . . . . . . . . . . . . . . . . . 59 4.4 Generalesed grey brownian motion . . . . . . . . . . . . . . . . . . . . . . 60 4.4.1 Generalized grey Brownian motion definition . . . . . . . . . . . . 62 4.4.2 The p-variation of generalized grey Brownian motion . . . . . . . . 65 4.5 The ggBm master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.6 Characterization of th ggBm . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.7 Representation of ggBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.8 ggBm trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Conclusion 70 Appendix 71 Bibliography 73 |
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