| Titre : | Fractional Stochastic Differential Equations |
| Auteurs : | Bendjebara Bouchra, Auteur ; Idrissi Soumia, Directeur de thèse |
| Type de document : | texte manuscrit |
| Editeur : | Université de Saida - Dr Moulay Tahar. Faculté des Sciences. Département de Mathématiques., 2018/2019 |
| Format : | 51 ص |
| Accompagnement : | CD |
| Langues: | Anglais |
| Index. décimale : | BUC-M 008343 |
| Catégories : |
Master Mathématiques:Analyse stochastique, statistique des processus et applications (ASSPA) |
| Résumé : |
O ur main goal in this work is the study of the existence and the uniqueness of solu-
tion for a class of fractional stochastic differential equation. First, we discussed some fundamental notions of stochastic processes and stochastic in- tegration as well as stochastic differential equations. Next, we introduced the concept of fractional calculus; the branch of mathematics which explores fractional integrals and derivatives. Although the history of fractional calculus is three hundred years old, it is still receiving great interest and acceptance from the research community. In the recent years new alternative definitions of fractional operators have been introduced in the lit- erature: Coimbra derivative (2003), Jumarie derivative (2006), Chen derivative (2010), local fractional Yang derivative (2012). At last we investigated a global result on the existence and uniqueness of solutions for Caputo fractional stochastic differential equations of order α ∈ (1/2; 1) whose coefficients satisfy a standard Lipschitz condition, and using a temporally weighted norm called Bi- elecki norm. With respect to this norm, it was proved that the operator associated with the stochastic integral equation is globally contractive and its fixed point gives rise to the appropriate global solution of the system. Furthermore, we also show that the solutions depend continuously on the initial values. |
| Note de contenu : |
Acknowledgments 3
Abstract/Résumé 6 Introduction 8 1 Preliminary Background 11 1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.1 The basic examples of stochastic processes, The Brownian motion . 14 1.2 Introduction to stochastic integration . . . . . . . . . . . . . . . . . . . . . 17 1.2.1 Itô integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.2 Extensiens of Itô integral . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.3 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . 20 2 Fractional Calculus 23 2.1 Useful Mathematical Functions . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.2 The Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.3 The Mittag-Lefler Function . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Fractional Derivatives and Integrals . . . . . . . . . . . . . . . . . . . . . . 25 2.2.1 Gr¨unwald-Letnikove, 1867-1868 . . . . . . . . . . . . . . . . . . . . 25 2.2.2 Riemann-Liouville, 1832-1847 . . . . . . . . . . . . . . . . . . . . . 26 2.2.3 Caputo, 1969 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Other Fractional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 Marchaud fractional derivative . . . . . . . . . . . . . . . . . . . . . 28 4 2.3.2 Hilfer fractional derivative . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.3 Canavati fractional derivative . . . . . . . . . . . . . . . . . . . . . 29 2.4 Basic Properties of Fractional Derivatives . . . . . . . . . . . . . . . . . . . 29 2.4.1 Semigroup Properties of Fractional Integral Operators . . . . . . . . 29 2.4.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.3 Zero Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.4 Product Rule & Leibniz’s Rule . . . . . . . . . . . . . . . . . . . . 31 2.4.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.1 The Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.2 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . 35 2.5.3 The Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 36 2.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.1 Economic example . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 Stochastic Fractional Differential Equations 39 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Conclusion 47 Appendix 48 Bibliography 50 |
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