| Titre : | Forward-Backward Stochastic Differential Equations in Optimal Control |
| Auteurs : | Lakhach Hafsa, Auteur ; Bouanani Hafida, Directeur de thèse ; Kebiri Omar, Directeur de thèse |
| Type de document : | texte manuscrit |
| Editeur : | Université de Saida - Dr Moulay Tahar. Faculté des Sciences. Département de Mathématiques., 2025/2026 |
| Format : | 78ص |
| Accompagnement : | CD |
| Langues: | Anglais |
| Index. décimale : | BUC-M 008493 |
| Catégories : | |
| Mots-clés: | Forward-Backward Stochastic Differential Equations, Optimal Control, Stochastic Maximum Principle, Hamilton-Jacobi-Bellman Equation, Partial Differential Equations, Feynman- Kac formula. |
| Résumé : |
This thesis presents a rigorous and comprehensive investigation of Forward-Backward Stochastic
Differential Equations (FBSDEs) and their application to stochastic optimal control theory, effec- tively bridging abstract stochastic analysis, deterministic partial differential equations (PDEs), and numerical computing. We first construct the foundational probabilistic framework by developing necessary elements of stochastic calculus, forward Stochastic Differential Equations (SDEs), and Backward Stochastic Differential Equations (BSDEs). Building upon these components, we analyze the structural properties of coupled and decoupled FBSDE systems, establishing the existence and uniqueness of their solutions. We further explore the deep interplay between probabilistic trajecto- ries and analytical mechanics by mapping these stochastic processes to linear and non-linear PDEs through the classical and non-linear Feynman-Kac formulas. Leveraging this analytical foundation, we formulate the stochastic optimal control problem through two core paradigms: the local approach via the Stochastic Maximum Principle (SMP) and the global approach via Dynamic Programming and the Hamilton-Jacobi-Bellman (HJB) equation. An impor- tant theoretical aspect of this work is to illustrate the connection established in the original work between the Stochastic Maximum Principle (SMP) and the Hamilton-Jacobi-Bellman (HJB) frame- work, where the adjoint process corresponds to the spatial gradient of the value function. Finally, recognizing that analytical solutions to fully coupled FBSDEs are exceptionally rare, we transition from theory to computation. We implement advanced time-discretization schemes and backward numerical methods to resolve terminal-value constraints. The accuracy, convergence, and practical utility of these numerical algorithms are successfully validated through two distinct case studies: a non-linear, coupled controlled SIR epidemiological model, and a benchmark Linear-Quadratic (LQ) control problem |
| Note de contenu : |
Contents
Introduction 1 1 Forward-Backward Stochastic Differential Equations 6 1.1 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Backward Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Forward–Backward Stochastic Differential Equations . . . . . . . . . . . . . . . . . . 21 1.5 Connection with linear and Non-linear Partial Differential Equations . . . . . . . . . 28 2 Optimal Control Framework Using FBSDEs 31 2.1 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Stochastic Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Dynamic Programming and the HJB Equation . . . . . . . . . . . . . . . . . . . . . 35 2.3.1 Dynamic Programming Principle . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.2 The Hamilton-Jacobi-Bellman Equation . . . . . . . . . . . . . . . . . . . . . 36 2.3.3 Connection between SMP and DP equations . . . . . . . . . . . . . . . . . . 38 3 Numerical Methods and Simulations 41 3.1 Discretization schemes for FBSDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Backward numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Numerical methods for FBSDE (coupled and decoupled) . . . . . . . . . . . . . . . . 47 3.4 Linear-Quadratic (LQ) Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Notations 64 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Conclusion 66 Bibliography 71 |
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BUC-M 008493 Adobe Acrobat PDF |
