| Titre : | Optimal control for stochastic differential equations governed by normal martingales |
| Auteurs : | Imad Eddine Lakhdari, Auteur ; Farid Chighoub, Directeur de thèse |
| Type de document : | Monographie imprimée |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2018 |
| Format : | 1 vol. (112 p.) / 30 cm |
| Langues: | Anglais |
| Résumé : |
This thesis presents two research topics, the first one being divided into two parts. In the first part, we study an optimal control problem where the state equation is driven by a normal martingale. We prove a sufficient stochastic maximum and we also show the relationship between stochastic maximum principle and dynamic programming in which the control of the jump size is essential and the corresponding Hamilton-Jacobi-Bellman (HJB) equation in this case is a mixed second order partial differential-difference equation. As an application, we solve explicitly a mean-variance portfolio selection problem. In the second part, we study a non smooth version of the relationship between MP and DPP for systems driven by normal martingales in the situation where the control domain is convex.
The second topic, is to characterize sub-game perfect equilibrium strategy of a partially observed optimal control problems for mean-field stochastic differential equations (SDEs) with correlated noises between systems and observations, which is time-inconsistent in the sense that it does not admit the Bellman optimality principle. |
| Sommaire : |
Acknowledgement i Abstract ii Résumé iii Table des matières iii Introduction 1 1 Stochastic Control Problem 5 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Strong formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Existence of optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Dynamic programming principle (DPP) . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1 The Bellman principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.2 The Hamilton Jacobi Bellman equation . . . . . . . . . . . . . . . . . . . 12 1.3.3 Viscosity solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4 Stochastic maximum principle (SMP) . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4.1 Problem formulation and assumptions . . . . . . . . . . . . . . . . . . . . 29 1.4.2 The stochastic maximum principle . . . . . . . . . . . . . . . . . . . . . . 30 1.4.3 Necessary conditions of optimality . . . . . . . . . . . . . . . . . . . . . . 31 1.4.4 Variational equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.4.5 Variational inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.4.6 Su¢ cient conditions of optimality . . . . . . . . . . . . . . . . . . . . . . 37 1.5 Relation to dynamic programming principle . . . . . . . . . . . . . . . . . . . . . 39 2 Relationship Between Maximum Principle and Dynamic Programming for Systems Driven by Normal Martingales 42 2.1 Assumptions and problem formulation . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2 Su¢ cient stochastic maximum principle . . . . . . . . . . . . . . . . . . . . . . . 45 2.3 Relation to dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4 Application to mean-variance portfolio selection problem . . . . . . . . . . . . . . 55 2.4.1 Quadratic loss minimization problem . . . . . . . . . . . . . . . . . . . . . 57 2.4.2 The solution of the mean-variance problem . . . . . . . . . . . . . . . . . 60 3 Relationship Between MP and DPP for Systems Driven by Normal Martin- gales: viscosity solution 63 3.1 Problem statement and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4 A Characterization of Sub-game Perfect Equilibria for SDEs of Mean-Field Type Under Partial Information 75 4.1 Notation and statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Adjoint equations and the stochastic maximum principle . . . . . . . . . . . . . . 80 4.3 An application to linear-quadratic control problem . . . . . . . . . . . . . . . . . 89 4.4 Extension to Mean-Field Game Models . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4.1 The local limiting decision problem . . . . . . . . . . . . . . . . . . . . . . 97 4.5 Proof of Theorem 4.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.5.1 The performance estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.5.2 Proof of Theorem 4.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Conclusion 106 |
| En ligne : | http://thesis.univ-biskra.dz/3677/1/Th%C3%A8se%20de%20Doctorat%20Finale%202018%2C%20Imad.pdf |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/82 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
