| Titre : | Some stochastic control problems governed by a controlled McKean-Vlasov systems |
| Auteurs : | Hafida BEN BRAHIM, Auteur ; Boulakhras Gherbal, Directeur de thèse |
| Type de document : | Thése doctorat |
| Année de publication : | 2025 |
| Format : | 1 vol. (115 p.) |
| Langues: | Anglais |
| Résumé : |
The objective of this thesis is to address stochastic control problems where the system is governed by McKean- Vlasov stochastic differential equations (MV-SDEs) driven by spatially parameterized continuous local martingales. In the first part of this thesis, we investigate the existence of weak solutions and optimal feedback controls for these equations. We first establish the existence of weak solutions by employing Euler approximation and the martingale problem formulation. Next, we show that this Euler approximation has an optimal rate of strong convergence. Finally, using the martingale problem formulation, we establish the existence of an optimal feedback control. In the second part of this thesis, we focus on stochastic singular control problems involving this type of MV-SDEs.The control variable consists of two components: an absolutely continuous control and a singular one. First, under Lipschitz conditions, we establish the existence and uniqueness of strong solutions. Next, we derive the necessary conditions for optimal singular control, assuming that the control domain is convex. The proof of our results is based on the derivatives of the local martingale with respect to spatial parameters and the derivative of the drift coefficient with respect to the probability measure. |
| Sommaire : |
Contents v General Introduction 1 1 Preliminaries 8 1.1 Some basic stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.1 Convergence of random variables . . . . . . . . . . . . . . . . . . . 8 1.1.2 Some properties of stochastic processes . . . . . . . . . . . . . . . . 10 1.1.3 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.4 Continuous local martingale and quadratic variation . . . . . . . . . 16 1.1.5 Stochastic integral with respect to continuous local martingale . . . 20 1.2 Stochastic calculus with respect to spatial parameters local martingale . . 23 1.2.1 Useful notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2.2 Regularity of continuous local martingales with respect to the spatial parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2.3 Stochastic integrals based on spatial parameters continuous local martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2.4 SDEs driven by spatial parameters continuous local martingales . . 29 2 Existence of weak solutions and optimal feedback controls of MV-SDEs driven by spatial parameters local martingale 42 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2.1 Wasserstein metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2.2 Useful notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 iv Contents 2.3 Assumptions and Euler scheme . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4 Weak solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4.1 Weak solution and martingale problem . . . . . . . . . . . . . . . . 47 2.4.2 Weak existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.5 Rate of convergence for the Euler scheme . . . . . . . . . . . . . . . . . . . 58 2.6 Optimal feedback control problem . . . . . . . . . . . . . . . . . . . . . . . 66 2.6.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.6.2 Existence of optimal feedback control . . . . . . . . . . . . . . . . . 68 3 A necessary conditions for optimal singular control of MV-SDEs driven by spatial parameters local martingale 76 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2.1 Differentiability of functions defined on the Wasserstein space . . . 77 3.2.2 Useful notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3 Strong solution of MV-SDEs driven by spatial parameters local martingales 79 3.4 Strong solution of MV-BSDEs driven by spatial parameters local martingale 83 3.5 McKean-Vlasov singular control problem . . . . . . . . . . . . . . . . . . . 89 3.5.1 Formulation of the control problem and assumptions . . . . . . . . 89 3.5.2 Variational equation and adjoint equation . . . . . . . . . . . . . . 91 3.5.3 Necessary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.5.4 Singular mean-field linear quadratic control problem . . . . . . . . . 105 Conclusion 109 Appendices 111 Bibliography 115 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/166 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
