| Titre : | Some results on the stochastic control of backward doubly stochastic differential equations |
| Auteurs : | Abdelhakim Ninouh, Auteur ; Boulakhras Gherbal, 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, 2020 |
| Format : | 1 vol. (118 p.) / couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Résumé : |
The objective of this thesis is to proof the existence of optimal relaxed controls as well as optimal stricts controls for systems governed by non linear forward–backward stochastic differential equations (FBSDEs). In the first part, we study an singular control problem for systems of
forward-backward stochastic differential equations of mean-field type (MF-FBSDEs) in which the control variable consists of two components: an absolutely continuous control and a singular one. The coefficients depend on the states of the solution processes as well as their distribution via the expectation of some function. Moreover the cost functional is also of mean-field type. Our approach is based on weak convergence techniques in a space equipped with a suitable topological setting. We prove in first, the existence of optimal relaxed-singular controls,which are a couple of measure-valued processes and a singular control. Then, by using a convexity assumption and measurable selection arguments, the optimal regular (strict)-singular control are constructed from the optimal relaxed-singular one. In the second part of this thesis, we concentrate on the study of a class of optimal controls for problems governed by forward-backward doubly stochastic differential equations (FBDSDEs). We prove the existence of an optimal control in the class of relaxed controls, which are measure-valued processes, generalizing the usual strict controls. The proof is based on some tightness properties and weak convergence on the space of càdlàg functions, endowed with the viAbstract Jakubowsky S-topology. Furthermore, under some convexity assumptions, we show that the optimal relaxed control is realized by a strict control |
| Sommaire : |
Abstract vi Résumé viii Symbols and Abbreviations x Introduction xii 1 General Introduction of Stochastic Analysis. 2 1.1 Introductions of Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Conditional expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.2 Convergence of probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Tightness of the laws of processes . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.3 Itô’s stochastic integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Definition of Itô’s Stochastic Integral . . . . . . . . . . . . . . . . . . . . . 32 1.3.2 Itô’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.3.3 Martingale representation theorem . . . . . . . . . . . . . . . . . . . . . . 37 1.4 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.4.1 Strong solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.4.2 Weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.5 Backward Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . 43 1.5.1 Weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2 A mixed Relaxed-Singular Optimal Controls For Systems of MF-FBSDEs Type. 49 2.1 Formulation of the problems and assumptions . . . . . . . . . . . . . . . . . . . . 50 2.1.1 Regular-singular control problem . . . . . . . . . . . . . . . . . . . . . . . 50 2.1.2 Relaxed-singular control problem . . . . . . . . . . . . . . . . . . . . . . . 53 2.1.3 Notation and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.2 Existence of an optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.2.1 Existence of an optimal relaxed-singular control . . . . . . . . . . . . . . 58 2.2.2 Existence of an optimal strict-singular control . . . . . . . . . . . . . . . . 67 3 Existence of Optimal Controls For Systems of Controlled Forward-Backward DoublySDEs. 71 3.1 Introduction to Backward Doubly Stochastic Differential Equations BDSDEs . 72 3.1.1 Existence and Uniqueness of BDSDEs . . . . . . . . . . . . . . . . . . . . 74 3.2 Existence of optimal controls for nonlinear FBDSDEs . . . . . . . . . . . . . . . . 77 3.2.1 Statement of the problems and assumptions . . . . . . . . . . . . . . . . . 77 3.2.2 Existence of optimal relaxed controls . . . . . . . . . . . . . . . . . . . . . 85 3.2.3 Existence of optimal strict control . . . . . . . . . . . . . . . . . . . . . . . 93 Appendix: Topologies on the Skorokhod space 96 Conclusion 108 Bibliography 11 |
| En ligne : | http://thesis.univ-biskra.dz/5332/1/These%20-%20Dr.Abd%20Elhakim%20NINOUH.pdf |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/108 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
