| Titre : | Some contributions on stochastic optimal control for FBSDEs |
| Auteurs : | Radhia Benbrahim, 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, 2019 |
| Format : | 1 vol. (94 p.) / 30 cm |
| Langues: | Anglais |
| Mots-clés: | Champ moyen,équation différentielle stochastique progressive rétrograde,contrôle stochastique,contrôle relaxé,contrôle strict,existence,conditions nécessaires et suffisantes,tension,la S-topologie. |
| Résumé : |
In this thesis, we are concerned with stochastic optimal control of systems governed by forwardbackward stochastic differential equations of mean field type. The first part of this thesis is dedicated to the existence and uniqueness of solutions for systems of forward-backward stochastic
differential equation of mean-field type. We use here, the Picard’s iteration method. In the second part, we study the existence of optimal solution of optimal control problem driven by a linear backward stochastic differential equations of mean field type (MF-FBSDE) with non linear cost, where the control domain and the cost functions are assumed convex. The second main result established in this part is a necessary as well as sufficient conditions of optimality satisfied by an optimal control for this kind of stochastic control problem, the proof of this result is based on the convex optimization principle. In the third part, we consider stochastic control problems for a system of forward backward stochastic differential equations of mean field (MF-FBSDEs) with uncontrolled diffusion. Our interest goes particularity to the questions of existence of optimal relaxed control as well as existence of optimal strict control for a nonlinear MF-FBSDEs. We derive also necessary and sufficient optimality conditions for both relaxed and strict control problems. The second and the third parts of this thesis are project of paper submitted in international journal. In the last chapter, we prove the existence of optimal relaxed control as well as optimal strict control for nonlinear MF-FBSDEs with controlled diffusion. This part is published as paper: R.Benbrahim, B.Gherbal, Existence of Optimal Controls for Forward-Backward Stochastic Differential Equation of Mean-field Type, Journal of Numerical Mathematics and Stochastic, 9(1): 33-47, 2017. Key words: Mean-field, forward backward stochastic differential equation, stochastic control, relaxed , strict control, existence, necessary and sufficient conditions, tightness, S-topology |
| Sommaire : |
abstract
Résumé viii Symbols and Abbreviations x Introduction 1 1 Existence and uniqueness of solution of FBSDE of Mean field type 7 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Existence of an optimal strict control and optimality conditions for linear FBSDEs ofmean-field type 24 2.1 Formulation of the problem and assumptions . . . . . . . . . . . . . . . . . . . . . 25 2.2 Existence of optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Necessary and sufficient conditions of optimality for a linear MF-FBSDE . . . . . . 30 3 Existence of optimal solutions and optimality conditions for optimal control problems of MF-FBSDEs systems with uncontrolled diffusion 37 3.1 Statement of the problems and assumptions . . . . . . . . . . . . . . . . . . . . . . 38 3.1.1 Strict control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.2 Relaxed control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.3 Notation and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Existence of optimal relaxed controls . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.1 Proof of theorem 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Existence of optimal strict control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 Necessary and sufficient optimality conditions for relaxed and strict control problems 3.4.1 Necessary and sufficient optimality conditions for relaxed control . . . . . 51 3.4.2 Necessary and sufficient optimality conditions for strict control . . . . . . 66 4 Existence of optimal solutions for optimal control problems of MF-FBSDEs systems with controlled diffusion 72 4.1 Statement of the problems and assumptions . . . . . . . . . . . . . . . . . . . . . . 73 4.1.1 Strict control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 Existence of optimal controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.1 Proof of theorem 4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Appendix: S-topology 85 Conclusion 88 Bibliography |
| En ligne : | http://thesis.univ-biskra.dz/4348/1/these%20final.pdf |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/91 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
