| Titre : | Equation différentielles stochastiques rétrogrades et application au contrôle optimal |
| Auteurs : | Rafika GATT, Auteur ; Brahim Mezerdi, 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, 2016 |
| Langues: | Français |
| Mots-clés: | Backward doubly stochastic di§erential equation, Superlinear growth condition, Localization, Stochastic partial di§erential equation, Sobolev weak solution, Backward stochastic di§erential equation, Stochastic control, Relaxed control, Young measures,Tightness, Jakubowskiís topology S. |
| Résumé : |
This thesis presents two independent research topics. The Örst part of this dissertation deals with backward doubly stochastic di§erential equations (BDSDEs) with a superlinear growth generator and a square integrable terminal data. We introduce a new local condition on the generator, then we show that they ensure the existence and uniqueness as well as the stability of solutions. This work goes beyond the previous results on the subject. Although we are focused on multidimensional case, The uniqueness result is new for one dimensional BDSDEs. As application, we establish the existence and uniqueness of probabilistic solutions to some semilinear stochastic partial di§erential equations (SPDEís) with superlinear growth generator. By probabilistic solution, we mean a solution which is representable throughout a BDSDEs. The second part of this PhD thesis is concerned with the stochastic control problems where the system is governed by backward stochastic di§erential equations (BSDEís). We are interested with existence of optimal relaxed controls for this kind of systems. Instead of proving this problem with the help of Skorokhodís representation theorem, our techniques are based on construction of the optimal control on an extended probability space, using Young measures. |
| Sommaire : |
DÈdicace i Remerciements ii Abstract ii Introduction 1 1 Backward stochastic di§erential equations BSDEís 5 1.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 The main existence and uniqueness result . . . . . . . . . . . . . . . . . . . 6 1.2.1 BSDEís with linear generator . . . . . . . . . . . . . . . . . . . . . 12 1.3 The comparison theorem of BSDEís . . . . . . . . . . . . . . . . . . . . . . 13 2 Backward doubly stochastic di§erential equations (BDSDEís) 15 2.1 Notation and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The main existence and uniqueness result . . . . . . . . . . . . . . . . . . . 17 2.3 The comparison theorem of BDSDEís . . . . . . . . . . . . . . . . . . . . . 24 3 Backward Doubly SDEs and SPDEs with superlinear growth generators 28 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . 30 3.3 Some observations and examples . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4.1 Proofs of Theorem 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4.2 Proof of Theorem 3.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.5 Application to Sobolev solutions of SPDEs . . . . . . . . . . . . . . . . . . 51 4 Existence of optimal relaxed control for systems driven by backward stochastic di§erential equations (BSDEís) 60 4.1 The setting and its assumtions . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Construction of a weak solution . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2.1 Tightness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 Proof of the main result: theorem (4:2:1) . . . . . . . . . . . . . . . . . . . 72 Bibliography 74 |
| En ligne : | http://thesis.univ-biskra.dz/2520/1/Th%C3%A8se_34_2016.pdf |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/59 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
