| Titre : | The study of optimal controls for forward backward doubly stochastic di¤erential equations |
| Auteurs : | Nassima Berrouis, Auteur ; Boulakhras Gherbal, Directeur de thèse |
| Type de document : | Thése doctorat |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2023 |
| Format : | 1 vol. (121 p.) / couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Résumé : |
In this thesis, we are concerned with stochastic optimal control problems of systems governed by di¤erent types of forward-backward doubly stochastic di¤erential equations. In the …rst part, we prove existence of strong optimal control (that is adapted to the initial -algebra) for linear forward-backward doubly stochastic di¤erential equations, with random coe¢ cients and non linear functional cost. The control domain and the cost function were assumed convex. The proof is based on strong convergence techniques for the associated linear FBDSDEs and Mazur’s theorem. We derive also necessary and su¢ cient conditions for optimality for this strict control problem. This result is based on the convex optimization principle. In the second part of this thesis, we generelize the results of the …rst part to systems governed by linear forward-backward doubly stochastic di¤erential equations of mean …eld type, in which the coe¢ cients depend on the state process, and also on the distribution of the state process, via the expectation of some function of the state. In particularly, we establish the existence of strong optimal solutions of a control problem for dynamics driven by a linear forward-backward doubly stochastic di¤erential equations of mean- …eld type (MF-LFBDSDEs), with random coe¢ cients and non linear functional cost which is also of mean-…eld type. Moreover, we establish necessary as well as su¢ cient optimality conditions for this kind of control problem. In the last part, we establish necessary as well as su¢ cient optimality conditions for existence of both optimal relaxed control and optimal strict control for dynamics of nonlinear forward-backward doubly SDEs of mean-…eld type. |
| Sommaire : |
Contents Acknowledgement ii Abstract iii Résumé iv Symbols and Acronyms v Introduction 1 1 Some Mathematical Preliminaries 8 1.1 Probability . . . . . . . . . .. . . . . . . . . 8 1.1.1 Probability spaces . . . . . . . . . . . . 8 1.1.2 Random variable . . . . . . . . . . . 11 1.1.3 Modes of Convergence . . . . . . 14 1.1.4 Conditional expectation . . .. . . . 15 1.2 Stochastic Processes . . . . . . . . . . . . . . 18 1.2.1 The Brownian Motion . . . . . . . . . . 25. . . . 21 1.3 Stochastic Integral . . . . . . . 1.3.1 Construction of Itô’s Integral . . . .. . . . 25 1.3.2 Introduction to backward integrals . . . . . 31 1.3.3 Itô’s Formula . .. . 32 viiContents 2 Optimal control problem for a linear FBDSDEs 35 2.1 Formulation of the problem and assumptions . .. . . 35 2.2 Existence of optimal strict controls for linear FBDSDEs: . . . . . . . . . . 37 2.3 Necessary and su¢ cient conditions . . . . . . . . . .. . . 45 3 Optimal control problem for a linear MF-FBDSDEs 51 3.1 Formulation of the problem and assumptions . . . . . . . 51 3.2 Existence of a strong optimal control . . . . . .. . . . . 54 3.3 Necessary and su¢ cient conditions for optimality . . . . . . 60 4 Necessary and su¢ cient optimality conditions for both relaxed and strict control problems for nonlinear MF-FBDSDEs 66 4.1 Statement of the problems . . . . . . . . . . . . . 67 4.1.1 Strict control problem . . . . . . . . . . . . . . 67 4.1.2 Relaxed control problem . . . . . . . . . . . . . 68 4.2 Necessary and su¢ cient optimality conditions for relaxed control problems 70 4.2.1 The variational inequality . . . . . .. . . . 72 4.2.2 Necessary optimality conditions for relaxed control . . . . 86 4.2.3 Su¢ cient optimality conditions for relaxed control . . . . . . 89 4.3 Necessary and Su¢ cient optimality conditions for strict control . . . . . . . 93 4.3.1 Necessary optimality conditions for strict control . . . . . . . . . . 93 4.3.2 Su¢ cient optimality conditions for strict control . . . . . . . . . . . 96 Conclusion 98 Appendix 1 |
| En ligne : | http://thesis.univ-biskra.dz/id/eprint/6065 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/140 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
