| Titre : | Contribution to the Study of Backward SDEs and Their Applications to Stochastic Optimal Control |
| Auteurs : | Hanine Azizi, Auteur ; Nabil khelfallah, 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, 2024 |
| Format : | 1 vol. (62 p.) / ill., couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Mots-clés: | Backward stochastic differential equations, forward-backward stochastic differential equation, Optimal stochastic control, stochastic maximum principle, Locally Lipschitz coefficients, non-smooth coefficients. |
| Résumé : |
In this thesis, we aim to generalize some existing results in the literature that concern a stochastic maximum principle for backward stochastic differential equations (BSDEs) or forward-backward stochastic differential equation (FBSDEs), with two possible directions. The first direction is concerned with the stochastic control problem for BSDEs with locally Lipschitz generators, where the domain is not necessarily convex, we establish a necessary and sufficient condition for optimality satisfied by all optimal controls. These conditions are described by a linear locally Lipschitz SDE and a maximum condition on the Hamiltonian. We first prove, under some convenient conditions, the existence of a unique solution to the resulting adjoint equation. Then, with the help of an approximation argument on the coefficients, we define a family of control problems with globally Lipschitz coefficients whereby we derive a stochastic maximum principle for near optimality to such approximated systems. Thereafter, we turn back to the original control problem by passing to the limits. The second direction is devoted to the stochastic maximum principle in optimal control of possibly degenerate FBSDEs, with irregular coefficients. We assume that the coefficients satisfy the Lipschitz conditions, the control domain is non-convex and the control variable does not enter to the diffusion coefficient. We obtain the necessary conditions for optimality utilizing an adjoint process, which is the unique solution of a linear backward-forward stochastic differential equation and a maximal condition on the Hamiltonian. Thanks to the Bouleau-Hirsch flow property, we are able to define the adjoint process employing the derivatives of the coefficients in the sense of distributions. Moreover, the adjoint process does not depend on the choice of the representatives of the weak derivatives. |
| Sommaire : |
Contents Dedication . . . . . . . . . . . . . . . . . . . . . . i Aknowledgemnt . . . . . . . . . . . . . . . . . ii Abstract in Arabic . . . . . . . . . . . . . . . . . . . iv Abstract in French . . . . . . . . . . . . . . . . iv Abstract in English . . . . . . . . .. vi Symbols and Abbreviations x General Introduction xi 1 Stochastic Maximum Principle for Optimal Control Problem of Forward-Backward System 9 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Assumptions and Statement of the Problem . . . . . . . . . . . . . . . . . 9 1.3 Variational Equations and Variational Inequality . . . . . . . . . . . . . . . 10 1.4 The Maximum Principle in Global Form . . . . . . . . . . . . . . . . . . . 19 2 The Maximum Principle for Optimal Control of BSDEs with Locally Lipschitz Coefficients 22 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Problem Formulation and Assumptions . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Formulation of the Control Problem . . . . . . . . . . . . . . . . . 22 2.2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Some Existence and uniqueness Results . . . . . . . . . . . . . . . . . . . . 25 2.4 A family of Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 Maximum Principle for Optimality . . . . . . . . . . . . . . . . . . . . . . 33 2.5.1 Some Convergence Lemmas . . . . . . . . . . . . . . . . . . . . . . 33 2.5.2 Necessary Condition for Optimality . . . . . . . . . . . . . . . . . . 38 2.5.3 Sufficient Condition of Optimality . . . . . . . . . . . . . . . . . . . 41 3 A Stochastic Maximum Principle in Optimal Control of FBSDE with Irregular Coefficients 43 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Problem Statement and the Main Result . . . . . . . . . . . . . . . . . . . 43 3.2.1 Formulation of Control Problem . . . . . . . . . . . . . . . . . . . . 43 3.3 A Maximum Principle for a Family of Perturbed Control Problems . . . . 47 3.3.1 Estimation Between Two Solutions and some Technical Results . . 51 3.3.2 Maximum Principle for Optimality . . . . . . . . . . . . . . . . . . 57 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 |
| En ligne : | http://thesis.univ-biskra.dz/id/eprint/6402 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/152 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
