| Titre : | Malliavin Smoothness of Solutions of BSDE and Applications |
| Auteurs : | Salima Doubbakh, 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. (77 p.) / ill., couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Résumé : |
this thesis studies two different topics in the stochastic systems fields: The solutions’ Malliavin regularity and control theory. The first is related to the Malliavin smoothness of the solutions of a specific type of quadratic backward stochastic differential equation (chapter 1) and the convergence of their numerical approximating schemes (chapter 2). The second topic refers to optimal control problems for stochastic systems with non-smooth coefficients (chapter 3). Chapter one focuses on the Lq(q ≥ 2)-existence and uniqueness of the solutions of the one-dimensional quadratic backward stochastic differential equation (Q-BSDEs for short) and their properties. The Lp-Hölder continuity of the solutions for any (q > 4 and 2 ≤ p Chapter two uses some existing results on L-BSDEs literature to construct and study the convergence rates of different types of numerical schemes for the solution of Q-BSDE in different cases: explicit and implicit. Those schemes are not completely discrete with respect to the z-variable. However, under some restrictive conditions, a completely discrete scheme” is introduced and studied. The last chapter investigates the necessary and sufficient optimality conditions for a class of controlled stochastic differential equations where the coefficients are merely Lipschitz continuous in the state variable but not necessarily differentiable everywhere. |
| Sommaire : |
Dedication . . . . . . . . . . . . . . . . . . . . . i Aknowledgemnt . . . . . . . . . . . . . . . . . . . . . ii Abstract in Arabic . . . . . . . . . . . . . . . . iv Abstract in French . . . . . . . . . . . . . . . . . . . . . . iv Abstract in English . . . . . . . . . . . . . . . . . . vi General Introduction xii 1 Lp-Hölder Continuity of the Solutions of Q-BSDEs (2 ≤ p 1.1 Introduction . . . . . . . . . . . . 13 1.2 Malliavin Calculus . . . . . . . . . . . . . . . . . . . . . . 13 1.2.1 Notations and Preliminaries . . . . . . . . . . .. . . 14 1.3 Some Properties of L-BSDE . . . . . . . . . . . . . . . . 17 1.3.1 Estimates on the Solution of BSDEs . . . . . . . 17 1.3.2 The Malliavin Regularity for L-BSDE . . . . . . . . . . . 19 1.4 Lp-Hölder Continuity of the Solutions of Q-BSDEs (2 ≤ p 1.4.1 Lq(q ≥ 2)-Solutions of Q-BSDE . . . . . . . . . . . . . . . . . . . . 23 1.4.2 The Malliavin Regularity of non-Markovian Quadratic BSDE . . . . 27 2 The Numerical Schemes for Q-BSDEs and Their Rate of Convergence 36 2.1 Introduction . . . . . . . . . . .. . . . . . . 36 2.2 Numerical Schemes for L-BSDEs . . . . . . . . . . . . . . . . 36 2.3 The Rate of Convergence of Q-BSDEs . . . . . . . . . . . . . 40 2.3.1 An Explicit Scheme for Q-BSDE . . . . . . . . . . . 40 2.3.2 An Implicit Scheme for Q-BSDE . . . . . . . . . 42 2.3.3 A Fully Discrete Scheme for Q-BSDE . . . . . . . . . 46 viiiCONTENTS ix 2.4 Simulation results for Q-BSDE . . . . . . . . . . . . . . . . . 50 2.4.1 Examples . . . . . . . . . . . . . . . . . . . . 50 3 The Maximum Principle for Optimal Control of Diffusion with Non-Smooth Coefficients via Malliavin Calculus 55 3.1 Introduction . . . . . . . . 55 3.2 Problem Formulation and Auxiliary Lemmas . . . . . . 56 3.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . 56 3.2.2 Some Auxiliary Findings . . . . . . . . . . . . .. . 57 3.2.3 Near-Optimality Conditions for a Sequence of Perturbed Control Problems . . . . . . . . . . . . . . . . . . . . . . . 61 3.3 Stochastic Maximum Principle . . . . . . . . . . .. . . . 65 3.3.1 Some Convergence Results . . . . . . . . . . . . . . . . 66 3.3.2 Optimal Variational Principle . . . . . . . . . . . . 71 3.4 Application to Quadratic SDE . . . . . . . . . . .. . . . . 74 3.4.1 Necessary and Sufficient Conditions for Optimality . . .. 76 |
| En ligne : | http://thesis.univ-biskra.dz/6576/ |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/163 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
