| Titre : | Some contributions to the problems of stochastic control of di¤usions with jumps |
| Auteurs : | Hanane Ben Gherbal, 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, 2017 |
| Format : | 1 vol. (82 p.) / 30 cm |
| Langues: | Anglais |
| Mots-clés: | Stochastic control,Stochastic differential equation,jump process,optimal control,relaxed control - maximum principle. |
| Résumé : |
This thesis studies optimal control of systems driven by stochastic differential equations (SDEs), with jump processes, where the control variable appears in the drift and the jump term. We study the relaxed problem for which admissible controls are measure-valued processes and the state variable is governed by an SDE driven by a counting measure valued process which we call relaxed Poisson measure such that the compensator is a product measure. Under some conditions on the coefficients, we prove that every diffusion process associated to a relaxed control is a limit of a sequence of diffusion processes associated to strict controls. And we show that the strict and the relaxed control problems have the same value function. The existence of an optimal relaxed control is a consequence of the development. Moreover we derive a maximum principle for this type of relaxed problem.
In second step, we study optimal control problem of the same type of SDEs defined in the first one, but the control variable has two components, the first being absolutely continuous and the second singular. Our goal is to establish a stochastic maximum principle for relaxed controls for this type of relaxed problem, using strong perturbation on the absolutely continuous part of the control and a convex perturbation on the singular one. The proofs are based on the strict maximum principle, Ekeland's variational principle, and some stability properties of the trajectories and adjoint processes with respect to the control variable. |
| Sommaire : |
Abstract i
Résumé ii Acknowledgement iii Symbols and Abbreviations iv Table of Contents v General Introduction 2 1 Stochastic calculus with jump di¤usion 6 1.1 The Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Lévy process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Stochastic integral with respect to N . . . . . . . . . . . . . . . . . 11 1.2.2 Itô-Lévy process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Itô’s formula for Itô-Lévy processes . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Stochastic di¤erential equation driven by a Lévy process . . . . . . . . . . 15 1.5 Relaxed control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Stochastic maximum principle of controlled jump di¤usions 20 2.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 20 vi Table of Contents 2.2 The maximum principle for strict control . . . . . . . . . . . . . . . . . . . 22 2.2.1 Using convex perturbations . . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 Using strong perturbations . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 The maximum principle for near optimal controls . . . . . . . . . . . . . . 41 3 The relaxed maximum principle of controlled jump di¤usions 44 3.1 Formulation of the relaxed control problem . . . . . . . . . . . . . . . . . . 45 3.2 Approximations and existence of relaxed control . . . . . . . . . . . . . . . 49 3.2.1 Approximation of trajectories . . . . . . . . . . . . . . . . . . . . . 49 3.2.2 Existence of an optimal relaxed control . . . . . . . . . . . . . . . . 53 3.3 Maximum principle for relaxed control problems . . . . . . . . . . . . . . . 55 4 The relaxed maximum principle in singular optimal control of controlled jump di¤usions 61 4.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.1.1 Strict control problem . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.1.2 Relaxed-Singular control problem . . . . . . . . . . . . . . . . . . . 64 4.2 Approximation of trajectories . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Maximum principle for relaxed control problems . . . . . . . . . . . . . . . 65 4.3.1 The maximum principle for strict control . . . . . . . . . . . . . . . 66 4.3.2 The maximum principle for near optimal controls . . . . . . . . . . 73 4.3.3 The relaxed stochastic maximum principle . . . . . . . . . . . . . . 75 4.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Conclusion 78 Bibliography 78 |
| En ligne : | http://thesis.univ-biskra.dz/3435/1/thesis.pdf |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/75 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
