| Titre : | Pointwise Second Order Necessary Conditions for Stochastic Optimal Control with Jump Diffusion |
| Auteurs : | Abdelhak Ghoul, Auteur ; Imad Eddine Lakhdari, 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, 2022 |
| Format : | 1 vol. (110 p.) / couv. ill. en coul / 30 cm |
| Langues: | Français |
| Mots-clés: | Optimal control, Stochastic systems with jumps, Pointwise secondorder necessary condition, Maximum principle, Variational equation |
| Résumé : |
Stochastic maximum principle is one of the important majorapproaches to discuss stochastic control problems. A lot of work has been done on this kind of problem, see, for example, Bensoussan [3], Cadenillas and Karatzas [10], Kushner [31], Peng [41]. Recently, another kind of stochastic maximum principle, pointwise second order necessary conditions for stochastic optimal controls has been established and studied for its applications in the financial market by Zhang and Zhang [58] when the control region is assumed to be convex. In Zhang and Zhang [59], the authors extended the pointwise second order necessary conditions for stochastic optimal controls in the general cases when the control region is allowed to be non convex. Second order necessary conditions for optimal control with recursive utilities was proved by Dong and Meng [13]. In this thesis, we generalizes the work of Zhang and Zhang [58] for jump diffusions, we establish a second order necessary conditions where the controlled system is described by a stochastic differential systems driven by Poisson random measure and an independent Brownian motion. The control domain is assumed to be convex. Pointwise second order maximum principle for controlled jump diffusion in terms of the martingale with respect to the time variable is proved. The proof of the main result is based on variational approach using the stochastic calculus of jump diffusions and some estimates on the state processes. Our stochastic control problem provides also an interesting models in many applications such as economics and mathematical finance. |
| Sommaire : |
Contents Abstract Résumé Symbols and Acronyms Introduction 1 Introduction to stochastic calculus 5 1.1 Diffusion process 5 1.1.1 Brownian motion and martingales5 1.1.2 Quadratic variation 6 1.1.3 Stochastic integrals 7 1.1.4 Stochastic differential equations 10 1.1.5 Itô’s lemma 10 1.1.6 Some examples 13 1.2.1 Lévy processes 13 1.2.2 Itô Formula with Jumps 17 1.2.3 Stochastic differential equations with jumps 19 2 Stochastic optimal control problems 21 2.1 Problem formulation 21 2.2 Dynamic programming principle 23 2.3 Stochastic maximum principle 29 2.4 A General stochastic maximum principle for optimal control problems . . . 37 2.4.1 Problem formulation and assumptions 38 2.4.2 Second order expansion 38 2.4.3 Adjoint processes and variational inequality 42 2.4.4 Adjoint equations and the maximum principle 46 3 Pointwise second order necessary conditions for stochastic optimal control 47 3.1 Preliminaries and assumptions 47 3.2 Second order necessary condition in integral form 49 3.3 Pointwise second order maximum principle in terms of the martingale . . . 59 4 Pointwise second order necessary conditions for stochastic optimal control with jump diffusions 66 4.1 Preliminaries and assumptions 66 4.2 Second order necessary condition in integral form with jump Diffusions . . 69 4.3 Pointwise second order maximum principle in terms of the martingale with Jump Diffusions 81 Conclusion 86 Bibliographie 90 |
| En ligne : | http://thesis.univ-biskra.dz/id/eprint/5765 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/126 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
