| Titre : | Partially obseved optimal control problem for SDEs of Mckean-Vlasov type and Applications |
| Auteurs : | Hakima Miloudi, Auteur ; Imad Eddine Lakhdari, Directeur de thèse ; Mokhtar Hafayad, 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, 2022 |
| Format : | 1 vol. (110 p.) / couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Résumé : |
Partially observed control problems have received much attention and became a powerful tool in many fields, such as mathematical finance, optimal control, etc.From the viewpoint of reality, many situations, full information is not always available to controllers, but the partial one with noise. Furthermore, the recent work of Buckdahn, R. [7] and Hafayed, M. [24] on Mckean-Vlasov type stochastic differential equations and their optimal control opens a new avenue for the study of optimal control problems in general. The objective of this thesis is to extend these results of [7] and [24] to the case of a partially observed optimal control problem. More precisely, we study partially observed optimal control problems of general McKean-Vlasov differential equations, in which the coefficients depend on the state of the solution process as well as of its law and the control variable. By applying Girsanov’s theorem with a standard convex variational technique, we develop the stochastic maximum principle for our partially observed control problem where the control domain is convex. Also, in this thesis, we prove a new stochastic maximum principle for a class of partially observed optimal control problems of Mckean-Vlasov type with jumps. The stochastic system under consideration is governed by a stochastic differential equation driven by Poisson random measure and an independent Brownian motion. The derivatives with respect to probability measure and the associate Itô-formula are applied to prove our main results. And as an illustration, by applying our maximum principle, McKean-Vlasov type linear quadratic control problem with jump is discussed,where the partially observed optimal control is obtained explicitly in feedback form.Key words. Partially observed optimal control, Stochastic maximum principle, Derivatives with respect to the measure, McKean-Vlasov differential equations, McKean-Vlasov stochastic system with jumps, Probability measure, Girsanov’s theorem |
| Sommaire : |
Contents Introduction x 1 Stochastic optimal control problems 16 1.1 Stochastic processes 16 1.2 Natural fitration 16 1.3 Brownian motion 16 1.4 Integration by parts formula 17 1.5 Strong formulation17 1.6 Weak formulation 18 1.7 Stochastic maximum principle (SMP) 19 1.7.1 Problem formulation 19 1.7.2 The stochastic maximum principle20 1.7.3 Necessary conditions of optimality 20 1.7.4 Variational equation 21 1.7.5 Variational inequality 23 1.8 Partial observation control problem 26 1.8.1 Assumptions and Problem Formulation 26 1.8.2 Stochastic maximum principle for partially observed optimal control problem 29 1.9 Some classes of stochastic controls 35 1.9.1 Optimal control 36 1.9.2 Admissible control 36 1.9.3 Near-optimal control 36 1.9.4 Feedback control 36 1.9.5 Random horizon36 viCONTENTS vii 1.9.6 Relaxed control 37 2 Partially-Observed Optimal Control Problems for SDEs 38 2.1 Formulation of the Problem 38 2.2 Stochastic Maximum Principle for Partially Observed Optimal Control Problems 41 2.3 Application: Partially observed linear-quadratic control problem 49 3 Stochastic maximum principle for partially observed optimal control problems of general McKeanVlasov differential equations 3.1 Introduction 53 3.2 Assumptions and Problem Formulation 54 3.3 Stochastic Maximum Principle 60 3.4 Application:Partially observed linear-quadratic control problem 70 4 Necessary Conditions for Partially Observed Optimal Control of General McKean-Vlasov Stochastic Differential Equations with Jumps 75 4.1 Introduction 75 4.2 Formulation of the problem and preliminaries 77 4.3 Necessary Conditions of Optimality 83 4.4 Partially observed McKean-Vlasov linear quadratic control problem with jumps . 95 Conclusion 98 Bibliographie 101 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/124 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
