| Titre : | Consistent control of stochastic systems with delay |
| Auteurs : | Dounia Bahlali, Auteur ; Farid Chighoub, 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, 2025 |
| Format : | 1 vol. (78 p.) / ill., couv. ill. en coul / 30 cm |
| Langues: | Français |
| Mots-clés: | Time inconsistency, Mean-Variance, Extended HJB equations, Reinsurance and Investment, Insurer, Equilibrium Strategy, Stochastic Differential Equation With Delay. vi |
| Résumé : |
This thesis focuses on solving two research topics in distinct contexts using stochastic control methods. The first topic develops a theory addressing a broad class of time-inconsistent stochastic control problems characterized by stochastic differential delayed equations (SDDEs), indicating the absence of a Bellman optimality principle. The approach involves framing these problems within a game theoretic framework and seeking subgame perfect Nash equilibrium strategies. For a general controlled process with delay and a reasonably broad objective functional, we extend the standard Bellman equation into a system of nonlinear equations. This extension facilitates the determination of both the equilibrium strategy and the equilibrium value function. Importantly, to exemplify the theory’s applicability, we delve into specific example such mean-variance portfolio with state dependent risk aversion problem with delay. By extending the theoretical foundations, this analysis provides insights into addressing and resolving time inconsistencies in a practical example. In the second topic studies an equilibrium investment-reinsurance /new business and investment strategy for mean-variance insurers with state dependent risk aversion, the insurers are allowed to purchase proportional reinsurance, acquire new business and invest in a financial market, where both the surplus and the price process of risky stocks of the insurers are assumed to follow geometric Levy process. Under the influence of performance-related capital inflow/outflow, the wealth process of the investor is modeled by a stochastic differential delay equation (SDDE). |
| Sommaire : |
Résumé iv Abstract vi Introduction x 1 Preliminaries in Classical Stochastic Control Problems 8 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Classical Stochastic Control Problems . . . . . . . . . . . 8 1.2.1 Formulation of the control problem . . . . . . . . . . . . 1.2.2 Methods to solving optimal control problem . . . . . . . 11 1.3 Time-inconsistent problem . . . . . . . . . . . . . . . . . .. . . 14 1.3.1 Approaches to handle time inconsistency . . . . . . 15 2 A General Time-Inconsistent Stochastic Optimal Control Problem with Delay 18 2.1 Introduction . . . . . . . . . . . . . . . . . . . 18 2.2 Model and problem formulations . . . . . . . . . . . . . . . 18 2.3 Optimal time-consistent solution . . . . . . . . . . . . . . 20 2.4 Extended HJB equations and verification theorem . . . . .. . 23 2.5 Application in mean-variance portfolio with state dependent risk aversion with delay . . . . . . . . . . . . . . . . . . . .. . . . 33 2.5.1 Wealth process . . . . . . . . . . . . . . . . . . . . . . 33 2.5.2 Equilibrium investment strategy solution . . . . . . . . 2.6 Existence and Uniqueness of solutions for integral equatio . . . 42 3 Equilibrium Reinsurance-Investment Strategies for Mean-Variance Insurers with Delay 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Surplus process and financial market . . . . . . . .. . 46 3.2.1 Wealth process . . . . . . . . . . . . . . . . . . . . .. 48 3.3 Mean variance Criterion with state dependent risk aversion . . . 52 3.4 Optimal time-consistent solution . . . . . . . . . . . . . . . 53 3.5 Extended HJB equations and verification theorem . . . 54 Conclusion 75 Appendix 76 Bibliography |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/170 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
