| Titre : | Stochastic Differential Equations and Equilibrium Strategy |
| Auteurs : | Chaima Chekalbi, Auteur ; Farid Chighoub, Directeur de thèse |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2024 |
| Format : | 1 vol. (53 p.) / couv. ill. en coul / 30cm |
| Langues: | Français |
| Résumé : |
This dissertation looks into a certain type of stochastic linear quadratic dynamic decision problems that are different from the usual Bellman optimality principle because they are time inconsistent. Since general discounting coefficients and quadratic terms are used in both the running and terminal costs, the situation is time inconsistent. We use the stochastic maximum principle to choose open-loop Nash equilibrium controls over standard optimal controls. We make plans for equilibrium by using a flow of forward-backward stochastic differential equations with a maximum condition. We get an explicit representations of equilibrium strategies in feedback form by separating the flow of the adjoint process. This method is important because it gives both necessary and sufficient conditions for describin equilibrium strategies. This is different from works that was based on dynamic programming and extended Hamilton-Jacobi-Bellman equations, which mostly only gave sufficient conditions. |
| Sommaire : |
Contents Remerciements ii Table des matières iii Introduction 1 1 Introduction to Stochastic Processes 4 1.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Stochastic Integration and Itô’s Formula . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Wiener Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 The Itô’s integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Stochastic Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Backward Stochastic Differential Equation . . . . . . . . . . . . . . . . . . . 15 1.6 Stochastic Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6.1 Optimization Problem Formulation . . . . . . . . . . . . . . . . . . . 19 1.6.2 Stochastic Maximum Principle . . . . . . . . . . . . . . . . . . . . . 19 2 Stochastic Maximum Principle for Time-Inconsistent LQ Control Problem 21 2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Characterization of equilibrium strategies . . . . . . . . . . . . . . . . . . . 27 2.3 The flow of adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.1 Proof of Theorem 2.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Linear Feedback Stochastic Equilibrium Control . . . . . . . . . . . . . . . 41 3 Application: General Discounting Linear Quadratic Regulator 47 Conclusion 51 Bibliographie 52 Abbreviations and Notations 53 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| MM/1303 | Mémoire master | bibliothèque sciences exactes | Consultable |
