| Titre : | On Fractional Brownian Motion with Application to Risk Sensitive |
| Auteurs : | Ikram Hamed, Auteur ; Adel Chala, 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, 2024 |
| Format : | 1 vol. (86 p.) / ill., couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Mots-clés: | Fractional Brownian motion, Risk-sensitive control, SDE, SMP. |
| Résumé : |
This thesis expands upon Pontryagin’s stochastic maximum principle to accommodate systems modeled by fractional Brownian motion. we present two research topics. The FIRrst centers on an optimal control problem wherein the state equation is driven by fractional Brownian motion, and the cost functional follows a risk-neutral type. Initially, we present the optimal control problem and its underlying dynamics, followed by the convex perturbation method in which the set of admissible controls is convex. Subsequently, we establish both optimality conditions for this model. Finally, we demonstrate our …ndings through a linear quadratic problem, solving the associated Riccati type equation. The second topic focuses on characterizing optimal control problems within a risk-sensitive framework. The system dynamics are de…ned using only the backward stochastic di¤erential equations. However, the performance criterion is distinct; instead of directly minimizing costs, we aim to minimize a convex disutility function of the cost. As an initial step, we elucidate the relationship between riskneutral and risk-sensitive loss functionals. Next, we establish the equivalence between expected exponential utility and quadratic backward stochastic di¤erential equations. Further, we reformulate the risk-sensitive problem into a standard risk-neutral one by introducing an auxiliary term and demonstrate the determination of the adjoint equation. Thus, we derive the stochastic maximum principle using a standard application of risk-neutral results. Finally, we apply these concepts to a control problem with linear quadratic risk sensitivity. |
| Sommaire : |
contents Dedication i Acknowledgement ii Abstract iii Résumé iv Table of Contents v Symbols and Abbreviations viii Introduction 1 1 Basic Notations and Stochastic Control 9 1.1 Fractional Brownian motion . . . . . . . . 10 1.1.1 Stochastic processes . . . . . . . . . . . 10 1.1.2 Natural …ltration . . . . . . . . . . . . . . . . 10 1.1.3 Brownian motion . . . . . . . . . . . .. . . 11 1.1.4 Fractional Brownian motion . . . . . . 11 1.2 Approaches for resolving optimal control problems. . . . . . 17 v1.2.1 Dynamic programming method . . . . . . . . . . 17 1.2.2 Pontryagin’s maximum principle . . . . . . . . . . . 22 1.3 Some classes of stochastic controls . . . . . . . . . . . . . 27 1.3.1 Relaxed control . . . . . . . . . . . . . . . . 27 1.3.2 Random horizon . . . . . . . . . . . . . .. . 27 1.3.3 Admissible control . . . . . . . . . .. . . 28 1.3.4 Feedback control . . . . . . . . . . . . . . . . . 28 1.3.5 Optimal control . . . . . . . . . . . . . . . . . . . 28 1.3.6 Near-optimal control . . . . . . . . . 28 1.3.7 Ergodic control . . . . . . . . . . . . . . . . . 29 1.3.8 Robust control . . . . . . . . . . . . .. . 29 1.3.9 Partial observation control problem . . . . . . . . . . 30 1.3.10 Singular control . . . . . . . . . . . . . . . . . . 30 2 Stochastic Maximum Principle for Risk-Neutral Control Problem 32 2.1 Problem formulation . . . . . . . . . . . . . . . 33 2.2 Primary results . . . . . . . . . . . . . . . . . . . . . . 36 2.3 Stochastic maximum principle . . . . . . . . . . . . . . . . 49 2.3.1 A necessary maximum principle . . . . . . . . 51 2.3.2 A su¢ cient maximum principle . . . . . . . . . . . . 52 2.4 LQ problem . . . . . 55 2.4.1 Integrating Riccati and ordinary di¤erential equations . . . . . . 57 3 A Risk-Sensitive Stochastic Maximum Principle for Backward Stochastic Differential Equation with Application 59 3.1 Statement of risk-sensitive problem . . . . . . . 61 3.2 Risk-neutral SMP for fractional backward stochastic di¤erential equation 63 3.3 Results statment . . . . . . . . . . . . . . 66 3.3.1 Expected exponential utility and backward quadratic stochastic equation . . . . . . 66 3.3.2 New adjoint equations and risk-sensitive necessary optimality conditions . . . . . .. 69 3.4 Risk-sensitive su¢ cient optimality conditions . . . . . 77 3.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.5.1 A control problem with linear quadratic risk sensitivity . .. 79 3.5.2 Explicit solution of the Riccati equation . . . . . . . . .. 83 Conclusion and Perspectives 84 Bibliography 86 |
| En ligne : | http://thesis.univ-biskra.dz/id/eprint/6561 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/159 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
