| Titre : | On the Optimal Control of a System Governed by a Fractional Brownian Motion via Malliavin Calculu |
| Auteurs : | Tayeb Bouaziz, 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, 2022 |
| Format : | 1 vol. (70 p.) / couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Résumé : |
In this thesis, we use the Malliavin calculus to derive the Pontryagin’s stochastic maximum principle under the form of necessary and suffcient optimality conditions. In the introductory chapter 1, we state and build the framework that we use in the following chapters. We introduce the necessary tools from the Malliavin calculus, the Russo & Vallois integral, and apply the Doss-Sussmann transformation to our system, which is governed by backward doubly stochastic dynamics driven by standard Wiener and fractional Brownian motions. At the end of this chapter, we present important Girsanov theorems and uniqueness and existence result. In chapter 2, we derive the Pontryagin stochastic maximum principle for a system driven by standard and fractional Brownian motions, with Hurst parameter H 2 ?12, 1? . In chapter 3, we solve a stochastic optimization problem for backward stochastic differential equations driven by fractional Brownian motions, using the Malliavin calculus, where we minimize the cost functional, which is in the risk-sensitive type, with respect to the admissible control. In addition, we present the necessary and suffcient optimality conditions for this problem. Finally, we apply the pre-established results to an interesting linear-quadratic control problem. Our work is considered an extension of the approaches of Buckdahn et al. in [12, 13] and Zähle in [62, 63] and the risk neutral stochastic maximum principle established by Yong in [61] to backward stochastic differential equations driven by fractional Brownian motions. |
| Sommaire : |
Contents Cover i Dedication ii Abstract iii Résumé iv Contents vi Symbols and Abbreviations vii Abbreviations vii Symbols viii Introduction 1 1 Introduction to Malliavin Calculus 6 1.1 Malliavin Calculus with Respect to W (.) 1.2 Malliavin Calculus with Respect to BH (.) 9 1.2.1 Fractional Calculus 10 1.2.2 The Russo & Vallois Integral 15 1.2.3 Doss-Sussmann Transformation of Fractional BDSDE 16 1.3 Girsanov Theorems and Existence and Uniqueness Result for Systems Driven by Fractional Brownian Motions 21 1.3.1 Change of Probability Measures and Girsanov Transformations 22 vContents 1.4 Uniqueness and Existence Result 23 2 Pontryagin’s SMP for a System Driven by Fractional Brownian and Standard Wiener Motions via Malliavin Calculus 27 2.2 Variational Equality .. . . . . . 29 2.2.1 Assumptions and Definitions 29 2.3 Necessary Optimality Conditions 35 3 Malliavin Calculus Used to Derive Pontryagin’s Risk-Sensitive SMP for BSDEs Driven by Fractional Brownian Motion 43 3.1 Risk-Sensitive Stochastic Maximum Problem 44 3.2 Risk-Sensitive Necessary Optimality Conditions 45 3.3 Transformation of the Adjoint Equation 49 3.4 Risk-Sensitive Suffcient Optimality Conditions 55 3.5 Application 57 3.5.1 Linear Quadratic Risk-Sensitive Control Problem . 57 3.5.2 Explicit Solution of the Riccati Equation 62 Conclusion 64 Bibliography 65 |
| En ligne : | http://thesis.univ-biskra.dz/5981/1/Tayeb_BOUAZIZ_Thesis.pdf |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/132 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
