On the Optimal Control of a System Governed by a Fractional Brownian Motion via Malliavin Calculu

Abstract

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.