Stochastic Maximum Principle for the System Governed by Backward Doubly Stochastic Differential Equations with Risk-Sensitive Control Problem and Applications

Abstract

his thesis based on the study of the stochastic maximum principle with risk-sensitive for two different systems. We obtain these systems by generalizing the results of Chala [10; 11], and by using the paper of Djehiche et al. in [13]: The first system is driven by a backward doubly stochastic differential equation. We use the risk-neutral model for which an optimal solution exists as a preliminary step, this is an extension of the initial control problem. Our goal is to establish necessary and sufficient optimality conditions for the risk-sensitive performance functional control problem. We show for the second system which is driven by a fully coupled forward-backward stochastic differential equation of mean-field type, by using the same technique as in the first case, we get the necessary and sufficient optimality conditions for the risk-sensitive, where the set of admissible controls is convex in all the cases. Finally, we illustrate our main results by giving applied examples of risk-sensitive control problems.