Some contributions on stochastic optimal control for FBSDEs

Abstract

In this thesis, we are concerned with stochastic optimal control of systems governed by forwardbackward stochastic differential equations of mean field type. The first part of this thesis is dedicated to the existence and uniqueness of solutions for systems of forward-backward stochastic differential equation of mean-field type. We use here, the Picard’s iteration method. In the second part, we study the existence of optimal solution of optimal control problem driven by a linear backward stochastic differential equations of mean field type (MF-FBSDE) with non linear cost, where the control domain and the cost functions are assumed convex. The second main result established in this part is a necessary as well as sufficient conditions of optimality satisfied by an optimal control for this kind of stochastic control problem, the proof of this result is based on the convex optimization principle. In the third part, we consider stochastic control problems for a system of forward backward stochastic differential equations of mean field (MF-FBSDEs) with uncontrolled diffusion. Our interest goes particularity to the questions of existence of optimal relaxed control as well as existence of optimal strict control for a nonlinear MF-FBSDEs. We derive also necessary and sufficient optimality conditions for both relaxed and strict control problems. The second and the third parts of this thesis are project of paper submitted in international journal. In the last chapter, we prove the existence of optimal relaxed control as well as optimal strict control for nonlinear MF-FBSDEs with controlled diffusion. This part is published as paper: R.Benbrahim, B.Gherbal, Existence of Optimal Controls for Forward-Backward Stochastic Differential Equation of Mean-field Type, Journal of Numerical Mathematics and Stochastic, 9(1): 33-47, 2017.