Some results on the stochastic control of backward doubly stochastic differential equations

Abstract

The objective of this thesis is to proof the existence of optimal relaxed controls as well as optimal stricts controls for systems governed by non linear forward–backward stochastic differential equations (FBSDEs). In the first part, we study an singular control problem for systems of forward-backward stochastic differential equations of mean-field type (MF-FBSDEs) in which the control variable consists of two components: an absolutely continuous control and a singular one. The coefficients depend on the states of the solution processes as well as their distribution via the expectation of some function. Moreover the cost functional is also of mean-field type. Our approach is based on weak convergence techniques in a space equipped with a suitable topological setting. We prove in first, the existence of optimal relaxed-singular controls,which are a couple of measure-valued processes and a singular control. Then, by using a convexity assumption and measurable selection arguments, the optimal regular (strict)-singular control are constructed from the optimal relaxed-singular one. In the second part of this thesis, we concentrate on the study of a class of optimal controls for problems governed by forward-backward doubly stochastic differential equations (FBDSDEs). We prove the existence of an optimal control in the class of relaxed controls, which are measure-valued processes, generalizing the usual strict controls. The proof is based on some tightness properties and weak convergence on the space of càdlàg functions, endowed with the viAbstract Jakubowsky S-topology. Furthermore, under some convexity assumptions, we show that the optimal relaxed control is realized by a strict control.