The study of optimal controls for forward backward doubly stochastic di¤erential equations

Abstract

In this thesis, we are concerned with stochastic optimal control problems of systems gov- erned by di¤erent types of forward-backward doubly stochastic di¤erential equations. In the rst part, we prove existence of strong optimal control (that is adapted to the initial -algebra) for linear forward-backward doubly stochastic di¤erential equations, with random coe¢ cients and non linear functional cost. The control domain and the cost function were assumed convex. The proof is based on strong convergence techniques for the associated linear FBDSDEs and Mazur s theorem. We derive also necessary and su¢ cient conditions for optimality for this strict control problem. This result is based on the convex optimization principle. In the second part of this thesis, we generelize the results of the rst part to systems governed by linear forward-backward doubly stochastic di¤erential equations of mean eld type, in which the coe¢ cients depend on the state process, and also on the distribution of the state process, via the expectation of some function of the state. In particularly, we establish the existence of strong optimal solutions of a control problem for dynamics driven by a linear forward-backward doubly stochastic di¤erential equations of mean- eld type (MF-LFBDSDEs), with random coe¢ cients and non linear functional cost which is also of mean- eld type. Moreover, we establish necessary as well as su¢ cient optimality conditions for this kind of control problem. In the last part, we establish necessary as well as su¢ cient optimality conditions for exis- tence of both optimal relaxed control and optimal strict control for dynamics of nonlinear forward-backward doubly SDEs of mean- eld type.