Stochastic optimal control for dynamics of forward backward doubly SDEs of mean-field type

Abstract

In this paper we establish in first the existence of strong optimal solutions of a control problem for dynamics driven by a linear forward-backward doubly stochastic differential equations of mean-field type (MF-FBDSDEs), with random coefficients and non linear functional cost. Moreover, we establish necessary as well as sufficient optimality conditions for this kind of control problem. In the second part of this paper, we establish necessary as well as sufficient optimality conditions for existence of both optimal relaxed control and optimal strict control for dynamics of nonlinear mean-field forward-backward doubly stochastic differential equations.