Existence of optimal controls for systems of controlled forward-backward doubly SDEs

We wish to study a class of optimal controls for problems governed by forward-backward doubly stochastic differential equations (FBDSDEs). Firstly, we prove existence of optimal relaxed controls, which are measure-valued processes for nonlinear FBDSDEs, by using some tightness properties and weak convergence techniques on the space of Skorokhod {\mathbb{D}} equipped with the S -topology of Jakubowski. Moreover, when the Roxin-type convexity...