Weak compactness of weak solutions sets of forward-backward stochastic differential inclusions

The main result of the article deals with weak compactness of weak solutions sets of forward-backward stochastic differential inclusions. The main result is preceded by existence theorem for considered forward-backward stochastic differential inclusions. More precisely, we shall prove that by given set-valued mappings there exists a system consisting of a complete probability space (omega,F,P), an d-dimensional continuous stochastic process X, an l-dimensional cadlag process Y, and an m-dimensional Brownian motion B defined on the space (omega,F,P) satisfying considered forward-backward stochastic differential inclusion and that each FX-martingale is an FXY-martingale.