  We study a Mean Field Game system in a weak formulation through its
associated McKean-Vlasov Forward-Backward Stochastic Differential Equation
(FBSDE). Our main goal is to obtain existence and regularity results of this
FBSDE using the techniques of Malliavin calculus, in particular, classical and
Malliavin differentiability. Using these results, we characterize the
decoupling (master) field.

Optimization and Control;Probability