Weak solutions and a Yamada–Watanabe theorem for FBSDEs

We prove a Yamada–Watanabe theorem for forward-backward stochastic differential equations (FBSDEs). As a consequence, we show that weak existence and pathwise uniqueness of the solution imply the existence of a strong solution. We use this result to establish existence and uniqueness of weak solutions for a large class of FBSDEs. We also give a probabilistic interpretation of the solutions of certain nonlinear partial differential equations (PDEs).