On weak solutions of forward–backward SDEs

In this paper we continue exploring the notion of weak solution of forward–backward stochastic differential equations (FBSDEs) and associated forward–backward martingale problems (FBMPs). The main purpose of this work is to remove the constraints on the martingale integrands in the uniqueness proofs in our previous work (Ma et al. in Ann Probab 36(6):2092–2125, 2008). We consider a general class of non-degenerate FBSDEs in which all the coefficients are assumed to be essentially only bounded and uniformly continuous, and the uniqueness is proved in the space of all the square integrable adapted solutions, the standard solution space in the FBSDE literature. A new notion of semi-strong solution is introduced to clarify the relations among different definitions of weak solution in the literature, and it is in fact instrumental in our uniqueness proof. As a by-product, we also establish some a priori estimates of the second derivatives of the solution to the decoupling quasilinear PDE.