Weak existence and uniqueness for forward–backward SDEs

We aim to establish the existence and uniqueness of weak solutions to a suitable class of non-degenerate deterministic FBSDEs with a one-dimensional backward component. The classical Lipschitz framework is partially weakened: the diffusion matrix and the final condition are assumed to be space Hölder continuous whereas the drift and the backward driver may be discontinuous in x . The growth of the backward driver is allowed to be at most quadratic with respect to the gradient term. rategy holds in three different steps. We first build a well controlled solution to the associated PDE and as a by-product a weak solution to the forward–backward system. We then adapt the “decoupling strategy” introduced in the four-step scheme of Ma, Protter and Yong [J. Ma, P. Protter, J. Yong, Solving forward–backward stochastic differential equations explicitly — a four step scheme, Probab. Theory Related Fields 98 (1994) 339–359] to prove uniqueness.