Weak Solutions of Forward–Backward SDE's

Abstract

In this note we study a class of forward–backward stochastic differential equations (FBSDE for short) with functional-type terminal conditions. In the case when the time duration and the coefficients are “compatible” (e.g., the time duration is small), we prove the existence and uniqueness of the strong adapted solution in the usual sense. In the general case we introduce a notion of weak solution for such FBSDEs, as well as two notions of uniqueness. We prove the existence of the weak solution under mild conditions, and we prove that the Yamada–Watanabe Theorem, that is, pathwise uniqueness implies uniqueness in law, as well as the Principle of Causality also hold in this context.