Weak Solutions for Forward-Backward SDEs and Their Uniqueness

Backward stochastic differential equation (BSDE) and its generalized form, Forward-backward stochastic differential equation (FBSDE), have become a ubiquitous tool in mathematical finance during the past decade. The areas that such equations have been found useful include, but are not limited to: option pricing, hedging, portfolio optimizations, stochastic recursive utilities, term structure of interest rates, risk measures, and many others. However, verifying the well-posedness of an FBSDE is usually not easy. It often involves some stringent regularity conditions, or the special structures on the coefficients. Numerous efforts have been made in recent years to enlarge the class of solvable BSDEs/FBSDEs, mostly in the paradigm of the "strong solution".

In this talk we introduce the new notion of "Forward-Backward Martingale Problem" (FBMP), and study its relationship with the "weak solution" to an FBSDE. The FBMP extends the idea of the well-known (forward) martingale problem of Stroock and Varadhan, but it is structured specifically to fit the nature of an FBSDE. We prove the existence of the solution to the FBMP in the cases where general theory of ``strong" solutions cannot be applied. More importantly, we show that, in the Markovian case when all the coefficients are bounded and Holder continuous, then the uniqueness of the FBMP (whence the uniqueness of the weak solution) is determined by the uniqueness of the viscosity solution of a related quasilinear PDE. This talk is based on the joint works with Jianfeng Zhang and Ziyu Zheng.