Weak solutions for forward--backward SDEs--a martingale problem approach

TL;DR

This paper introduces a new Forward-Backward Martingale Problem (FBMP) framework to analyze weak solutions of FBSDEs, establishing conditions for existence and uniqueness linked to viscosity solutions of associated PDEs.

Contribution

It extends the classical martingale problem to FBSDEs, providing a novel approach to existence and uniqueness analysis through a structured martingale problem formulation.

Findings

Existence of solutions under general conditions

Weak solutions exist in the Markovian case with continuous coefficients

Uniqueness is characterized by viscosity solutions of PDEs

Abstract

In this paper, we propose a new notion of Forward--Backward Martingale Problem (FBMP), and study its relationship with the weak solution to the forward--backward stochastic differential equations (FBSDEs). 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 first prove a general sufficient condition for the existence of the solution to the FBMP. In the Markovian case with uniformly continuous coefficients, we show that the weak solution to the FBSDE (or equivalently, the solution to the FBMP) does exist. Moreover, we prove that the uniqueness of the FBMP (whence the uniqueness of the weak solution) is determined by the uniqueness of the viscosity solution of the corresponding quasilinear PDE.