## Weak solutions for forward-backward SDEs-a martingale problem approach.(English)Zbl 1154.60045

Summary: 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.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 35K55 Nonlinear parabolic equations 60H30 Applications of stochastic analysis (to PDEs, etc.)
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### References:

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