Annals of Probability

Weak solutions for forward–backward SDEs—a martingale problem approach

Jin Ma, Jianfeng Zhang, and Ziyu Zheng

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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.

Article information

Ann. Probab., Volume 36, Number 6 (2008), 2092-2125.

First available in Project Euclid: 19 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 35K55: Nonlinear parabolic equations 60H30: Applications of stochastic analysis (to PDE, etc.)

Forward–backward stochastic differential equations weak solutions martingale problems viscosity solutions uniqueness


Ma, Jin; Zhang, Jianfeng; Zheng, Ziyu. Weak solutions for forward–backward SDEs—a martingale problem approach. Ann. Probab. 36 (2008), no. 6, 2092--2125. doi:10.1214/08-AOP0383.

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