Open Access
November 2008 Weak solutions for forward–backward SDEs—a martingale problem approach
Jin Ma, Jianfeng Zhang, Ziyu Zheng
Ann. Probab. 36(6): 2092-2125 (November 2008). DOI: 10.1214/08-AOP0383


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.


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Jin Ma. Jianfeng Zhang. Ziyu Zheng. "Weak solutions for forward–backward SDEs—a martingale problem approach." Ann. Probab. 36 (6) 2092 - 2125, November 2008.


Published: November 2008
First available in Project Euclid: 19 December 2008

zbMATH: 1154.60045
MathSciNet: MR2478677
Digital Object Identifier: 10.1214/08-AOP0383

Primary: 60H10
Secondary: 35K55 , 60H30

Keywords: forward–backward stochastic differential equations , Martingale problems , uniqueness , viscosity solutions , weak solutions

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 6 • November 2008
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