The Annals of Probability

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

Jin Ma, Jianfeng Zhang, and Ziyu Zheng

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 19 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1229696597

Digital Object Identifier
doi:10.1214/08-AOP0383

Mathematical Reviews number (MathSciNet)
MR2478677

Zentralblatt MATH identifier
1154.60045

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

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

Citation

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. https://projecteuclid.org/euclid.aop/1229696597


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