WEAK SOLUTIONS FOR FORWARD-BACKWARD SDES-A MARTINGALE PROBLEM APPROACH

• 成果类型: Article
• 署名作者: Ma, Jin; Zhang, Jianfeng; Zheng, Ziyu
• 署名单位: University of Southern California
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP0383
• 发表日期: 2008
• 页码: 2092-2125
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; uniqueness; EXISTENCE

摘要

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.