The Annals of Applied Probability

On well-posedness of forward–backward SDEs—A unified approach

Jin Ma, Zhen Wu, Detao Zhang, and Jianfeng Zhang

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In this paper, we study the well-posedness of the Forward–Backward Stochastic Differential Equations (FBSDE) in a general non-Markovian framework. The main purpose is to find a unified scheme which combines all existing methodology in the literature, and to address some fundamental longstanding problems for non-Markovian FBSDEs. An important device is a decoupling random field that is regular (uniformly Lipschitz in its spatial variable). We show that the regulariy of such decoupling field is closely related to the bounded solution to an associated characteristic BSDE, a backward stochastic Riccati-type equation with superlinear growth in both components $Y$ and $Z$. We establish various sufficient conditions for the well-posedness of an ODE that dominates the characteristic BSDE, which leads to the existence of the desired regular decoupling random field, whence the solvability of the original FBSDE. A synthetic analysis of the solvability is given, as a “User’s Guide,” for a large class of FBSDEs that are not covered by the existing methods. Some of them have important implications in applications.

Article information

Ann. Appl. Probab., Volume 25, Number 4 (2015), 2168-2214.

Received: August 2013
Revised: April 2014
First available in Project Euclid: 21 May 2015

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Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H30: Applications of stochastic analysis (to PDE, etc.) 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

Forward–backward SDEs decoupling random fields characteristic BSDEs backward stochastic Riccati equations comparison theorem


Ma, Jin; Wu, Zhen; Zhang, Detao; Zhang, Jianfeng. On well-posedness of forward–backward SDEs—A unified approach. Ann. Appl. Probab. 25 (2015), no. 4, 2168--2214. doi:10.1214/14-AAP1046.

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