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

By Jin Ma, Zhen Wu, Detao Zhang, Jianfeng Zhang. 2011

170 citations recorded by the source corpus.

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.

Subfields: Demography, Finance

View Paper: https://www.semanticscholar.org/paper/a379c12658697d44b18a90fe9ed3dc2f0e34e92d

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Citations in the corpus

Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data (2019, background): “As illustrated in [28, 57, 58], to extend from a local (in time) solution of an FBSDE to a global solution, the key is the uniform Lipschitz continuity of the decoupling field, which is exactly implied by (4.”
Selection of equilibria in a linear quadratic mean-field game (2018, methodology): “4) is uniquely solvable, we can use Ma-Protter-Yong’s four-step-scheme [41] to represent the solution, see also [42].”
A Probabilistic Approach to Mean Field Games with Major and Minor Players (2014, background): “Its non-Markovian counterpart corresponding to non-Markovian FBSDEs was introduced in [21].”