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

TL;DR

This paper develops a unified framework for analyzing the well-posedness of non-Markovian forward-backward stochastic differential equations, introducing a regular decoupling field linked to a characteristic BSDE with superlinear growth.

Contribution

It presents a comprehensive scheme that unifies existing methods and addresses longstanding open problems in the well-posedness of non-Markovian FBSDEs, including new conditions for solvability.

Findings

Established conditions for the existence of a regular decoupling field.

Linked the regularity of the decoupling field to solutions of a characteristic BSDE.

Provided a 'User's Guide' for solvability of a broad class of FBSDEs.

Abstract

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.