On path-dependent multidimensional forward-backward SDEs

TL;DR

This paper develops a theoretical framework for analyzing path-dependent multidimensional forward-backward stochastic differential equations, establishing existence, uniqueness, and stability results crucial for stochastic control applications.

Contribution

It extends existing results to path-dependent cases, introduces a decoupling random field on path space, and constructs a global solution via characteristic BSDEs.

Findings

Existence and uniqueness of decoupling fields on small intervals

Global decoupling field construction by patching local solutions

Stability results for path-dependent FBSDEs

Abstract

This paper extends the results of Ma, Wu, Zhang, Zhang [11] to the context of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the forward-backward SDE at time t can depend on the whole path of the forward process up to time t. Such a situation appears when solving path-dependent stochastic control problems by means of variational calculus. At the heart of our analysis is the construction of a decoupling random field on the path space. We first prove the existence and the uniqueness of decoupling field on small time interval. Then by introducing the characteristic BSDE, we show that a global decoupling field can be constructed by patching local solutions together as long as the solution of the characteristic BSDE remains bounded. Finally, we provide a stability result for path-dependent forward-backward SDEs.