Forward Backward SDEs in Weak Formulation

TL;DR

This paper explores a weak formulation of forward-backward stochastic differential equations (FBSDEs), establishing a Feynman-Kac formula and offering a new framework that improves efficiency in certain cases.

Contribution

It introduces a weak formulation for FBSDEs driven by the forward component, extending theoretical understanding and providing a novel approach compared to traditional strong formulations.

Findings

Established Feynman-Kac formula for weak FBSDEs in classical and viscosity sense.

Demonstrated improved efficiency when the diffusion involves the Z-component.

Provided a new theoretical framework for analyzing FBSDEs.

Abstract

Although having been developed for more than two decades, the theory of forward backward stochastic differential equations is still far from complete. In this paper, we take one step back and investigate the formulation of FBSDEs. Motivated from several considerations, both in theory and in applications, we propose to study FBSDEs in weak formulation, rather than the strong formulation in the standard literature. That is, the backward SDE is driven by the forward component, instead of by the Brownian motion noise. We establish the Feyman-Kac formula for FBSDEs in weak formulation, both in classical and in viscosity sense. Our new framework is efficient especially when the diffusion part of the forward equation involves the $Z$-component of the backward equation.