Coupling of forward-backward stochastic differential equations on the Wiener space, and application on regularity

TL;DR

This paper extends the coupling method to forward-backward stochastic differential equations with random, Lipschitz coefficients, establishing their regularity in time and Malliavin differentiability.

Contribution

It introduces a novel application of the coupling method to forward-backward SDEs, proving regularity and Malliavin differentiability of solutions.

Findings

Established time regularity of solutions.

Proved Malliavin Sobolev ${ m D}_{1,2}$ differentiability.

Extended coupling method to a broader class of SDEs.

Abstract

S. Geiss and J. Ylinen proposed the coupling method \cite{Geiss:Ylinen:21} to investigate the regularity for the solution to the backward stochastic differential equations with random coefficients. In this paper, we explore this method in setting for the forward-backward stochastic differential equation with random and Lipschitz coefficients, We obtain the regularity in time, and the Malliavin Sobolev ${\mathbb D}_{1,2}$ differentiability for the solution.