Zhang $L^2$-Regularity for the solutions of Backward Doubly Stochastic Differential Equations under globally Lipschitz continuous assumptions

TL;DR

This paper extends $L^2$-regularity results from FBSDEs to FBDSDEs under globally Lipschitz conditions, using Malliavin calculus, and applies this to analyze the convergence rate of a numerical scheme.

Contribution

It generalizes regularity results to the doubly stochastic setting and provides convergence rate analysis for Euler discretization of FBDSDEs.

Findings

Established $L^2$-regularity for FBDSDE solutions.

Derived convergence rate for Euler scheme under Lipschitz assumptions.

Extended Zhang's regularity results to a broader stochastic framework.

Abstract

We prove an $L^2$-regularity result for the solutions of Forward Backward Doubly Stochastic Differentiel Equations (FBDSDEs in short) under globally Lipschitz continuous assumptions on the coefficients. Therefore, we extend the well known regularity results established by Zhang (2004) for Forward Backward Stochastic Differential Equations (FBSDEs in short) to the doubly stochastic framework. To this end, we prove (by Malliavin calculus) a representation result for the martingale component of the solution of the F-BDSDE under the assumption that the coefficients are continuous in time and continuously differentiable in space with bounded partial derivatives. As an (important) application of our $L^2$-regularity result, we derive the rate of convergence in time for the (Euler time discretization based) numerical scheme for FBDSDEs proposed by Bachouch et al. (2016) under only globally Lipschitz continuous assumptions.