Numerical scheme for backward doubly stochastic differential equations

TL;DR

This paper introduces a numerical scheme for approximating solutions to decoupled forward-backward doubly stochastic differential equations, proving convergence with a rate of ||^{1/2} under Lipschitz conditions.

Contribution

It develops a discrete-time approximation method for FBDSDEs and proves its convergence and rate, extending previous regularity results to this context.

Findings

Convergence of the scheme as step size approaches zero

Rate of convergence is ||^{1/2}

Extension of regularity results to backward doubly stochastic equations

Abstract

We study a discrete-time approximation for solutions of systems of decoupled forward-backward doubly stochastic differential equations (FBDSDEs). Assuming that the coefficients are Lipschitz-continuous, we prove the convergence of the scheme when the step of time discretization, $|\pi|$ goes to zero. The rate of convergence is exactly equal to $|\pi|^{1/2}$. The proof is based on a generalization of a remarkable result on the $^{2}$-regularity of the solution of the backward equation derived by J. Zhang