A Numerical scheme for backward doubly stochastic differential equations

TL;DR

This paper introduces a new numerical scheme for backward doubly stochastic differential equations with path-dependent terminal conditions, proving its strong convergence and deriving a novel regularity property of solutions.

Contribution

It presents a novel numerical scheme for BDSDEs, establishes its strong convergence, and introduces an $L^2$-type regularity concept for solutions.

Findings

Scheme converges in the strong $L^2$-sense

Derived an $L^2$-type regularity of solutions

Established the rate of convergence of the scheme

Abstract

In this paper we propose a numerical scheme for the class of backward doubly stochastic (BDSDEs) with possible path-dependent terminal values. We prove that our scheme converge in the strong $L^2$-sense and derive its rate of convergence. As an intermediate step we derive an $L^2$-type regularity of the solution to such BDSDEs. Such a notion of regularity which can be though of as the modulus of continuity of the paths in an $L^2$-sense, is new.