An Euler scheme for BSDEs via the Wiener chaos decomposition

TL;DR

This paper introduces a novel Euler scheme for backward stochastic differential equations (BSDEs) that uses Wiener chaos decomposition, allowing for flexible terminal conditions and providing convergence analysis with numerical demonstrations.

Contribution

The paper develops a new Euler scheme for BSDEs based on Wiener chaos decomposition, accommodating arbitrary terminal conditions unlike previous methods.

Findings

The scheme converges under broad conditions.

Numerical examples demonstrate effectiveness.

Method handles non-Markovian terminal conditions.

Abstract

The Euler scheme is a standard time discretization for BSDEs, but its implementation hinges on approximating conditional expectations and the associated martingale terms at each time step. We propose an implementation based on the Wiener chaos decomposition to approximate these quantities. In contrast to many numerical schemes that rely on a forward-backward (Markovian) structure, our approach accommodates arbitrary $\mathcal{F}_T$-measurable square-integrable terminal conditions. We provide a comprehensive convergence analysis and illustrate the method on several numerical examples.