Strong Approximations of BSDEs in a domain

TL;DR

This paper analyzes the strong approximation of backward SDEs with finite stopping times, specifically the first exit time from a domain, achieving error bounds of order h^{1/4-ε} and h^{1/2-ε} under certain conditions.

Contribution

It provides new error bounds for the Euler scheme approximation of BSDEs with exit times, without requiring uniform ellipticity.

Findings

Error bound of order h^{1/4-ε} for exit time approximation

Error bound improves to h^{1/2-ε} when exit time is exactly simulated

Results hold without uniform ellipticity condition

Abstract

We study the strong approximation of a Backward SDE with finite stopping time horizon, namely the first exit time of a forward SDE from a cylindrical domain. We use the Euler scheme approach of Bouchard and Touzi, Zhang 04}. When the domain is piecewise smooth and under a non-characteristic boundary condition, we show that the associated strong error is at most of order $h^{\frac14-\eps}$ where $h$ denotes the time step and $\eps$ is any positive parameter. This rate corresponds to the strong exit time approximation. It is improved to $h^{\frac12-\eps}$ when the exit time can be exactly simulated or for a weaker form of the approximation error. Importantly, these results are obtained without uniform ellipticity condition.