Strong approximations of BSDEs in a domain

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 (Stochastic Process. Appl. 111 (2004) 175–206 and Ann. Appl. Probab. 14 (2004) 459–488). 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 $h^{1/4−ɛ}$, where $h$ denotes the time step and $ɛ$ is any positive parameter. This rate corresponds to the strong exit time approximation. It is improved to $h^{1/2−ɛ}$ 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.