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 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.Comment: 35 page