Strong Approximations of BSDEs in a domain

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 $h^{\frac{1}{4}-\text{eps}}$ where $h$ denotes the time step and $\text{eps}$ is any positive parameter. This rate corresponds to the strong exit time approximation. It is improved to $h^{\frac{1}{2}-\text{eps}}$ 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.