  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.
