Approximation for non-smooth functionals of stochastic differential equations with irregular drift

We find upper bounds for the rate of convergence when the Euler-Maruyama approximation is used in order to compute the expectation of non-smooth functionals of some stochastic differential equations whose diffusion coefficient is constant, whereas the drift coefficient may be very irregular. As a by product of our method, we establish the weak order of the Euler-Maruyama approximation for a diffusion processes killed when it leaves an open set. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is H\"older continuous.