On irregular functionals of SDEs and the Euler scheme

We prove a sharp upper bound for the approximation error E|g(X) − g(X̂)|p in terms of moments of X − X̂, where X and X̂ are random variables and the function g is a function of bounded variation. We apply the results to the approximation of a solution of a stochastic differential equation at time T by the Euler scheme, and show that the approximation of the payoff of the binary option has asymptotically sharp strong convergence rate 1/2. This has consequences for multilevel Monte Carlo methods.