Abstract. We prove a sharp upper bound for the approximation error |g(X) − g(X)| p in terms of moments of X −X, where X andX 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.
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