On irregular functionals of SDEs and the Euler scheme

AbstractWe prove a sharp upper bound for the error $\mathbb {E}|g(X)-g(\hat{X})|^{p}$ in terms of moments of $X-\hat{X}$ , where X and $\hat{X}$ are random variables and the function g is a function of bounded variation. We apply the results to the approximation of a solution to 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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