Exact convergence rate of the Euler-Maruyama scheme, with application to sampling design

Abstract

We consider regular partitions of the interval [0,T] into n subintervals based on a sampling density h. If xt is the solution of the original stochastic differential equation and is the approximate solution using the Euler-Maruyama scheme corresponding to a time discretization partition π, we find the limit as a functional of the sampling density h and the parameters of the stochastic differential equation. We then find the best sampling density and thus the best way to discretize regularly the interval for integrated mean square approximation error. The linear case is also considered in detail. We also apply the same idea to obtain optimal asymptotically efficient schemes.