Abstract

A formal meansquare error expansion (MSE) is derived for Euler–Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise, a posteriori, adaptive time-stepping Euler–Maruyama algorithm for numerical solutions of SDE, and the resulting algorithm is incorporated into a multilevel Monte Carlo (MLMC) algorithm for weak approximations of SDE. This gives an efficient MSE adaptive MLMC algorithm for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC algorithm is shown to outperform the uniform time-stepping MLMC algorithm by orders of magnitude, producing output whose error with high probability is bounded by TOL>0 at the near-optimal MLMC cost rate O(TOL^−2log(TOL)^4) that is achieved when the cost of sample generation is O(1).