A formal mean square 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 б(TOL−2 log(TOL)4) that is achieved when the cost of sample generation is б(1).
CITATION STYLE
Hoel, H., Häppölä, J., & Tempone, R. (2016). Construction of a mean square error adaptive euler-maruyama method with applications in multilevel monte carlo. In Springer Proceedings in Mathematics and Statistics (Vol. 163, pp. 29–86). Springer New York LLC. https://doi.org/10.1007/978-3-319-33507-0_2
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