Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo

Hakon Hoel, Juho Häppölä, Raul Tempone

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Scopus citations

Abstract

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 log(TOL)) that is achieved when the cost of sample generation is б(1).
Original languageEnglish (US)
Title of host publicationSpringer Proceedings in Mathematics & Statistics
PublisherSpringer Nature
Pages29-86
Number of pages58
ISBN (Print)9783319335056
DOIs
StatePublished - Jun 14 2016

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