Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

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This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic diffe- rential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.
Original languageEnglish
Number of pages20
JournalProceedings of the Royal Society A
Issue number2176
Early online date18 Mar 2015
Publication statusPublished - Apr 2015



  • numericalsolutionofstochasticdifferential equations,modifiedequations,geometrical integrators,weakapproximation, extrapolation

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