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This paper applies several well-known tricks from the numerical treatment ofdeterministic differential equations to improve the efficiency of theMultilevel Monte Carlo (MLMC) method for stochastic differential equations(SDEs) and especially the Langevin equation. We use modified equations analysisto circumvent the need for a strong-approximation theory for the integrator,and we apply this to introduce MLMC for Langevin-type equations withintegrators based on operator splitting. We combine this with extrapolation andinvestigate the use of discrete random variables in place of the Gaussianincrements, which is a well-known technique for the weak approximation of SDEs.We show that, for small-noise problems, discrete random variables can lead toan increase in efficiency of almost two orders of magnitude for practicallevels of accuracy.

Improving MLMC for SDEs with application to the Langevin equation