2014/09/08 by Eike H. Mueller, Robert Scheichl, Mueller, Eike H. +3
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1409.2342
openalex publication_date 2014/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis to circumvent the need for a 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.