Multilevel Monte Carlo for stochastic differential equations with small noise

David F. Anderson, Desmond J. Higham, Yu Sun

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the problem of numerically estim ating expectations of solutions to stochastic differential equations driven by Brownian motions in the commonly occurring small noise regime. We consider (i) standard Monte Carlo methods combined with numerical discretization algorithms tailored to the small noise setting, and (ii) a multilevel Monte Carlo method combined with a standard Euler-Maruyama implementation. Under the assumptions we make on the underlying model, the multilevel method combined with Euler-Maruyama is often found to be the most efficient option. Moreover, under a wide range of scalings the multilevel method is found to give the same asymptotic complexity that would arise in the idealized case where we have access to exact samples of the required distribution at a cost of O(1) per sample. A key step in our analysis is to analyze the variance between two coupled paths directly, as opposed to their L2 distance. Careful simulations are provided to illustrate the asymptotic results.
Original languageEnglish
Pages (from-to)505-529
Number of pages25
JournalSiam journal on numerical analysis
Volume54
Issue number2
DOIs
Publication statusPublished - 3 Mar 2016

Keywords

  • Monte Carlo
  • multilevel Monte Carlo
  • small noise
  • stochastic differential equations

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