Stochastic Runge-Kutta methods for Itô sodes with small noise

Evelyn Buckwar, Andreas Rößler, Renate Winkler

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)


We consider stochastic Runge-Kutta methods for Itô stochastic ordinary differential equations, and study their mean-square convergence properties for problems with small multiplicative noise or additive noise. First we present schemes where the drift part is approximated by well-known methods for deterministic ordinary differential equations, and a Maruyama term is used to discretize the diffusion. Further, we suggest improving the discretization of the diffusion part by taking into account also mixed classical-stochastic integrals, and we present a suitable class of fully derivativefree methods. We show that the relation of the applied step-sizes to the smallness of the noise is essential to decide whether the new methods are worth the effort. Simulation results illustrate the theoretical findings. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Original languageEnglish
Pages (from-to)1789-1808
Number of pages20
JournalSIAM Journal on Scientific Computing
Issue number4
Publication statusPublished - 2010


  • Itô stochastic differential equations
  • Mean-square convergence
  • Small noise
  • Stochastic Runge-Kutta methods


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