Stochastic Runge-Kutta Methods for Itô SODEs with Small Noise

Authors:
Evelyn Buckwar;Andreas Rößler;Renate Winkler
Affiliations:
E.Buckwar@hw.ac.uk;roessler@mathematik.tu-darmstadt.de;winkler@math.uni-wuppertal.de
Venue:
SIAM Journal on Scientific Computing
Year:
2010

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Abstract

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 derivative-free 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.