A stepsize control algorithm for SDEs with small noise based on stochastic Runge-Kutta Maruyama methods

A variable stepsize control algorithm for solution of stochastic differential equations (SDEs) with a small noise parameter epsilon is presented. In order to determine the optimal stepsize for each stage of the algorithm, an estimate of the global error is introduced based on the local error of the Stochastic Runge-Kutta Maruyama (SRKM) methods. Based on the relation of the stepsize and the small noise parameter, the local mean-square stochastic convergence order can be different from stage to stage. Using this relation, a strategy for producing and controlling the stepsize in the numerical integration of SDEs is proposed. Numerical experiments on several standard SDEs with small noise are presented to illustrate the effectiveness of this approach.