Stochastic Runge-Kutta Rosenbrock type methods for SDE systems

In this paper we introduce a family of stochastic Runge-Kutta Rosenbrock (SRKR) type methods for multi-dimensional Itô stochastic differential equations (SDEs). The presented class of semi-implicit methods need less computational effort in comparison with some implicit ones. General order conditions for the coefficients and the random variables of the SRKR methods are obtained. Then a set of order conditions for a subclass of stochastic weak second order is given. Numerical examples are presented to demonstrate the efficiency and accuracy of the new schemes.