Second Order Runge–Kutta Methods for Itô Stochastic Differential Equations

A new class of stochastic Runge–Kutta methods for the weak approximation of the solution of Itô stochastic differential equation systems with a multidimensional Wiener process is introduced. As the main innovation, the number of stages of the methods does not depend on the dimension of the driving Wiener process, and the number of necessary random variables which have to be simulated is reduced considerably. Compared to well-known schemes, this reduces the computational effort significantly. Order conditions for the stochastic Runge–Kutta methods assuring weak convergence with order two are calculated by applying the colored rooted tree analysis due to the author. Further, some coefficients for explicit second order stochastic Runge–Kutta schemes are presented.