Runge–Kutta methods for numerical solution of stochastic differential equations

The way to obtain deterministic Runge–Kutta methods from Taylor approximations is generalized for stochastic differential equations, now by means of stochastic truncated expansions about a point for sufficiently smooth functions of an Itô process. A class of explicit Runge–Kutta schemes of second order in the weak sense for systems of stochastic differential equations with multiplicative noise is developed. Also two Runge–Kutta schemes of third order have been obtained for scalar equations with constant diffusion coefficients. Numerical examples that compare the proposed schemes to standard ones are presented.