Y. Komori (2007), Weak order stochastic Runge-Kutta methods for commutative stochastic differential equations, Journal of Computational and Applied Mathematics, 203 (1), 57-79.

Abstract
A new explicit stochastic Runge-Kutta scheme of weak order 2 is proposed under a commutativity condition, which is derivative-free and which attains order 4 for ordinary differential equations. The weak order conditions are derived by utilizing multi-colored rooted tree analysis and a solution is found in a transparent way. The scheme is compared with other derivative-free and weak second order schemes in numerical experiments.

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