  A general class of stochastic Runge-Kutta methods for the weak approximation
of It\^o and Stratonovich stochastic differential equations with a
multi-dimensional Wiener process is introduced. Colored rooted trees are used
to derive an expansion of the solution process and of the approximation process
calculated with the stochastic Runge-Kutta method. A theorem on general order
conditions for the coefficients and the random variables of the stochastic
Runge-Kutta method is proved by rooted tree analysis. This theorem can be
applied for the derivation of stochastic Runge-Kutta methods converging with an
arbitrarily high order.
