  In the present paper, a class of stochastic Runge-Kutta methods containing
the second order stochastic Runge-Kutta scheme due to E. Platen for the weak
approximation of It\^o stochastic differential equation systems with a
multi-dimensional Wiener process is considered. Order one and order two
conditions for the coefficients of explicit stochastic Runge-Kutta methods are
solved and the solution space of the possible coefficients is analyzed. A full
classification of the coefficients for such stochastic Runge-Kutta schemes of
order one and two with minimal stage numbers is calculated. Further, within the
considered class of stochastic Runge-Kutta schemes coefficients for optimal
schemes in the sense that additionally some higher order conditions are
fulfilled are presented.
