  A new class of third order Runge-Kutta methods for stochastic differential
equations with additive noise is introduced. In contrast to Platen's method,
which to the knowledge of the author has been up to now the only known third
order Runge-Kutta scheme for weak approximation, the new class of methods
affords less random variable evaluations and is also applicable to SDEs with
multidimensional noise. Order conditions up to order three are calculated and
coefficients of a four stage third order method are given. This method has
deterministic order four and minimized error constants, and needs in addition
less function evaluations than the method of Platen. Applied to some examples,
the new method is compared numerically with Platen's method and some well known
second order methods and yields very promising results.
