  The class of stochastic Runge-Kutta methods for stochastic differential
equations due to R\"o{\ss}ler is considered. Coefficient families of diagonally
drift-implicit stochastic Runge-Kutta (DDISRK) methods of weak order one and
two are calculated. Their asymptotic stability as well as mean-square stability
(MS-stability) properties are studied for a linear stochastic test equation
with multiplicative noise. The stability functions for the DDISRK methods are
determined and their domains of stability are compared to the corresponding
domain of stability of the considered test equation. Stability regions are
presented for various coefficients of the families of DDISRK methods in order
to determine step size restrictions such that the numerical approximation
reproduces the characteristics of the solution process.
