  We analyze the Landau-Lifshitz-Gilbert equation when the precession motion of
the magnetic moments is additionally subjected to an uniaxial anisotropy and is
driven by a multiplicative coupled stochastic field with a finite correlation
time $\tau$. The mean value for the spin wave components offers that the
spin-wave dispersion relation and its damping is strongly influenced by the
deterministic Gilbert damping parameter $\alpha$, the strength of the
stochastic forces $D$ and its temporal range $\tau$. The spin-spin-correlation
function can be calculated in the low correlation time limit by deriving an
evolution equation for the joint probability function. The stability analysis
enables us to find the phase diagram within the $\alpha-D$ plane for different
values of $\tau$ where damped spin wave solutions are stable. Even for zero
deterministic Gilbert damping the magnons offer a finite lifetime. We detect a
parameter range where the deterministic and the stochastic damping mechanism
are able to compensate each other leading to undamped spin-waves. The onset is
characterized by a critical value of the correlation time. An enhancement of
$\tau$ leads to an increase of the oscillations of the correlation function.
