  Casimir preserving integrators for stochastic Lie-Poisson equations with
Stratonovich noise are developed extending Runge-Kutta Munthe-Kaas methods. The
underlying Lie-Poisson structure is preserved along stochastic trajectories. A
related stochastic differential equation on the Lie algebra is derived. The
solution of this differential equation updates the evolution of the Lie-Poisson
dynamics by means of the exponential map. The constructed numerical method
conserves Casimir-invariants exactly, which is important for long time
integration. This is illustrated numerically for the case of the stochastic
heavy top and the stochastic sine-Euler equations.
