  We propose a class of numerical integration methods for stochastic Poisson
systems (SPSs) of arbitrary dimensions. Based on the Darboux-Lie theorem, we
transform the SPSs to their canonical form, the generalized stochastic
Hamiltonian systems (SHSs), via canonical coordinate transformations found by
solving certain PDEs defined by the Poisson brackets of the SPSs. An
a-generating function approach with \alpha\in [0,1] is then used to create
symplectic discretizations of the SHSs, which are then transformed back by the
inverse coordinate transformation to numerical integrators for the SPSs. These
integrators are proved to preserve both the Poisson structure and the Casimir
functions of the SPSs. Applications to a three-dimensional stochastic rigid
body system and a three-dimensional stochastic Lotka-Volterra system show
efficiency of the proposed methods.
