  Variational integrators are derived for structure-preserving simulation of
stochastic Hamiltonian systems with a certain type of multiplicative noise
arising in geometric mechanics. The derivation is based on a stochastic
discrete Hamiltonian which approximates a type-II stochastic generating
function for the stochastic flow of the Hamiltonian system. The generating
function is obtained by introducing an appropriate stochastic action functional
and its corresponding variational principle. Our approach permits to recast in
a unified framework a number of integrators previously studied in the
literature, and presents a general methodology to derive new
structure-preserving numerical schemes. The resulting integrators are
symplectic; they preserve integrals of motion related to Lie group symmetries;
and they include stochastic symplectic Runge-Kutta methods as a special case.
Several new low-stage stochastic symplectic methods of mean-square order 1.0
derived using this approach are presented and tested numerically to demonstrate
their superior long-time numerical stability and energy behavior compared to
nonsymplectic methods.
