Modified equations for weak stochastic symplectic schemes via their generating functions

In this paper, a systematic approach of constructing modified equations for weak stochastic symplectic methods of stochastic Hamiltonian systems is given via using the generating functions of the stochastic symplectic methods. This approach is valid for stochastic Hamiltonian systems with either additive noises or half-multiplicative noises, and we prove that the modified equation of the weak stochastic symplectic methods are perturbed stochastic Hamiltonian systems of the original systems, which reveals in certain sense the reason for the good long time numerical behavior of stochastic symplectic methods.