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

Modified equations for weak stochastic symplectic schemes via their generating functions