Modified equations for weakly convergent stochastic symplectic schemes via their generating functions

Abstract

In this paper, an approach of constructing modified equations of weak $$k+k'$$ order ( $$k'\ge 1$$ ) apart from the k-th order weakly convergent stochastic symplectic methods, i.e., stochastic symplectic methods with respect to weak convergence and of weak order k, is given using the underlying generating functions of them. This approach is valid for stochastic Hamiltonian systems with additive noises, and those with multiplicative noises but for which the Hamiltonian functions $$H_r(p,q),\,\,r\ge 1$$ associated to the diffusion parts depend only on p or only on q. In such cases, we find that the modified equations of the weakly convergent stochastic symplectic methods are perturbed stochastic Hamiltonian systems of the original systems.