Symplectic Schemes for Stochastic Hamiltonian Systems Preserving Hamiltonian Functions

Abstract

We present high-order symplectic schemes for stochastic Hamiltonian systems preserving Hamiltonian functions. The approach is based on the generating function method, and we prove that the coefficients of the generating function are invariant under permutations for this class of systems. As a consequence, the proposed high-order symplectic weak and strong schemes are computationally efficient because they require less stochastic multiple integrals than the Taylor expansion schemes with the same order.