We present high-order symplectic schemes for stochastic Hamiltonian systems preserving Hamiltonian functions. The approach is based on the generating function method, and we show that for the stochastic Hamiltonian systems, the coefficients of the generating function are invariant under permutations. As a consequence, the high-order symplectic schemes have a simpler form than the explicit Taylor expansion schemes with the same order. Moreover, we demonstrate numerically that the symplectic schemes are effective for long time simulations.
['Cristina E. Rizo Antón', 'Yau Shu Wong', 'Jian Deng']
Symplectic Numerical Schemes for Stochastic Systems Preserving Hamiltonian Functions