Stochastic symplectic methods based on the Padé approximations for linear stochastic Hamiltonian systems

In this article, we propose a kind of numerical methods based on the Padé approximations, for two kinds of stochastic Hamiltonian systems. For the general linear stochastic Hamiltonian systems, it is shown that the applied Padé approximations P(k,k)P(k,k) produce numerical solutions that are symplectic, and the proposed numerical schemes based on P(r,s)P(r,s) are of root-mean-square convergence order r+s2. For a special kind of linear stochastic Hamiltonian systems with additive noises, the numerical methods using two kinds of Padé approximations, P(rˆ,sˆ) and P(ř,1), possess root-mean-square convergence order ř+2 when rˆ+sˆ=ř+3, and are symplectic if rˆ=sˆ. These generalize the Padé approximation approaches for symplectic integration of linear Hamiltonian systems to the stochastic context.