Symplectic methods based on Pad$\acute{e}$ approximation for some stochastic Hamiltonian systems

In this article, we introduce a kind of numerical schemes, based on Pad$\acute{e}$ approximation, for two stochastic Hamiltonian systems which are treated separately. For the linear stochastic Hamiltonian systems, it is shown that the applied Pad$\acute{e}$ approximations $P_{(k,k)}$ give numerical solutions that inherit the symplecticity and the proposed numerical schemes based on $P_{(r,s)}$ are of mean-square order $\frac{r+s}{2}$ under appropriate conditions. In case of the special stochastic Hamiltonian systems with additive noises, the numerical method using two kinds of Pad$\acute{e}$ approximation $P_{(\widehat{r},\widehat{s})}$ and $P_{(\check{r},\check{s})}$ has mean-square order $\check{r}+\check{s}+1$ when $\widehat{r}+\widehat{s}=\check{r}+\check{s}+2$. Moreover, the numerical solution is symplectic if $\widehat{r}=\widehat{s}$.