Asymptotic error distributions of symplectic and non-symplectic methods for stochastic Hamiltonian system with additive noise

This paper studies the asymptotic error distributions of several symplectic and non-symplectic methods for stochastic Hamiltonian systems. Focusing on stochastic Hamiltonian systems driven by additive noise, we obtain the asymptotic limit of the normalized error distribution of the $ \theta $-method $ (\theta\in[0,1]) $ that is symplectic if and only if $ \theta = \frac12 $. The upper bound for the second moment of the asymptotic error distribution suggests that the midpoint method may minimize the error constant of the $ \theta $-method over a large time horizon $ T $. Furthermore, we take the linear stochastic oscillator as a test equation and investigate exact asymptotic error constants of several symplectic and non-symplectic methods. Our result implies that in the long-time computation, the probability that the error deviates from zero decays exponentially faster for the symplectic methods than for the non-symplectic ones.