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

Abstract

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 θ-method (θ ∈ [0,1]) that is symplectic if and only if (Formula presented). The upper bound for the second moment of the asymptotic error distribution suggests that the midpoint method may minimize the error constant of the θ-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.