Symplectic numerical integration for Hamiltonian stochastic differential equations with multiplicative Lévy noise in the sense of Marcus

Abstract

In this paper, we propose a symplectic numerical integration method for a class of Hamiltonian stochastic differential equations with multiplicative Lévy noise in the sense of Marcus. We first construct a general symplectic Euler scheme for these equations, then we prove its convergence. In addition, we provide realizable numerical implementations for the proposed symplectic Euler scheme in detail. Some numerical experiments are conducted to demonstrate the effectiveness and superiority of the proposed method by the simulations of its orbits, Hamiltonian and convergence order over a long time interval. The results show the applicability of the methods considered.