Symplectic method for Hamiltonian stochastic differential equations with multiplicative L\'{e}vy noise in the sense of Marcus

TL;DR

This paper develops a symplectic Euler scheme for Hamiltonian stochastic differential equations with multiplicative Lévy noise, proving its convergence and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a novel symplectic Euler scheme specifically designed for Hamiltonian SDEs with Lévy noise in the Marcus sense, including convergence proof and implementation methods.

Findings

The scheme converges with proven order.

Numerical experiments show the scheme's effectiveness.

The method outperforms existing approaches in long-term simulations.

Abstract

A class of Hamiltonian stochastic differential equations with multiplicative L\'{e}vy noise in the sense of Marcus, and the construction and numerical implementation methods of symplectic Euler scheme, are considered. A general symplectic Euler scheme for this kind of Hamiltonian stochastic differential equations is devised, and its convergence theorem is proved. The second part presents realizable numerical implementation methods for this scheme in details. Some numerical experiments are conducted to demonstrate the effectiveness and superiority of the proposed method by the simulations of its orbits, Hamlitonian,and convergence order over a long time interval.