Symplectic Euler scheme for Hamiltonian stochastic differential equations driven by Levy noise

TL;DR

This paper introduces a symplectic Euler numerical scheme for Hamiltonian stochastic differential equations driven by Levy noise, demonstrating its convergence, effective implementation, and advantages through numerical experiments.

Contribution

It presents a novel symplectic Euler scheme tailored for Levy noise-driven Hamiltonian SDEs, including convergence analysis and practical implementation details.

Findings

The scheme converges for the class of Hamiltonian SDEs considered.

Numerical experiments show the scheme preserves symplectic structure.

The method outperforms existing approaches in stability and accuracy.

Abstract

This paper proposes a general symplectic Euler scheme for a class of Hamiltonian stochastic differential equations driven by L$\acute{e}$vy noise in the sense of Marcus form. The convergence of the symplectic Euler scheme for this Hamiltonian stochastic differential equations is investigated. Realizable numerical implementation of this scheme is also provided in details. Numerical experiments are presented to illustrate the effectiveness and superiority of the proposed method by the simulations of its orbits, symplectic structure and Hamlitonian.