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

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.

[1]  Zhiming Wang,et al.  Marcus canonical integral for non-Gaussian processes and its computation: pathwise simulation and tau-leaping algorithm. , 2013, The Journal of chemical physics.

[2]  Jinqiao Duan,et al.  Hamiltonian systems with Lévy noise: Symplecticity, Hamilton’s principle and averaging principle , 2018, Physica D: Nonlinear Phenomena.

[3]  G. N. Milstein,et al.  Symplectic Integration of Hamiltonian Systems with Additive Noise , 2001, SIAM J. Numer. Anal..

[4]  Qingyi Zhan.  Shadowing orbits of a class of random differential equations , 2019, Applied Numerical Mathematics.

[5]  E. Helfand.  Numerical integration of stochastic differential equations , 1979, The Bell System Technical Journal.

[6]  Jinqiao Duan.  An Introduction to Stochastic Dynamics , 2015 .

[7]  Zhang Qingyi.  Shadowing orbits of stochastic differential equations , 2016 .

[8]  Xiao Wang,et al.  Numerical methods for the mean exit time and escape probability of two-dimensional stochastic dynamical systems with non-Gaussian noises , 2015, Appl. Math. Comput..

[9]  Qingyi Zhan,et al.  Numerical study on (ω, Lδ)-Lipschitz shadowing of stochastic differential equations , 2020, Appl. Math. Comput..

[10]  M. Qin,et al.  Symplectic Geometric Algorithms for Hamiltonian Systems , 2010 .

[11]  Mean-square numerical approximations to random periodic solutions of stochastic differential equations , 2015 .

[12]  Lijin Wang,et al.  Predictor-corrector methods for a linear stochastic oscillator with additive noise , 2007, Math. Comput. Model..

[13]  L. Einkemmer.  Structure preserving numerical methods for the Vlasov equation , 2016, 1604.02616.