How do Monte Carlo estimates affect stochastic geometric numerical integration?

Abstract

In this work, we investigate the numerical conservation of characteristic properties of stochastic Hamiltonian problems driven by additive noise under time discretizations, in the spirit of stochastic geometric numerical integration. In particular, we aim to understand how Monte Carlo and multilevel Monte Carlo estimators for the expected values eventually affect the conservation properties of the numerical scheme. Specifically, our analysis is focused on the role of random number generation in the conservation of invariant laws for stochastic Hamiltonian problems under time discretization by the drift-preserving numerical scheme introduced in C. Chen, D. Cohen, R. D'Ambrosio and A. Lang, [Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math. 46(2) (2020), pp. 27.] and the stochastic perturbation of a symplectic Runge–Kutta method, introduced in K. Burrage and P.M. Burrage, [Low rank Runge–Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise, J. Comput. Appl. Math. 236(16) (2012), pp. 3920–3930.]. The numerical evidence confirms the aforementioned theoretical analysis.

Keywords:

• Stochastic differential equations
• Monte Carlo methods
• stochastic Hamiltonian problems
• geometric numerical integration
• additive noise

AMS Subject Classifications:

• 65C30
• 65P10
• 60H10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by Istituto Nazionale di Alta Matematica ‘Francesco Severi’ GNCS-INDAM project and by Ministero dell'Istruzione, dell'Università e della Ricerca PRIN2017-MIUR project 2017JYCLSF ‘Structure preserving approximation of evolutionary problems’. The authors are member of the INDAM Research group GNCS. The authors are grateful to the anonymous referees for their precious suggestions.