In this talk, we focus our investigation on providing long-term estimates of the Hamiltonian deviation computed along numerical approximations to the solutions of stochastic Hamiltonian systems, both of It\(\hat{\rm o}\) and Statonovich types. It is well-known that the expected Hamiltonian of an It\(\hat{\rm o}\) Hamiltonian system with additive noise exhibits a linear drift in time [2], while the Hamiltonian function is conserved along the exact flow of a Stratonovich Hamiltonian system [3, 4]. Here, we address our attention to providing modified differential equations associated to suitable discretizations for above problems, by means of weak backward error analysis arguments [1, 5, 6]. Then, long-term estimates are provided both for It\(\hat{\rm o}\) and Stratonovich Hamiltonian systems, revealing the presence of parasitic terms affecting the overall conservation accuracy. Finally, selected numerical experiments are provided to confirm the theoretical analysis. This talk is based on a joint work with Raffaele D’Ambrosio (University of L’Aquila).

References

1. C. Anton, Weak backward error analysis for stochastic Hamiltonian Systems, BIT Numer. Math., 59, 613–646 (2019).

2. C. Chen, D. Cohen, R. D’Ambrosio, A. Lang, Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math. 46(2), 27 (2020).

3. T. Misawa, Energy Conservative Stochastic Difference Scheme for Stochastic Hamiltonian Dynamical Systems, Japan J. Indust. App!. Math., 17, 119-128 (2000).

4. T. Misawa,Symplectic Integrators to Stochastic Hamiltonian Dynamical Systems Derived from Composition Methods, Mathematical Problems in Engineering, 2010, 384937 (2010).

5. T. Shardlow, Modified equations for stochastic differential equations, BIT Numer. Math., 46, 111–125 (2006).

6. K.C. Zygalakis, On the existence and the applications of modified equations for stochastic differential equations, SIAM J. Sci. Comput. 33, 102–130 (2011).