Perturbative analysis of stochastic Hamiltonian problems under time discretizations

In this work we focus on the study of stochastic Hamiltonian problem driven by additive Wiener noise. In particular, we aim to analyse the behaviour of discretizations to these problems, motivated by some results on stochastic Runge–Kutta methods (SRK) developed by Burrage and Burrage (2012). In fact, SRK methods present a remarkable error that increases with the parameter of the diffusive part of the problem. Through a perturbative analysis, we investigate the reason of this behaviour, leading to a negative answer: retaining the main features of stochastic Hamiltonian problems does not happen straightforwardly for any time discretization. This analysis is also numerically confirmed.