The talk is focused on the analysis of numerical methods solving stochastic Hamiltonian problems of Ito type, for which a linear drift of the expected energy is visible along the exact dynamics. We study the behaviour of stochastic Runge-Kutta methods through epsilon-expansions of the solutions, where epsilon is the amplitude of the stochastic fluctuation. A drift-preserving scheme is also provided and analyzed, whose effectiveness is also checked through a selection of test problems.
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