  The paper deals with numerical discretizations of separable nonlinear
Hamiltonian systems with additive noise. For such problems, the expected value
of the total energy, along the exact solution, drifts linearly with time. We
present and analyze a time integrator having the same property for all times.
Furthermore, strong and weak convergence of the numerical scheme along with
efficient multilevel Monte Carlo estimators are studied. Finally, extensive
numerical experiments illustrate the performance of the proposed numerical
scheme.
