Error Expansion for a Symplectic Scheme for Stochastic Hamiltonian Systems

Abstract

We consider a stochastic autonomous Hamiltonian system for which the flow preserves the symplectic structure. Numerical simulations show that for stochastic Hamiltonian systems symplectic schemes produce more accurate results for long term simulations than non-sysmplectic numerical schemes. We study the approximation error corresponding to a symplectic weak scheme of order one. A backward error analysis is done at the level of the Kolmogorov equation associated with the initial stochastic Hamiltonian system. We obtain an expansion of the error in terms of powers of the discretization step size and the solutions of the modified Kolmogorov equation.