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We investigate the stochastic modified equation which plays an important rolein the stochastic backward error analysis for explaining the mathematicalmechanism of a numerical method. The contribution of this paper is threefold.First, we construct a new type of stochastic modified equation, which is aperturbation of the Wong--Zakai approximation of the rough differentialequation. For a symplectic method applied to a rough Hamiltonian system, theassociated stochastic modified equation is proved to have a Hamiltonianformulation. Second, the pathwise convergence order of the truncated modifiedequation to the numerical method is obtained by techniques in the rough paththeory. Third, if increments of noises are simulated by truncated randomvariables, we show that the one-step error can be made exponentially small withrespect to the time step size. Numerical experiments verify our theoreticalresults.

Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems based on the Wong--Zakai approximation