  We investigate the stochastic modified equation which plays an important role
in the stochastic backward error analysis for explaining the mathematical
mechanism of a numerical method. The contribution of this paper is threefold.
First, we construct a new type of stochastic modified equation, which is a
perturbation of the Wong--Zakai approximation of the rough differential
equation. For a symplectic method applied to a rough Hamiltonian system, the
associated stochastic modified equation is proved to have a Hamiltonian
formulation. Second, the pathwise convergence order of the truncated modified
equation to the numerical method is obtained by techniques in the rough path
theory. Third, if increments of noises are simulated by truncated random
variables, we show that the one-step error can be made exponentially small with
respect to the time step size. Numerical experiments verify our theoretical
results.
