Discrete analogue of the Krylov–Veretennikov expansion

We consider a difference analogue of the stochastic flow with interaction in ${\mathbb R}.$ The discrete-time flow is given by a difference equation with random perturbation which is defined by a sequence of stationary Gaussian processes. We obtain the Itô–Wiener expansion for a solution to the stochastic difference equation which can be regarded as a discrete analogue of the Krylov–Veretennikov representation for a solution to the stochastic differential equation.