Stochastic version of the averaging principle for diffusion type processes

Abstract

We consider the family of stochastic processes where x ε is the solution of the Ito stochastic differential equation whose drift coefficient depends also on the strictly stationary and ergodic process . Using a transformation which is similar to the classical averaging principle we find explicitly first and second order approximations of (in the sense of weak convergence). The approximations are diffusion Markov processes.