The Averaging Principle for Stochastic Differential Equations

The averaging principle established by Krylov and Bogolyubov is a powerful tool for investigating the properties of dynamic systems involving a small parameter. Its extension to Markov processes is described in papers [R. Z. Khas'minskii, Teor. Veroyatn. Primen., 1963, vol. 8, no. 1, pp. 3–25; I. I.  Gikhman, in Winter School on the Theory of Probability and Mathematical Statistics, Kiev, 1964; I. Vrkoc, Czech. Math. J., 1966, vol. 16, pp. 518–544]. In this note it is supposed that both the “slow” and “fast” motions are components of a Markov process of diffusion type.