Averaging principle for two-time-scale stochastic differential equations with correlated noise

This article is devoted to studying the averaging principle for two-time-scale stochastic differential equations with correlated noise. By the technique of multiscale expansion of the solution to the backward Kolmogorov equation and consequent elimination of variables, we obtain the Kolmogorov equation corresponding to the reduced simplified system. The approximation of the slow component of the original system to the solution of the corresponding averaged equation is in the weak sense. An example is also provided to illustrate our result.