Averaging Principle For Multiscale Stochastic Differential Equations

In this paper,we study the averaging principle for multiscale stochastic differential equations driven by Brownian motion under non-Lipschitz condition.Firstly,we use the classical method picard iteration to prove the existence and uniqueness of the system solution.Secondly,we construct the auxiliary process((?)tε,(?)tε)by utilizing Khasminskii techniques.We can prove the moment estimation of the solution of the equation and estimate |Xtε-(?)tε| this term through Jensen inequality,B-D-G inequality,Gronwall inequality,etc.Then,based on the exponential ergodicity of the fast equation,we estimate the error between the auxiliary process (?)tε and the (?)t of the averaged equation solution.Finally,it is proved that the slow component Lp(p≥2)converges to the solution of the averaged equation.