Strong convergence of multi-scale stochastic differential equations with a full dependence

This paper considers the strong convergence of multi-scale stochastic differential equations, where diffusion coefficient of the slow component depends on fast process. In this situation, it is well-known that strong convergence in the averaging principle does not hold in general.

We propose a new approximation equation, and prove that the order of strong convergence is $1/2$ via the technique of Poisson equation. In particular, when diffusion coefficient of the slow component does not depend on fast process, the approximation equation is exactly the averaged equation. This provides us a new perspective to study the strong convergence of multi-scale stochastic differential equations with a full dependence.