The Poisson Equation and Application to Multi-Scale SDEs with State-Dependent Switching

This paper study the Poisson equation associated with a Markov chain. By investigating the differentiability of the corresponding transition probability matrix with respect to parameters, we establish the regularity of the Poisson equation solution. As an application, we further study the averaging principle for a class of multi-scale stochastic differential equations with state-dependent switching, ultimately achieving an optimal strong convergence order of 1/2.Comment: 19 page

Asymptotic behavior for multi-scale SDEs with monotonicity coefficients driven by L\'evy processes

In this paper, we study the asymptotic behavior for multi-scale stochastic differential equations driven by L\'evy processes. The optimal strong convergence order 1/2 is obtained by studying the regularity estimates for the solution of Poisson equation with polynomial growth coefficients, and the optimal weak convergence order 1 is got by using the technique of Kolmogorov equation. The main contribution is that the obtained results can be applied to a class of multi-scale stochastic differential equations with monotonicity coefficients, as well as the driven processes can be the general L\'evy processes, which seems new in the existing literature.Comment: 39 pages. To appear in Potential Analysi