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

TL;DR

This paper investigates the asymptotic behavior of multi-scale stochastic differential equations driven by Lévy processes, establishing optimal convergence orders and extending applicability to equations with monotonicity coefficients and general Lévy processes.

Contribution

It introduces new results on convergence orders for multi-scale SDEs driven by Lévy processes, applicable to equations with monotonicity coefficients and general Lévy processes.

Findings

Optimal strong convergence order 1/2 achieved.

Optimal weak convergence order 1 obtained.

Results applicable to a broad class of multi-scale SDEs with Lévy noise.

Abstract

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.