Strong and weak convergence rates for fully coupled multiscale stochastic differential equations driven by $\alpha$-stable processes

TL;DR

This paper establishes strong and weak convergence rates for multiscale stochastic differential equations driven by alpha-stable processes, using nonlocal Poisson equations to analyze different scaling regimes.

Contribution

It provides new convergence rate results for multiscale SDEs with alpha-stable noise, including the derivation of averaged equations under various scaling regimes.

Findings

Strong convergence orders relate to the optimal order 1 - 1/alpha.

Weak convergence orders are established as 1 under certain regularity conditions.

Four averaged equations are derived for different scaling regimes.

Abstract

We first establish strong convergence rates for multiscale systems driven by $\alpha$-stable processes, with analyses constructed in two distinct scaling regimes. When addressing weak convergence rates of this system, we derive four averaged equations with respect to four scaling regimes. Notably, under sufficient H\"{o}lder regularity conditions on the time-dependent drifts of slow process, the strong convergence orders are related to the known optimal strong convergence order $1-\frac{1}{\alpha}$, and the weak convergence orders are 1. Our primary approach involves employing nonlocal Poisson equations to construct ``corrector equations" that effectively eliminate inhomogeneous terms.