Averaging principle for SDEs with singular drifts driven by $\alpha$-stable processes

TL;DR

This paper studies the convergence rate of the averaging principle for SDEs with singular drifts driven by alpha-stable processes, deriving Schauder estimates and establishing optimal convergence rates under various parameter conditions.

Contribution

It provides new Schauder estimates for nonlocal PDEs and establishes the strong convergence rate of the averaging principle for SDEs with singular drifts driven by alpha-stable processes.

Findings

Established Schauder estimates for nonlocal PDEs in Besov-H"older spaces.

Derived the optimal strong convergence rate for the averaging principle.

Proved convergence of martingale solutions under certain parameter regimes.

Abstract

In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with $\beta$-H\"older drift driven by $\alpha$-stable processes. More specifically, we first derive the Schauder estimate for nonlocal partial differential equations (PDEs) associated with the aforementioned SDEs, within the framework of Besov-H\"older spaces. Then we consider the case where $(\alpha,\beta)\in(0,2)\times(1-\tfrac{\alpha}{2},1)$. Using the Schauder estimate, we establish the strong convergence rate for the averaging principle. In particular, under suitable conditions we obtain the optimal rate of strong convergence when $(\alpha,\beta)\in(\tfrac{2}{3},1]\times(2-\tfrac{3\alpha}{2},1)\cup(1,2)\times(\tfrac{\alpha}{2},1)$. Furthermore, when $(\alpha,\beta)\in(0,1]\times(1-\alpha,1-\tfrac{\alpha}{2}]\cup(1,2)\times(\tfrac{1-\alpha}{2},1-\tfrac{\alpha}{2}]$, we show the convergence of the martingale solutions of original systems to that of the averaged equation. When $\alpha\in(1,2)$, the drift can be a distribution.