Averaging principle for SDEs with singular drifts driven by $\hat{I}\pm$-stable processes

Abstract

In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with ${\hat{I}}^2$-Hölder drift driven by $\hat{I}\pm$-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-Hölder spaces. Then we consider the case where $(\hat{I}\pm ,{\hat{I}}^2)\in (0,2)\times (1-\frac{\hat{I}}{\pm }2,1)$. 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 $(\hat{I}\pm ,{\hat{I}}^2)\in (\frac{2}{3},1]\times (2-\frac{3\hat{I}\pm }{2},1)\cup (1,2)\times (\frac{\hat{I}}{\pm }2,1)$. Furthermore, when $(\hat{I}\pm ,{\hat{I}}^2)\in (0,1]\times (1-\hat{I}\pm ,1-\frac{\hat{I}}{\pm }2]\cup (1,2)\times (\frac{1-\hat{I}\pm }{2},1-\frac{\hat{I}}{\pm }2]$, we show the convergence of the martingale solutions of original systems to that of the averaged equation. When $\hat{I}\pm \in (1,2)$, the drift can be a distribution.

30 pages