Derivative formula and gradient estimate for SDEs driven by $\alpha$-stable processes

TL;DR

This paper establishes a derivative formula and gradient estimates for SDEs driven by alpha-stable processes, and applies these results to demonstrate the strong Feller property for certain SPDEs.

Contribution

It introduces a Bismut-Elworthy-Li type derivative formula and gradient estimates specifically for SDEs with alpha-stable noise, extending existing methods to non-Gaussian stable processes.

Findings

Derived a new derivative formula for alpha-stable driven SDEs

Established gradient estimates for these SDEs

Proved the strong Feller property for related SPDEs

Abstract

In this paper we prove a derivative formula of Bismut-Elworthy-Li's type as well as gradient estimate for stochastic differential equations driven by $\alpha$-stable noises, where $\alpha\in(0,2)$. As an application, the strong Feller property for stochastic partial differential equations driven by subordinated cylindrical Brownian motions is presented.