Derivative Formula and Harnack Inequality for Linear SDEs Driven by L\'evy Processes

TL;DR

This paper develops derivative formulas and Harnack inequalities for linear SDEs driven by Lévy processes, providing explicit gradient estimates, heat kernel inequalities, and a new Girsanov theorem for Lévy processes.

Contribution

It introduces new derivative formulas and Harnack inequalities for Lévy-driven SDEs, along with explicit estimates and a novel Girsanov theorem for Lévy processes.

Findings

Derived derivative formulas and Harnack inequalities for Lévy-driven SDEs.

Established explicit gradient estimates and heat kernel inequalities.

Proposed a new Girsanov theorem for Lévy processes.

Abstract

By using lower bound conditions of the L\'evy measure, derivative formulae and Harnack inequalities are derived for linear stochastic differential equations driven by L\'evy processes. As applications, explicit gradient estimates and heat kernel inequalities are presented. As byproduct, a new Girsanov theorem for L\'evy processes is derived.