Gradient estimates for SDEs Driven by Multiplicative L\'evy Noise

TL;DR

This paper establishes sharp gradient estimates for semigroups of SDEs driven by multiplicative Lévy noise, using a novel Bismut-Elworthy-Li type formula and Malliavin calculus, advancing understanding of stochastic processes with jumps.

Contribution

It introduces the first gradient estimates for such SDEs with multiplicative Lévy noise, employing a new derivative formula and finite-jump approximation techniques.

Findings

Gradient estimates are sharp for α-stable noises.

A new Bismut-Elworthy-Li type formula is developed.

Malliavin calculus is effectively applied to jump processes.

Abstract

Gradient estimates are derived, for the first time, for the semigroup associated to a class of stochastic differential equations driven by multiplicative L\'evy noise. In particular, the estimates are sharp for $\alpha$-stable type noises. To derive these estimates, a new derivative formula of Bismut-Elworthy-Li's type is established for the semigroup by using the Malliavin calculus and a finite-jump approximation argument.