Estimates of transition densities and their derivatives for jump L\'evy processes

TL;DR

This paper provides explicit upper and lower bounds for the densities and their derivatives of convolution semigroups associated with jump Lévy processes, based on assumptions on the Lévy measure and exponent.

Contribution

It introduces new explicit estimates for densities and their derivatives of jump Lévy processes under specific conditions, advancing understanding of their probabilistic behavior.

Findings

Derived upper and lower bounds for densities of jump Lévy processes.

Provided estimates for derivatives of these densities.

Enhanced theoretical understanding of Lévy process distributions.

Abstract

We give upper and lower estimates of densities of convolution semigroups of probability measures under explicit assumptions on the corresponding Levy measure and the Levy--Khinchin exponent. We obtain also estimates of derivatives of densities.