Transition densities of spectrally positive L\'evy processes

TL;DR

This paper analyzes the asymptotic behavior of transition densities for a broad class of spectrally positive Lévy processes, providing precise estimates under certain conditions on their exponents.

Contribution

It establishes new asymptotic results and sharp estimates for transition densities of spectrally positive Lévy processes with unbounded variation, extending previous knowledge.

Findings

Asymptotic behavior of transition densities proved for a large class of processes.

Sharp two-sided estimates obtained for processes without Gaussian component.

Results apply under mild conditions on the Laplace exponent's second derivative.

Abstract

We prove asymptotic behaviour of transition density for a large class of spectrally one-sided L\'evy processes of unbounded variation satisfying mild condition imposed on the second derivative of the Laplace exponent, or equivalently, on the real part of the characteristic exponent. We also provide sharp two-sided estimates on the density when restricted additionally to processes without Gaussian component.