Probability measures, L\'{e}vy measures and analyticity in time

TL;DR

This paper explores the relationship between the probability density of Lévy processes and their Lévy measures, introducing three methods for subordinators and examining the smoothness of the density over time.

Contribution

It presents three novel methods to compute semigroup densities from Lévy measures for subordinators and analyzes their smoothness properties.

Findings

Three methods for computing semigroup densities from Lévy measures.

Analytic continuation of Lévy density enables contour integration.

Smoothness of the semigroup density in time is characterized.

Abstract

We investigate the relation of the semigroup probability density of an infinite activity L\'{e}vy process to the corresponding L\'{e}vy density. For subordinators, we provide three methods to compute the former from the latter. The first method is based on approximating compound Poisson distributions, the second method uses convolution integrals of the upper tail integral of the L\'{e}vy measure and the third method uses the analytic continuation of the L\'{e}vy density to a complex cone and contour integration. As a by-product, we investigate the smoothness of the semigroup density in time. Several concrete examples illustrate the three methods and our results.