Occupation times of spectrally negative L\'evy processes with applications

TL;DR

This paper derives formulas for the Laplace transform of occupation times of spectrally negative Lévy processes, extending known results for Brownian motion and jump-diffusion, with applications to insurance risk modeling.

Contribution

It provides new explicit expressions for occupation times of spectrally negative Lévy processes using scale functions, broadening the analytical tools available for these processes.

Findings

Derived Laplace transform formulas for occupation times

Extended results to spectrally negative Lévy processes

Applied findings to insurance risk models

Abstract

In this paper, we compute the Laplace transform of occupation times (of the negative half-line) of spectrally negative L\'evy processes. Our results are extensions of known results for standard Brownian motion and jump-diffusion processes. The results are expressed in terms of the so-called scale functions of the spectrally negative L\'evy process and its Laplace exponent. Applications to insurance risk models are also presented.