# Poissonian occupation times of spectrally negative L\'evy processes with applications

## TL;DR

This paper introduces Poissonian occupation times for spectrally negative Lévy processes, extending known occupation time concepts, and connects these to insurance risk models with delays, providing new analytical tools.

## Contribution

It defines Poissonian occupation times for spectrally negative Lévy processes and links them to insurance risk models with Parisian delays, extending existing occupation time theories.

## Key findings

- Extended occupation time concepts to Poissonian observation schemes.
- Established a connection between Poissonian occupation times and insurance risk models.
- Provided analytical results for spectrally negative Lévy processes with applications.

## Abstract

In this paper, we introduce the concept of \emph{Poissonian occupation times} below level $0$ of spectrally negative L\'evy processes. In this case, occupation time is accumulated only when the process is observed to be negative at arrival epochs of an independent Poisson process. Our results extend some well known continuously observed quantities involving occupation times of spectrally negative L\'evy processes. As an application, we establish a link between Poissonian occupation times and insurance risk models with Parisian implementation delays.