# On occupation times in the red of L\'evy risk models

## TL;DR

This paper derives explicit formulas for the distribution of occupation times below zero in spectrally negative Lévy risk models, enhancing understanding of risk processes and enabling new applications in finance and insurance.

## Contribution

It provides the first analytical expressions for occupation times in these models, extending beyond Laplace transform results to explicit distributions.

## Key findings

- Explicit distribution formulas for occupation times in Lévy risk models
- Applications to drawdown, ruin, and maximum-related quantities
- Distribution results for refracted Brownian and Cramér-Lundberg models

## Abstract

In this paper, we obtain analytical expression for the distribution of the occupation time in the red (below level $0$) up to an (independent) exponential horizon for spectrally negative L\'{e}vy risk processes and refracted spectrally negative L\'{e}vy risk processes. This result improves the existing literature in which only the Laplace transforms are known. Due to the close connection between occupation time and many other quantities, we provide a few applications of our results including future drawdown, inverse occupation time, Parisian ruin with exponential delay, and the last time at running maximum. By a further Laplace inversion to our results, we obtain the distribution of the occupation time up to a finite time horizon for refracted Brownian motion risk process and refracted Cram\'{e}r-Lundberg risk model with exponential claims.