L\'evy insurance risk processes with parisian type severity of debt

TL;DR

This paper introduces a novel bankruptcy definition for spectrally negative Lévy insurance risk processes, analyzing the Gerber-Shiu distribution with a new Parisian-type severity of debt using excursion theory and scale functions.

Contribution

It proposes a new ruin model with Parisian-type severity levels and derives its Gerber-Shiu distribution using advanced excursion theory and scale functions.

Findings

Derived explicit formulas for the Gerber-Shiu distribution.

Extended the classical ruin theory to include Parisian-type severity.

Utilized recent excursion measure descriptions for Lévy processes.

Abstract

In this article, we introduce a new definition of bankruptcy for a spectrally negative L\'evy insurance risk process. More precisely, we study the Gerber-Shiu distribution for a ruin model where at each time the surplus goes negative, an independent negative random level is considered. If a negative excursion of the surplus exceeds such random level then the insurance company goes out of business. Our methodology uses excursion theory and relies on the description of the excursion measure away from 0 which was recently obtained by the authors in Pardo et al. (see arXiv:1507.05225). Our results are given in terms of the so-called scale functions.