Discounted Penalty Function at Parisian Ruin for L\'evy Insurance Risk Process

TL;DR

This paper investigates Parisian ruin in Lévy insurance risk processes, deriving joint Laplace transforms and potential measures that generalize existing results and facilitate computation of expected discounted penalties.

Contribution

It provides new semi-explicit formulas for Parisian ruin probabilities and related quantities using scale functions and Lévy process distributions.

Findings

Joint Laplace transform of ruin-time and ruin-position derived

$q$-potential measure of process killed at Parisian ruin obtained

Results expressed in terms of $q$-scale function and Lévy distribution

Abstract

In the setting of a L\'evy insurance risk process, we present some results regarding the Parisian ruin problem which concerns the occurrence of an excursion below zero of duration bigger than a given threshold $r$. First, we give the joint Laplace transform of ruin-time and ruin-position (possibly killed at the first-passage time above a fixed level $b$), which generalises known results concerning Parisian ruin. This identity can be used to compute the expected discounted penalty function via Laplace inversion. Second, we obtain the $q$-potential measure of the process killed at Parisian ruin. The results have semi-explicit expressions in terms of the $q$-scale function and the distribution of the L\'evy process.