Ruin probability with Parisian delay for a spectrally negative L\'evy risk process

TL;DR

This paper derives explicit formulas and asymptotic behaviors for the probability of Parisian ruin in spectrally negative Lévy risk processes, providing insights into risk assessment over extended periods.

Contribution

It introduces a general framework for calculating Parisian ruin probabilities for spectrally negative Lévy processes, including explicit formulas and asymptotic results.

Findings

Derived explicit expressions for ruin probability

Established asymptotic behaviors as reserves grow large

Analyzed specific examples to illustrate results

Abstract

In this paper we analyze so-called Parisian ruin probability that happens when surplus process stays below zero longer than fixed amount of time $\zeta>0$. We focus on general spectrally negative L\'{e}vy insurance risk process. For this class of processes we identify expression for ruin probability in terms of some other quantities that could be possibly calculated explicitly in many models. We find its Cram\'{e}r-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples.