On the longest/shortest negative excursion of a L\'evy risk process and related quantities

TL;DR

This paper investigates the distributions of the longest and shortest negative excursions of spectrally negative Lévy processes, applying binomial expansion to address ruin problems and stochastic ordering.

Contribution

It introduces a novel binomial expansion method to analyze negative excursions and related quantities in Lévy processes, advancing understanding of ruin probabilities and stochastic comparisons.

Findings

Derived distributions of negative excursions and their joint behavior

Applied results to Parisian ruin and distress period analysis

Demonstrated the effectiveness of the binomial expansion approach

Abstract

In this paper, we analyze some distributions involving the longest and shortest negative excursions of spectrally negative L\'evy processes using the binomial expansion approach. More specifically, we study the distributions of such excursions and related quantities such as the joint distribution of the shortest and longest negative excursion and their difference (also known as the range) over a random and infinite horizon time. Our results are applied to address new Parisian ruin problems, stochastic ordering and the number near-maximum distress periods showing the superiority of the binomial expansion approach for such cases.