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On the longest/shortest negative excursion of a Lévy risk process and related quantities

M. A. Lkabous, Z. Palmowski

Abstract

In this paper, we analyze some distributions involving the longest and shortest negative excursions of spectrally negative Lévy 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.

On the longest/shortest negative excursion of a Lévy risk process and related quantities

Abstract

In this paper, we analyze some distributions involving the longest and shortest negative excursions of spectrally negative Lévy 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.

Paper Structure

This paper contains 16 sections, 4 theorems, 115 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on spectrally negative Lévy processes
  3. Fluctuation identities
  4. Cramér-Lundberg process with exponential jumps
  5. Main results
  6. Distribution of longest negative duration
  7. Shortest duration up to an infinite time horizon
  8. Joint Distribution of $\underline{U}_{\infty}$ and $\bar{U}_{\infty}$
  9. The range of negative excursions
  10. Applications
  11. New Parisian ruin models
  12. Parisian ruins under the same roof
  13. Parisian ruin under a peak-to-sum constraint
  14. Stochastic ordering of Parisian ruin probabilities
  15. On the number of near-maximum distress periods
  16. ...and 1 more sections

Key Result

Proposition 1

For $r>0$, $x\in \mathbb{R}$ and $\mathbb{E}[X_1]>0$, we have where

Figures (5)

  • Figure 1: Sample path of $U$ (red line) and corresponding running longest $\bar{U}$ (dashed black line).
  • Figure 2: Values of $\mathbb{P}_{x}\left( \underline{U}_{\infty }>r,\tau _{0}^{-}<\infty \right)$ for the Cramér-Lundberg process with $\alpha=1/2$, $\eta=2$ and $x=1$.
  • Figure 3: Values of the joint cumulative distribution function of $(\underline{U}_{\infty},\bar{U}_{\infty})$ for the Cramér-Lundberg process with $\alpha=1/2$, $\eta=2$, $c=5.5$ and $x=1$.
  • Figure 4: Values of $\mathbb{P}\left( \mathcal{R}_{\infty} < r, \tau^-_0 <\infty \right)$ for the Cramér-Lundberg process with $\alpha=1/2$, $\eta=2$.
  • Figure 5: Values of $\mathbb{E}\left[\mathcal{E}_a\right]$ for the Cramér-Lundberg process with ($c=8.5$, $\eta=1$), ($c=9.5$, $\alpha=1/3$) and ($\eta=2$, $\alpha=1/2$).

Theorems & Definitions (11)

  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Remark 4
  • Theorem 5
  • proof
  • Remark 6
  • Proposition 7
  • ...and 1 more