Draw-down Parisian ruin for spectrally negative L\'{e}vy process

TL;DR

This paper investigates the draw-down Parisian ruin problem for spectrally negative Lévy processes, providing new analytical tools and results for risk processes with dividend barriers and Parisian ruin times.

Contribution

It introduces the draw-down Parisian ruin time, solves the two-sided exit problem using excursion theory, and derives the potential measure for the killed process, advancing risk process analysis.

Findings

Derived explicit expressions for the draw-down Parisian ruin time.

Solved the two-sided exit problem using excursion theory.

Obtained new results for risk processes with dividend barriers.

Abstract

In this paper we study the draw-down related Parisian ruin problem for spectrally negative L\'{e}vy risk processes. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit time via excursion theory. We also obtain an expression of the potential measure for the process killed at the draw-down Parisian time. As applications, new results are obtained for spectrally negative L\'{e}vy risk process with dividend barrier and Parisian ruin.