Parisian excursion below a fixed level from the last record maximum of Levy insurance risk process

TL;DR

This paper investigates Parisian ruin in Levy insurance risk processes, focusing on drawdown-based ruin, providing joint Laplace transforms of ruin time and position, and establishing new fluctuation identities.

Contribution

It introduces new results on Parisian ruin under Levy processes with drawdown conditions, using recent fluctuation theory developments.

Findings

Ruin occurs in finite time with probability one under drawdown.

Joint Laplace transforms of ruin time and position are derived.

Semi-explicit identities involve scale functions and Levy process laws.

Abstract

This paper presents some new results on Parisian ruin under Levy insurance risk process, where ruin occurs when the process has gone below a fixed level from the last record maximum, also known as the high-water mark or drawdown, for a fixed consecutive periods of time. The law of ruin-time and the position at ruin is given in terms of their joint Laplace transforms. Identities are presented semi-explicitly in terms of the scale function and the law of the Levy process. They are established using recent developments on fluctuation theory of drawdown of spectrally negative Levy process. In contrast to the Parisian ruin of Levy process below a fixed level, ruin under drawdown occurs in finite time with probability one.