A unified approach to ruin probabilities with delays for spectrally negative L\'evy processes

TL;DR

This paper introduces a unified framework for Parisian ruin with delays in spectrally negative Lévy processes, combining deterministic and stochastic delays, and derives the joint distribution of ruin time and deficit.

Contribution

It generalizes existing models by unifying deterministic and exponential delays in ruin theory for spectrally negative Lévy processes.

Findings

Derived the joint distribution of ruin time and deficit.

Unified deterministic and stochastic delay models.

Generalized previous specific delay results.

Abstract

In this paper, we unify two popular approaches for the definition of actuarial ruin with implementation delays, also known as Parisian ruin. Our new definition of ruin includes both deterministic delays and exponentially distributed delays: ruin is declared the first time an excursion in the red zone lasts longer than an implementation delay with a deterministic and a stochastic component. For this Parisian ruin with mixed delays, we identify the joint distribution of the time of ruin and the deficit at ruin, therefore providing generalizations of many results previously obtained, such as in \cite{baurdoux_et_al_2015} and \cite{loeffenetal2017} for the case of an exponential delay and that of a deterministic delay, respectively.