On the time spent in the red by a refracted L\'evy risk process

TL;DR

This paper models an insurance risk process with adaptive premiums that increase when in deficit, analyzing the duration spent in the red zone and deriving related ruin probabilities using spectrally negative Lévy processes.

Contribution

It introduces a novel refracted Lévy risk model with adaptive premiums and derives the distribution of occupation times and ruin probabilities within this framework.

Findings

Derived the distribution of time spent in the red zone.

Computed bankruptcy and Parisian ruin probabilities.

Extended existing Lévy process results to the refracted risk context.

Abstract

In this paper, we introduce an insurance ruin model with adaptive premium rate, thereafter refered to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished. In this model, the premium rate is increased as soon as the wealth process falls into the red zone and is brought back to its regular level when the process recovers. The analysis is mainly focused on the time a refracted L\'evy risk process spends in the red zone (analogous to the duration of the negative surplus). Building on results from Kyprianou and Loeffen (2010) and Loeffen et al. (2012), we identify the distribution of various functionals related to occupation times of refracted spectrally negative L\'evy processes. For example, these results are used to compute the probability of bankruptcy and the probability of Parisian ruin in this model with restructuring.