Parisian ruin for a refracted L\'evy process

TL;DR

This paper extends the analysis of Parisian ruin probabilities to a refracted Lévy process with an adaptive premium rate, providing a generalized and compact formula applicable to more complex risk models.

Contribution

It generalizes previous results on Parisian ruin probabilities from standard Lévy processes to refracted Lévy processes with deterministic delays.

Findings

Derived a compact formula for Parisian ruin probability in refracted Lévy processes.

Extended existing models to include adaptive premium rates and delays.

Provided examples illustrating the applicability of the results.

Abstract

In this paper, we investigate Parisian ruin for a L\'evy surplus process with an adaptive premium rate, namely a refracted L\'evy process. More general Parisian boundary-crossing problems with a deterministic implementation delay are also considered. Our main contribution is a generalization of the result in Loeffen et al. (2013) for the probability of Parisian ruin of a standard L\'evy insurance risk process. Despite the more general setup considered here, our main result is as compact and has a similar structure. Examples are provided.