Fractional Risk Process in Insurance

TL;DR

This paper extends classical insurance risk models by incorporating the fractional Poisson process, capturing dependence in claim arrivals, and analyzes its impact on ruin probabilities and capital requirements.

Contribution

It introduces the fractional Poisson process into ruin probability models, highlighting its effects on initial surplus stress and risk measures in insurance.

Findings

Fractional Poisson process models claim arrivals with dependence.

Average capital needed for recovery remains unchanged under fractional Poisson.

Simplified evaluation of risk measures in fractional Poisson environment.

Abstract

Important models in insurance, for example the Carm{\'e}r--Lundberg theory and the Sparre Andersen model, essentially rely on the Poisson process. The process is used to model arrival times of insurance claims. This paper extends the classical framework for ruin probabilities by proposing and involving the fractional Poisson process as a counting process and addresses fields of applications in insurance. The interdependence of the fractional Poisson process is an important feature of the process, which leads to initial stress of the surplus process. On the other hand we demonstrate that the average capital required to recover a company after ruin does not change when switching to the fractional Poisson regime. We finally address particular risk measures, which allow simple evaluations in an environment governed by the fractional Poisson process.