The Cram\'er-Lundberg model with a fluctuating number of clients

TL;DR

This paper extends the classical Cramér-Lundberg risk model by incorporating a fluctuating client population, establishing a large-deviation principle to analyze rare-event behaviors and ruin probabilities.

Contribution

It introduces a novel model with stochastic client numbers and derives a large-deviation principle for joint reserve and client population processes.

Findings

Provides decay rate of ruin probability over time.

Identifies how client fluctuations influence ruin risk.

Determines the most likely path to ruin under rare events.

Abstract

This paper considers the Cram\'er-Lundberg model, with the additional feature that the number of clients can fluctuate over time. Clients arrive according to a Poisson process, where the times they spend in the system form a sequence of independent and identically distributed non-negative random variables. While in the system, every client generates claims and pays premiums. In order to describe the model's rare-event behaviour, we establish a sample-path large-deviation principle. This describes the joint rare-event behaviour of the reserve-level process and the client-population size process. The large-deviation principle can be used to determine the decay rate of the time-dependent ruin probability as well as the most likely path to ruin. Our results allow us to determine whether the chance of ruin is greater with more or with fewer clients and, more generally, to determine to what extent a large deviation in the reserve-level process can be attributed to an unusual outcome of the client-population size process.