Ruin Probabilities in a Markovian Shot-Noise Environment

TL;DR

This paper models risk using a Markovian shot-noise process for the intensity, enabling more realistic ruin probability estimates and asymptotic analysis compared to traditional models with constant jump intensity.

Contribution

It introduces a risk model with a Markovian shot-noise intensity, applying PDMP theory and change of measure techniques to derive ruin probabilities and their asymptotic behavior.

Findings

Derived an upper bound for ruin probability.

Established asymptotic behavior of ruin probability.

Applied PDMP theory to a new risk model.

Abstract

We consider a risk model with a counting process whose intensity is a Markovian shot-noise process, to resolve one of the disadvantages of the Cram\'er-Lundberg model, namely the constant jump intensity of the Poisson process. Due to this structure, we can apply the theory of PDMPs on a multivariate process containing the intensity and the reserve process, which allows us to identify a family of martingales. Eventually, we use change of measure techniques to derive an upper bound for the ruin probability in this model. Exploiting a recurrent structure of the shot-noise process, even the asymptotic behaviour of the ruin probability can be determined.