Phase-type approximations perturbed by a heavy-tailed component for the Gerber-Shiu function of risk processes with two-sided jumps

TL;DR

This paper develops a perturbation expansion for the Gerber-Shiu function in risk processes with phase-type gains and heavy-tailed claim perturbations, using fluid embedding and Markov-additive process theory.

Contribution

It introduces a novel approximation method for the Gerber-Shiu function when claim sizes are phase-type perturbed by heavy tails, expanding the function in powers of the perturbation parameter.

Findings

Derived the expansion of the Gerber-Shiu function in powers of the perturbation parameter.

Constructed practical approximations using the first two terms of the expansion.

Utilized fluid embedding and fluctuation theory of spectrally negative Markov-additive processes.

Abstract

We consider in this paper a risk reserve process where the claims and gains arrive according to two independent Poisson processes. While the gain sizes are phase-type distributed, we assume instead that the claim sizes are phase-type perturbed by a heavy-tailed component; that is, the claim size distribution is formally chosen to be phase-type with large probability $1-\epsilon$ and heavy-tailed with small probability $\epsilon$. We analyze the seminal Gerber-Shiu function coding the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin. We derive its value as an expansion with respect to powers of $\epsilon$ with known coefficients and we construct approximations from the first two terms of the aforementioned series. The main idea is based on the so-called fluid embedding that allows to put the considered risk process into the framework of spectrally negative Markov-additive processes and use its fluctuation theory developed in Ivanovs and Palmowski (2012).