Ruin problems for risk processes with dependent phase-type claims

TL;DR

This paper introduces a new class of dependent phase-type claim distributions for risk processes, providing recursive methods to analyze ruin probabilities and related metrics, enhancing modeling flexibility and computational tractability.

Contribution

It proposes a novel class of dependent phase-type distributions for claims, enabling simple recursive calculations of ruin-related probabilities using a Markov-modulated approach.

Findings

Recursive procedures for joint distribution of ruin time, deficit, and claims

Bounds for ultimate ruin probability

Numerical illustrations with multivariate phase-type distributions

Abstract

We consider continuous time risk processes in which the claim sizes are dependent and non-identically distributed phase-type distributions. The class of distributions we propose is easy to characterize and allows to incorporate the dependence between claims in a simple and intuitive way. It is also designed to facilitate the study of the risk processes by using a Markov-modulated fluid embedding technique. Using this technique, we obtain simple recursive procedures to determine the joint distribution of the time of ruin, the deficit at ruin and the number of claims before the ruin. We also obtain some bounds for the ultimate ruin probability. Finally, we provide a few examples of multivariate phase-type distributions and use them for numerical illustration.