Ruin probabilities with dependence on the number of claims within a fixed time window

TL;DR

This paper studies ruin probabilities in an insurance risk model where claim inter-arrival times depend on recent claims, using regenerative structures, asymptotic analysis, and numerical methods including importance sampling.

Contribution

It introduces a novel dependence structure based on recent claims, derives asymptotic results, and provides closed-form and simulation-based methods for computing ruin probabilities.

Findings

Asymptotic ruin probability regimes identified

Closed-form ruin probability formulas derived

Simulation results demonstrate effectiveness of importance sampling

Abstract

We analyse the ruin probabilities for a renewal insurance risk process with inter-arrival time distributions depending on the claims that arrived within a fixed (past) time window. This dependence could be explained through a regenerative structure. The main inspiration of the model comes from the Bonus-Malus feature. We discuss first asymptotic results of ruin probabilities for different regimes of claim distributions. For numerical results, we recognise an embedded Markov additive process. Via an appropriate change of measure, ruin probabilities could be computed to a closed form formulae. Additionally, we present simulated results via the importance sampling method, which further permit an in-depth analysis of a few concrete cases.