Asymptotics of Ruin Probabilities in a Subordinated Cram\'er-Lundberg Model

TL;DR

This paper analyzes how subordinating a classical ruin model with a Lévy process affects the probability of ruin, revealing that such transformations can significantly increase ruin probabilities despite unchanged expected claims.

Contribution

It introduces a subordinated Cramér-Lundberg model and demonstrates that this transformation can arbitrarily slow the decay of ruin probabilities, highlighting a new impact of claim clustering.

Findings

Subordination increases ruin probability.

Expected total claims remain unchanged.

Ruin probability decay can be arbitrarily slow.

Abstract

We study a dynamic model of a non-life insurance portfolio. The foundation of the model is a compound Poisson process that represents the claims side of the insurer. To introduce clusters of claims appearing, e.g. with catastrophic events, this process is time-changed by a L\'evy subordinator. The subordinator is chosen so that it evolves, on average, at the same speed as calendar time, creating a trade-off between intensity and severity. We show that such a transformation always has a negative impact on the probability of ruin. Despite the expected total claim amount remaining invariant, it turns out that the probability of ruin as a function of the initial capital falls arbitrarily slowly depending on the choice of the subordinator.