Iterated Poisson Processes for Catastrophic Risk Modeling in Ruin Theory

TL;DR

This paper introduces the Multiply Iterated Poisson Process (MIPP) for modeling clustered catastrophic claims in ruin theory, providing explicit formulas for jump probabilities and ruin probabilities, enhancing risk assessment accuracy.

Contribution

It develops a novel MIPP-based ruin model with explicit scale functions, capturing claim clustering effects not addressed by classical models.

Findings

Explicit jump size probabilities derived

Closed-form scale function obtained

Enhanced ruin probability calculations for clustered claims

Abstract

This paper studies the properties of the Multiply Iterated Poisson Process (MIPP), a stochastic process constructed by repeatedly time-changing a Poisson process, and its applications in ruin theory. Like standard Poisson processes, MIPPs have exponentially distributed sojourn times (waiting times between jumps). We explicitly derive the probabilities of all possible jump sizes at the first jump and obtain the Laplace transform of the joint distribution of the first jump time and its corresponding jump size. In ruin theory, the classical Cram\'er-Lundberg model assumes that claims arrive independently according to a Poisson process. In contrast, our model employs an MIPP to allow for clustered arrivals, reflecting real-world scenarios, such as catastrophic events. Under this new framework, we derive the corresponding scale function in closed form, facilitating accurate calculations of the probability of ruin in the presence of clustered claims. These results improve the modeling of extreme risks and have practical implications for insurance solvency assessments, reinsurance pricing, and capital reserve estimation.