Iterated Poisson Processes for Catastrophic Risk Modeling in Ruin Theory

TL;DR

This work develops the Multiply Iterated Poisson Process (MIPP) by iterating Poisson processes in time and embeds it into ruin theory to capture clustered claim arrivals. It derives explicit first-jump distributions, the joint Laplace transform of jump times and sizes, and a closed-form ruin-scale function for clustered risk, enabling analytic ruin probabilities. Beyond MIPP, the paper studies Multiple Subordinated Poisson Processes (MSPP) and their moments, martingale structure, long-range dependence, and hitting-time properties, then extends to multiple subordinated compound Poisson settings and subordinated risk/financial models (e.g., Merton-type applications). The results provide tractable expressions for scale functions, moments, MGFs, and crossing times, with clear implications for solvency assessment, risk pricing, and long-memory behavior in systems experiencing bursty events. Overall, the framework yields a cohesive toolkit for modeling catastrophic risk and clustered insurance/credit events with explicit analytic tractability.

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 Cramér-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.

Figures (10)

• Figure 1: Probability distribution (\ref{['71']}) for $\lambda = 0.5$ and (a) $n=1$, (b) $n=2$, and (c) $n=3$.
• Figure 2: Probability distribution (\ref{['71']}) for $\lambda = 2$ and (a) $n=1$, (b) $n=2$, and (c) $n=3$.
• Figure 3: Probability distribution (\ref{['71']}) as $n$ increases for $\lambda = 1$ and $t=20$.
• Figure 4: Sample paths for $N_t^1$, $N_t^2$, and $N_t^3$ and the corresponding subordinated Poisson processes $V_t^{(2)}$ and $V_t^{(3)}$.
• Figure 5: Ten Sample paths for $M_t$ as in Proposition 2.4 for (a) $n=2$ and (b) $n=3$.

Theorems & Definitions (32)

• Remark 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Remark 2.4
• Proposition 2.5
• proof
• Proposition 2.6
• proof
• Corollary 2.7