Iterated poisson processes for catastrophic risk modeling in ruin theory
研究了多重迭代泊松过程（MIPP）的性质，将其用于破产理论中允许索赔聚集到达的模型，推导了首次跳跃的联合分布拉普拉斯变换和尺度函数，有助于保险偿付能力评估和再保险定价。
This paper studies the properties of the Multiple 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, pricing reinsurance, and the estimation of capital reserves.