Ruin theory incorporating MIPP-type jumps

Abstract

The paper investigates the ruin probability of an insurer's surplus process when claims follow a Multiply Iterated Poisson Process (MIPP). This setting extends the classical Cramer-Lundberg model by allowing for clustered claim arrivals, making it particularly appropriate for modeling catastrophic insurance losses. The primary contribution is the derivation of explicit criteria for ultimate ruin, showing that the ruin probability is largely determined by the jump intensity parameter and the iteration count. Furthermore, we establish integro-differential equations for the survival and ruin probabilities and compute their Laplace transforms. These transforms are then linked via recursive formulas that relate ruin probabilities at consecutive iteration levels. We also introduce a Cramer-Lundberg-style approximation, which provides asymptotic expressions for ruin probabilities along with a Lundberg-type inequality. Numerical examples, including comparisons with the classical model, are provided to validate the theoretical results and to highlight the impact of claim clustering on the insurer's risk of ruin.