Ruin theory incorporating MIPP-type jumps

Abstract

The paper studies the ruin probability for an insurer's surplus process when claim arrivals are governed by a Multiply Iterated Poisson Process (MIPP). This framework generalizes the classical Cramér-Lundberg model by permitting claims to arrive in clusters, which makes it well suited for representing catastrophic insurance events. The main result is the development of explicit conditions for ultimate ruin, demonstrating that the ruin probability is driven chiefly by the jump intensity and the number of iterations. In addition, we derive integro-differential equations for both survival and ruin probabilities and obtain their Laplace transforms. These transforms are connected through recursive relations that tie ruin probabilities at successive iteration levels. We further propose a Cramér-Lundberg-type approximation, yielding asymptotic formulas for ruin probabilities together with a Lundberg-style bound. Finally, numerical experiments - along with comparisons to the classical model - are included to support the theory and to illustrate how claim clustering affects an insurer's likelihood of ruin.