Modified ruin probability for a Cram\'er-Lundberg model driven by a compound mixed Poisson process

TL;DR

This paper analyzes modified ruin probabilities in a Cramér-Lundberg model with a compound mixed Poisson process, providing asymptotic equivalences and explicit results in both heavy-tailed and light-tailed regimes.

Contribution

It introduces new asymptotic results for modified ruin probabilities under different tail regimes and clarifies their relationship with classical ruin probabilities.

Findings

In heavy-tailed regime, modified and classical ruin probabilities are asymptotically equivalent.

In light-tailed regime, a fixed-intensity ratio theorem is established.

Explicit constants and asymptotic behaviors are derived for endpoint and density results.

Abstract

We study modified ruin probabilities in a Cram\'er-Lundberg model driven by a compound mixed Poisson process. In the heavy-tailed regime, if the integrated claim-size distribution is subexponential and the upper endpoint of the mixing distribution stays below the net-profit boundary, the modified and classical ruin probabilities are asymptotically equivalent. In the light-tailed regime, we prove a fixed-intensity ratio theorem and obtain both an endpoint-atom result and a sharp endpoint-density asymptotic with an explicit constant.