Some mathematical properties of the premium function and ruin probability of a generalized Cramér–Lundberg model driven by mixed poisson processes

Abstract

This paper derives several mathematical properties of the generalized Cramér–Lundberg model proposed by Tomita et al. (J. Appl. Probab. 59(3):849-859, 2022). The model extends the Bayesian-estimator model of Dubey. (Versicherungsmathematiker. 2:130-141, 1977) to the case of multiple insurance policies. We study the instantaneous premium function and the dependence structure of the ruin probability on the intensity of the driving mixed Poisson process. In particular, we show that the conditional ruin probability is monotonic with respect to the intensity value under certain assumptions. Monte Carlo simulations suggest that, without these assumptions, the monotonicity does not generally hold. Our study contributes to the risk management of insurance companies in the sense that it reveals how the difference between the assumed and true distribution of the risk factor affects the ruin probability.