On multivariate modifications of Cramer Lundberg risk model with constant intensities

TL;DR

This paper extends the Cramer-Lundberg risk model to a multivariate setting with group arrivals, dependent processes, and various claim types, providing a unified framework that generalizes several known models.

Contribution

It introduces a broad multivariate modification of the Cramer-Lundberg model with dependent group arrivals and shows equivalence to simpler models under certain conditions.

Findings

Model generalizes several known risk processes

Derived relations between process characteristics and distributions

Established stochastic equivalence with non-empty group models

Abstract

The paper considers very general multivariate modifications of Cramer-Lundberg risk model. The claims can be of different types and can arrive in groups. The groups arrival processes within a type have constant intensities. The counting groups processes are dependent multivariate compound Poisson processes of type I. We allow empty groups and show that in that case we can find stochastically equivalent Cramer-Lundberg model with non-empty groups. The investigated model generalizes the risk model with common shocks, the Poisson risk process of order k, the Poisson negative binomial, the Polya-Aeppli, the Polya-Aeppli of order k among others. All of them with one or more types of polices. The relations between the numerical characteristics and distributions of the components of the risk processes are proven to be corollaries of the corresponding formulae of the Cramer-Lundberg risk model.