Aggregate claims when their sizes and arrival times are dependent and governed by a general point process

TL;DR

This paper introduces a comprehensive method for analyzing aggregate insurance claims with dependent sizes and arrival times governed by a broad class of point processes, extending classical models and enabling complex dependence structures.

Contribution

It develops a general analytical framework for aggregate claims with dependent claim sizes and arrival times driven by a wide class of point processes, including new dependence structures.

Findings

Unified approach for various point process models

Closed-form formulas for specific cases

Potential for advanced future research

Abstract

We suggest a general method for analyzing aggregate insurance claims that arrive according to a very general point process, known in the literature as the order statistic point process, which includes as special cases the classical compound Poisson and the Sparre Andersen models, among many others. We also allow for the process to govern claim sizes via a general dependence structure that relates claim sizes to claim and/or inter-claim times. The obtained general results are supplemented with special closed-form illustrative formulas, that also exemplify the potential for future research in the area.