Asymptotic Correlation Structure of Discounted Incurred But Not Reported Claims under Fractional Poisson Arrival Process

TL;DR

This paper analyzes the long-term correlation structure of discounted IBNR claims and queue length in systems with fractional Poisson arrivals, deriving recursive formulas and asymptotic behaviors.

Contribution

It introduces recursive methods to compute joint moments and explores the asymptotic correlation structure under fractional Poisson processes.

Findings

Derived recursive formulas for joint moments.

Established asymptotic covariance and correlation behaviors.

Analyzed special cases with exponential and Pareto delays.

Abstract

This paper studies the joint moments of a compound discounted renewal process observed at different times with each arrival removed from the system after a random delay. This process can be used to describe the aggregate (discounted) Incurred But Not Reported claims in insurance and also the total number of customers in an infinite server queue. It is shown that the joint moments can be obtained recursively in terms of the renewal density, from which the covariance and correlation structures are derived. In particular, the fractional Poisson process defined via the renewal approach is also considered. Furthermore, the asymptotic behaviour of covariance and correlation coefficient of the aforementioned quantities is analyzed as the time horizon goes to infinity. Special attention is paid to the cases of exponential and Pareto delays.