Time-changed Poisson processes of order $k$

TL;DR

This paper introduces and analyzes time-changed Poisson processes of order k, exploring their distributional properties, dependence structures, limit theorems, and applications in ruin theory, with specific focus on inverse Gaussian subordination.

Contribution

It develops new models of time-changed Poisson processes of order k, deriving their distributional, differential equations, and applying them to insurance ruin probabilities.

Findings

Derived distributional properties and limit theorems for TCPPoK-I and TCPPoK-II.

Established governing differential equations for specific subordinator cases.

Applied processes to model ruin probabilities in insurance.

Abstract

In this article, we study the Poisson process of order k (PPoK) time-changed with an independent L\'evy subordinator and its inverse, which we call respectively, as TCPPoK-I and TCPPoK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, we give a governing differential equation of ruin probability in insurance ruin using these processes. Finally, we present some simulated sample paths of both the processes.