Asymptotic Analysis of Discrete-Time Hawkes Process

TL;DR

This paper analyzes the asymptotic behavior of the discrete-time Hawkes process, establishing large deviation principles and demonstrating its application in modeling insurance claims with self-exciting properties.

Contribution

It provides the first large deviation analysis for the discrete-time Hawkes process and applies it to insurance claim modeling.

Findings

Established large deviation principles for the process.

Demonstrated the process's applicability to insurance claims.

Provided asymptotic behavior insights for self-exciting processes.

Abstract

In a discrete-time setting, we consider an arrival process $\left\{\xi_n \, \middle| \, n = 1, 2, \ldots \right\}$, which models the occurrence of events, and a corresponding point process $\left\{H_n \, \middle| \, n = 1, 2, \ldots \right\}$, known as the discrete-time Hawkes process. These two stochastic processes are related by $H_n = \sum_{i=1}^n \xi_i$, and exhibit a self-exciting property. In particular, we study the limiting behavior of the arrival process and establish the Large Deviation Principle for the discrete-time Hawkes process. We also illustrate an application in which insurance claims are modeled using the discrete-time Hawkes process and analyze its behavior.