Asymptotic Analysis of Discrete-Time Hawkes Process

Abstract

In a discrete-time setting, we consider an arrival process $\left\{ξ_n \, \middle| \, n = 1, 2, \ldots \right\}$, 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 $H_n = \sum_{i=1}^n ξ_i$, 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.

Figures (3)

• Figure 1: Feasible region for the limit function $\Gamma(t)$ bounded by $L(t)$ (in blue) and $U(t)$ (in red).
• Figure 2: Sample paths of the DTHP $\left\{H_n \, \middle| \, n=1,2,\ldots\right\}$ and the corresponding surplus process $\left\{U_n \, \middle| \, n=1,2,\ldots\right\}$.
• Figure 3: Monte Carlo simulations of the surplus process $\left\{U_n \, \middle| \, n=1,2,\ldots\right\}$ with $100,000$ sample paths.

Theorems & Definitions (45)

• Remark 2.1
• Remark 2.2
• Definition 2.1: Rate function
• Definition 2.2: Large Deviation Principle
• Theorem 2.1: Gärtner-Ellis Theorem
• proof
• Definition 2.3: Exponentially tight measure/distribution
• Theorem 2.2: Bryc's Theorem
• proof
• Definition 2.4: Well-separating functions