Tempered Fractional Hawkes Process and Its Generalization

TL;DR

This paper introduces the tempered fractional Hawkes process (TFHP) by applying a time change with an inverse tempered stable subordinator, generalizing the fractional Hawkes process to include semi-heavy tailed decay, and also proposes the generalized fractional Hawkes process (GFHP) with broader inverse Le9vy subordinator time changes.

Contribution

It generalizes the fractional Hawkes process to a tempered version with semi-heavy tails and introduces the GFHP encompassing all inverse Le9vy subordinator time changes, deriving key properties and equations.

Findings

Derived mean, variance, and covariance of TFHP.

Established governing fractional difference-differential equations.

Explored distributional characteristics of GFHP.

Abstract

Hawkes process (HP) is a point process with a conditionally dependent intensity function. This paper defines the tempered fractional Hawkes process (TFHP) by time-changing the HP with an inverse tempered stable subordinator. We obtained results that generalize the fractional Hawkes process defined in Hainaut (2020) to a tempered version which has \textit{semi-heavy tailed} decay. We derive the mean, the variance, covariance and the governing fractional difference-differential equations of the TFHP. Additionally, we introduce the generalized fractional Hawkes process (GFHP) by time-changing the HP with the inverse L\'evy subordinator. This definition encompasses all potential (inverse L\'evy) time changes as specific instances. We also explore the distributional characteristics and the governing difference-differential equation of the one-dimensional distribution for the GFHP.