Tempered Fractional Hawkes Process and Its Generalization

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.