Tempered fractional Hawkes process and its generalizations

Hawkes process (HP) is a point process with a conditionally dependent intensity function. This paper defines the generalized fractional Hawkes process (GFHP) by time-changing the HP with an inverse Lévy subordinator. This definition encompasses all potential (inverse Lévy) 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. Furthermore, we focus on the specific tempered fractional Hawkes process (TFHP), which is derived by time-changing the Hawkes process (HP) using an inverse-tempered stable subordinator. Our results generalize the fractional Hawkes process introduced in [20] to a tempered version, which exhibits semi-heavy-tailed decay. We derive the mean, the variance, covariance and the governing fractional difference-differential equations of the TFHP. Finally, we present simulated sample paths of the HP and the TFHP.