Fractional Hawkes processes

Hawkes processes have a self-excitation mechanism used for modeling the clustering of events observed in natural or social phenomena. In the first part of this article, we find the forward differential equations ruling the probability density function and the Laplace's transform of the intensity of a Hawkes process, with an exponential decaying kernel. In the second part, we study the properties of the fractional version of this process. The fractional Hawkes process is obtained by subordinating the point process with the inverse of a α-stable Lévy process. This process is not Markov but the probability density function of its intensity is solution of a fractional Fokker-Planck equation. Finally, we present closed form expressions for moments and autocovariance of the fractional intensity.