A Fractional Hawkes process

Abstract

We modify ETAS models by replacing the Pareto-like kernel proposed by Ogata with a Mittag-Leffler type kernel. Provided that the kernel decays as a power law with exponent $\beta +1\in (1,2]$, this replacement has the advantage that the Laplace transform of the Mittag-Leffler function is known explicitly, leading to simpler calculation of relevant quantities.

13 pages, 3 figures, submitted to Nonlocal and Fractional Operators, Proceedings of a Conference held in Rome (12-13 April 2019), Festschrift in honour of Renato Spigler, published in SEMA SIMAI Springer Series and edited by Roberto Garrappa (University of Bari), Francesco Mainardi (University of Bologna) and Luisa Beghin (University of Rome)