A fractional Hawkes process II: Further characterization of the process

TL;DR

This paper provides a detailed analysis of a fractional Hawkes process with Mittag-Leffler kernel, including analytical results, validation algorithms, and distribution derivations through simulations.

Contribution

It introduces new analytical characterizations of a fractional Hawkes process with Mittag-Leffler kernel and validates these with numerical algorithms and Monte Carlo simulations.

Findings

Analytical expressions for expected intensity and event count.

Validated algorithms for Laplace transform inversion.

Monte Carlo simulations for full event distribution.

Abstract

We characterize a Hawkes point process with kernel proportional to the probability density function of Mittag-Leffler random variables. This kernel decays as a power law with exponent $\beta +1 \in (1,2]$. Several analytical results can be proved, in particular for the expected intensity of the point process and for the expected number of events of the counting process. These analytical results are used to validate algorithms that numerically invert the Laplace transform of the expected intensity as well as Monte Carlo simulations of the process. Finally, Monte Carlo simulations are used to derive the full distribution of the number of events. The algorithms used for this paper are available at {\tt https://github.com/habyarimanacassien/Fractional-Hawkes}.