Hawkes process with tempered Mittag-Leffler kernel

TL;DR

This paper introduces a novel Hawkes process extension using a tempered Mittag-Leffler kernel, providing analytical insights and empirical comparisons to existing models.

Contribution

It generalizes previous Hawkes process models by incorporating a tempered Mittag-Leffler kernel and derives new analytical results for its properties.

Findings

Analytical expressions for the expectation of the conditional intensity

Expected number of events in the process

Empirical comparison with limiting special cases

Abstract

In this paper, we propose an extension of the Hawkes process by incorporating a kernel based on the tempered Mittag-Leffler distribution. This is the generalization of the work presented in [10]. We derive analytical results for the expectation of the conditional intensity and the expected number of events in the counting process. Additionally, we investigate the limiting behavior of the expectation of the conditional intensity. Finally, we present an empirical comparison of the studied process with its limiting special cases.