Markovian multivariate Hawkes population processes: Efficient evaluation of moments

TL;DR

This paper introduces a novel analytical framework for multivariate Hawkes processes that enables efficient computation of moments and covariances, along with insights into their asymptotic behavior.

Contribution

It provides a closed-form characterization of the joint transform and a nested matrix structure for explicit moment calculations, improving computational efficiency.

Findings

Explicit formulas for moments and covariances

Efficient computational methods demonstrated

Asymptotic behavior in nearly unstable regimes analyzed

Abstract

We provide probabilistic and computational results on Markovian multivariate Hawkes processes and induced population processes. By applying the Markov property, we characterize in closed form a joint transform, bijective to the probability distribution, of the population process and its underlying intensity process. We demonstrate a method that exploits the transform to obtain analytic expressions for transient and stationary multivariate moments of any order, as well as auto- and cross-covariances. We reveal a nested sequence of block matrices that yields the moments in explicit form and brings important computational advantages. We also establish the asymptotic behavior of the intensity of the multivariate Hawkes process in its nearly unstable regime, under a specific parameterization. In extensive numerical experiments, we analyze the computational complexity, accuracy and efficiency of the established results.