Exact and Asymptotic Analysis of General Multivariate Hawkes Processes and Induced Population Processes

TL;DR

This paper provides a comprehensive analysis of multivariate Hawkes processes, deriving exact and asymptotic distributions, and enabling the evaluation of moments and tail behaviors in complex population models.

Contribution

It offers the first full characterization of joint distributions and tail behavior for general multivariate Hawkes processes with non-Markovian features.

Findings

Exact joint transform characterization achieved

Asymptotic tail behavior analyzed for heavy-tailed jumps

Method enables evaluation of moments and distributional properties

Abstract

This paper considers population processes in which general, not necessarily Markovian, multivariate Hawkes processes dictate the stochastic arrivals. We establish results to determine the corresponding time-dependent joint probability distribution, allowing for general intensity decay functions, general intensity jumps, and general sojourn times. We obtain an exact, full characterization of the time-dependent joint transform of the multivariate population process and its underlying intensity process in terms of a fixed-point representation and corresponding convergence results. We also derive the asymptotic tail behavior of the population process and its underlying intensity process in the setting of heavy-tailed intensity jumps. By exploiting the results we establish, arbitrary joint spatial-temporal moments and other distributional properties can now be readily evaluated using standard transform differentiation and inversion techniques, and we illustrate this in a few examples.