Exact and Asymptotic Analysis of General Multivariate Hawkes Processes and Induced Population Processes
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.
Keywords
Cite
@article{arxiv.2106.03560,
title = {Exact and Asymptotic Analysis of General Multivariate Hawkes Processes and Induced Population Processes},
author = {Raviar Karim and Roger J. A. Laeven and Michel Mandjes},
journal= {arXiv preprint arXiv:2106.03560},
year = {2021}
}