A Hawkes Microfoundation for Multitype Inverse Gaussian Subordinators

2026-10-07

AI Summary

This study addresses the lack of a microfoundation based on multi-type inverse Gaussian random clocks in existing event-level Hawkes processes, where high-intensity assumptions limit theoretical applicability. To overcome this, it introduces multi-type inverse Gaussian subordinators and replaces the high-intensity assumption with a weak cumulative activity condition. The analysis integrates near-critical branching processes, first-passage representations of Brownian additive fields, and Riccati identification techniques. This work proves that near-critical multivariate Hawkes processes converge to pure-jump subordinators, elucidating cluster mechanisms and establishing connections with existing scaling theories. Furthermore, it achieves strong convergence of counting processes and compensators, extending scalar results to finite-variance age-dependent branching clusters. By recovering boundary model results and resolving the uniqueness problem for coupled limits, this research significantly advances the theoretical framework of self-exciting point processes.

Abstract

We provide an event-level Hawkes microfoundation for a multitype inverse-Gaussian stochastic clock. We show that the event counts and integrated intensities of nearly critical multivariate linear Hawkes processes converge jointly to a multivariate pure-jump subordinator when reproduction delays have finite mean and immigration is balanced so that rare large families remain visible. The dependence among the limiting coordinates is inherited from microscopic cross-excitation, and the limit admits a Brownian additive-field first-passage representation. We call this process the multitype inverse-Gaussian subordinator. Its diagonal case consists of independent classical inverse-Gaussian subordinators, with the univariate model as a further special case. The square-root specialization recovers the inverse-Gaussian clock obtained at the boundary of hyper-rough square-root models by Abi Jaber--Attal--Rosenbaum (\textit{Ann. Appl. Probab.} \textbf{36}(4): 3635--3660, 2026). Our cluster proof exposes the underlying mechanism: finite-variance near-critical branching creates rare but macroscopic families, while their internal timing disappears on the observation scale, so each family becomes a jump. We also clarify the relation with the multivariate Hawkes scaling theory of Xu (arXiv:2412.14459): its diagonal-atom condition yields the diagonal specialization, whereas the genuinely coupled limit requires a separate uniqueness argument. In addition, under a common tilted-stability condition, we replace the high-intensity assumption in that theory by a weaker accumulated-activity condition and provide the Riccati identification needed when the potential has an atom. We strengthen convergence of the count and compensator and extend the scalar conclusion to finite-variance age-dependent branching clusters.

Problem

• Hawkes process
• inverse-Gaussian subordinator
• microfoundation
• scaling limit
• branching process

Innovation

• Hawkes process
• inverse-Gaussian subordinator
• near-critical branching
• scaling limit
• microfoundation