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Scaling limits for supercritical nearly unstable Hawkes processes

Journal Article (2025)
Author(s)

C. Liu (TU Delft - Transport Engineering and Logistics)

Liping Xu (Beihang University)

Arno Zhang (Beihang University)

Research Group
Transport Engineering and Logistics
DOI related publication
https://doi.org/10.1017/jpr.2025.10047
More Info
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Publication Year
2025
Language
English
Research Group
Transport Engineering and Logistics
Bibliographical Note
Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/publishing/publisher-deals Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.@en
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Abstract

We investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has $L^1$ norm strictly greater than 1 and close to 1 as time goes to infinity. We find that the scaling size determines the scaling behavior of the processes as in Jaisson and Rosenbaum (2015). Specifically, after a suitable rescale of (Formula presented), the limit of the sequence of Hawkes processes is deterministic. Also, with another appropriate rescaling of 1/T2 , the sequence converges in law to an integrated Cox–Ingersoll–Ross-like process. This theoretical result may apply to model the recent COVID-19 outbreak in epidemiology and phenomena in social networks.

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File under embargo until 19-05-2026