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Scaling limits for supercritical nearly unstable Hawkes processes

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  19 November 2025

Chenguang Liu ,
Liping Xu  and
An Zhang
Chenguang Liu*
Affiliation:
TU Delft
Liping Xu*
Affiliation:
Beihang University
An Zhang*
Affiliation:
Beihang University
*
*Postal address: Delft Institute of Applied Mathematics, EEMCS, TU Delft, 2628 Delft, The Netherlands. Email: C.Liu-13@tudelft.nl
**Postal address: School of Mathematical Sciences, Beihang University, PR China.
**Postal address: School of Mathematical Sciences, Beihang University, PR China.
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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 $({a_T-1})/{T{\textrm{e}}^{b_TTx}}$, the limit of the sequence of Hawkes processes is deterministic. Also, with another appropriate rescaling of $1/T^2$, 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.

Keywords

Hawkes processeslimit theoremspoint processes

MSC classification

Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems

Information

Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 16
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J.-F. (2013). Some limit theorems for Hawkes processes and application to financial statistics. Stoch. Process. Appl. 123, 2475–2499.CrossRefGoogle Scholar
Bacry, E. and Muzy, J.-F. (2014). Hawkes model for price and trades high-frequency dynamics. Quant. Finance 14, 1147–1166.CrossRefGoogle Scholar
Bacry, E. and Muzy, J.-F. (2016). First- and second-order statistics characterization of Hawkes processes and non-parametric estimation. IEEE Trans. Inf. Theory 62, 2184–2202.CrossRefGoogle Scholar
Bordenave, C. and Torrisi, G. L. (2007). Large deviations of Poisson cluster processes. Stoch. Models 23, 593–625.CrossRefGoogle Scholar
Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563–1588.CrossRefGoogle Scholar
Cattiaux, P., Colombani, L. and Costa, M. (2022). Limit theorems for Hawkes processes including inhibition. Stoch. Process. Appl. 149, 404–426.CrossRefGoogle Scholar
Cox, J. C., Ingersoll, J. E. Jr. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385–407.CrossRefGoogle Scholar
Crane, R. and Sornette, D. (2008). Robust dynamic classes revealed by measuring the response function of a social system. Proc. Nat. Acad. Sci. 105, 15649–15653.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, 2nd edn., Vol. I. Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, 2nd edn., Vol. II. Springer, New York.CrossRefGoogle Scholar
Dankel, T. Jr. (1991). On the distribution of the integrated square of the Ornstein–Uhlenbeck process. SIAM J. Appl. Math. 51, 568–574.CrossRefGoogle Scholar
Delattre, S. and Fournier, N. (2016). Statistical inference versus mean field limit for Hawkes processes. Electron. J. Statist. 10, 1223–1295.CrossRefGoogle Scholar
Delattre, S., Fournier, N. and Hoffmann, M. (2016). Hawkes processes on large networks. Ann. Appl. Prob. 26, 216–261.CrossRefGoogle Scholar
Filipović, D. (2009). Term-Structure Models. Springer, Berlin.CrossRefGoogle Scholar
Gao, F. and Zhu, L. (2018). Some asymptotic results for nonlinear Hawkes processes. Stoch. Process. Appl. 128, 4051–4077.CrossRefGoogle Scholar
Gao, F. and Zhu, L. (2021), Precise deviations for Hawkes processes. Bernoulli 27, 221–248.CrossRefGoogle Scholar
Garetto, M., Leonardi, E. and Torrisi, G. L. (2021). A time-modulated Hawkes process to model the spread of COVID-19 and the impact of countermeasures. Ann. Rev. Control 51, 551–563.CrossRefGoogle ScholarPubMed
Hansen, N. R., Reynaud-Bouret, P. and Rivoirard, V. (2015). Lasso and probabilistic inequalities for multivariate point processes. Bernoulli 21, 83–143.CrossRefGoogle Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.CrossRefGoogle Scholar
Hawkes, A. G. (2018). Hawkes processes and their applications to finance: A review. Quant. Finance 18, 193–198.CrossRefGoogle Scholar
Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493–503.CrossRefGoogle Scholar
Horst, U. and Xu, W. (2024). Functional limit theorems for Hawkes processes. Preprint, arXiv:2401.11495.Google Scholar
Horst, U., Xu, W. and Zhang, R. (2023). Convergence of heavy-tailed Hawkes processes and the microstructure of rough volatility. Preprint, arXiv:2312.08784.Google Scholar
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd edn. (North-Holland Math. Library 24). North-Holland, Amsterdam.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes (Grundlehren der mathematischen Wissenschaften 288). Springer, Berlin.CrossRefGoogle Scholar
Jaisson, T. and Rosenbaum, M. (2015). Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Prob. 25, 600–631.CrossRefGoogle Scholar
Karabash, D. and Zhu, L. (2015). Limit theorems for marked Hawkes processes with application to a risk model. Stoch. Models 31, 433–451.CrossRefGoogle Scholar
Liu, C. (2020). Statistical inference for a partially observed interacting system of Hawkes processes. Stoch. Process. Appl. 130, 5636–5694.CrossRefGoogle Scholar
Rizoi, M.-A., Mishra, S., Kong, Q., Carman, M. and Xie, L. (2018). SIR-Hawkes: Linking epidemic models and Hawkes processes to model diffusions in finite populations. In WWW ’18: Proc. 2018 World Wide Web Conf., pp. 419–428.CrossRefGoogle Scholar
Xu, W. (2024). Diffusion approximations for self-excited systems with applications to general branching processes. Ann. Appl. Prob. 34, 2650–2713.CrossRefGoogle Scholar
Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. J. Appl. Prob. 50, 760–771.CrossRefGoogle Scholar
Zhu, L. (2013). Moderate deviations for Hawkes processes. Statist. Prob. Lett. 83, 885–890.CrossRefGoogle Scholar
Zhu, L. (2013). Nonlinear Hawkes processes. PhD thesis, New York University.Google Scholar
Zhu, L. (2014). Process-level large deviations for nonlinear Hawkes point processes. Ann. Inst. H. Poincaré Prob. Statist. 50, 845–871.CrossRefGoogle Scholar
Zhu, L. (2015). Large deviations for Markovian nonlinear Hawkes processes. Ann. Appl. Prob. 25, 548–581.CrossRefGoogle Scholar