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Expectations of Linear and Nonlinear Hawkes Processes Using a Field-Theoretical Approach

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  • Lirong Cui

    (Qingdao University)

  • Didier Sornette

    (Southern University of Science and Technology)

Abstract

Moments play a crucial role for understanding the mathematical properties and practical applications of Hawkes processes. Here, we derive expectations of Hawkes processes and their intensity functions using a recently introduced Markovian embedding of (generally non-Markovian) linear and nonlinear Hawkes processes via a field-theoretical approach. The necessary and sufficient conditions for the stability of the Hawkes processes are also given by using the expectations of intensity functions directly via some matrix manipulations. Two kinds of Hawkes processes are considered, the standard linear Hawkes process with non-Markovian memory function expressed as a finite sum of exponentials, and the nonlinear Hawkes process with an intensity function that is quadratic as a function of an internal variable (“tension”) itself expressed as the sum over all past events with memory function given as a finite sum of exponentials and with zero mean random marks. All results obtained for the quadratic Hawkes processes are new contributions to the literature. The results obtained for linear Hawkes processes recover already known conclusions, while providing a novel alternative approach to existing methods. The matrix method presented in this paper gives a new way for finding the necessary and sufficient conditions for the stability of Hawkes processes.

Suggested Citation

  • Lirong Cui & Didier Sornette, 2025. "Expectations of Linear and Nonlinear Hawkes Processes Using a Field-Theoretical Approach," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 30(1), pages 63-88, March.
    • Handle: RePEc:spr:jagbes:v:30:y:2025:i:1:d:10.1007_s13253-024-00644-8
    DOI: 10.1007/s13253-024-00644-8
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    References listed on IDEAS

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    1. V. Filimonov & D. Sornette, 2015. "Apparent criticality and calibration issues in the Hawkes self-excited point process model: application to high-frequency financial data," Quantitative Finance, Taylor & Francis Journals, vol. 15(8), pages 1293-1314, August.
    2. Gao, Fuqing & Zhu, Lingjiong, 2018. "Some asymptotic results for nonlinear Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4051-4077.
    3. Kiyoshi Kanazawa & Didier Sornette, 2021. "Ubiquitous power law scaling in nonlinear self-excited Hawkes processes," Papers 2102.00242, arXiv.org, revised Oct 2021.
    4. Zhongping Li & Lirong Cui, 2020. "Numerical method for means of linear Hawkes processes," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(15), pages 3681-3697, August.
    5. Vladimir Filimonov & Didier Sornette, 2012. "Quantifying reflexivity in financial markets: towards a prediction of flash crashes," Papers 1201.3572, arXiv.org, revised Apr 2012.
    6. A. Saichev & D. Sornette, 2011. "Generating functions and stability study of multivariate self-excited epidemic processes," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 83(2), pages 271-282, September.
    7. Vladimir Filimonov & Didier Sornette, 2012. "Quantifying Reflexivity in Financial Markets: Towards a Prediction of Flash Crashes," Swiss Finance Institute Research Paper Series 12-02, Swiss Finance Institute.
    8. Spencer Wheatley & Alexander Wehrli & Didier Sornette, 2019. "The endo–exo problem in high frequency financial price fluctuations and rejecting criticality," Quantitative Finance, Taylor & Francis Journals, vol. 19(7), pages 1165-1178, July.
    9. Alexander Saichev & Thomas Maillart & Didier Sornette, 2013. "Hierarchy of temporal responses of multivariate self-excited epidemic processes," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 86(4), pages 1-19, April.
    10. A. I. Saichev & D. Sornette, 2010. "Generation-by-generation dissection of the response function in long memory epidemic processes," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 75(3), pages 343-355, June.
    11. Lirong Cui & Bei Wu & Juan Yin, 2022. "Moments for Hawkes Processes with Gamma Decay Kernel Functions," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1565-1601, September.
    12. Filimonov, Vladimir & Bicchetti, David & Maystre, Nicolas & Sornette, Didier, 2014. "Quantification of the high level of endogeneity and of structural regime shifts in commodity markets," Journal of International Money and Finance, Elsevier, vol. 42(C), pages 174-192.
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