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Limit theorems for Markovian Hawkes processes with a large initial intensity

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  • Gao, Xuefeng
  • Zhu, Lingjiong

Abstract

Hawkes process is a simple point process that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we study the linear Hawkes process with an exponential exciting function in the asymptotic regime where the initial intensity of the Hawkes process is large. We derive limit theorems under this asymptotic regime as well as the regime when both the initial intensity and the time are large.

Suggested Citation

  • Gao, Xuefeng & Zhu, Lingjiong, 2018. "Limit theorems for Markovian Hawkes processes with a large initial intensity," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3807-3839.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:11:p:3807-3839
    DOI: 10.1016/j.spa.2017.12.001
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    References listed on IDEAS

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    1. Thibault Jaisson & Mathieu Rosenbaum, 2013. "Limit theorems for nearly unstable Hawkes processes," Papers 1310.2033, arXiv.org, revised Mar 2015.
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    3. José Da Fonseca & Riadh Zaatour, 2014. "Hawkes Process: Fast Calibration, Application to Trade Clustering, and Diffusive Limit," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 34(6), pages 548-579, June.
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    5. repec:dau:papers:123456789/5478 is not listed on IDEAS
    6. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    7. Gourieroux, Christian & Jasiak, Joanna & Le Fol, Gaelle, 1999. "Intra-day market activity," Journal of Financial Markets, Elsevier, vol. 2(3), pages 193-226, August.
    8. Zhu, Lingjiong, 2013. "Moderate deviations for Hawkes processes," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 885-890.
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    Cited by:

    1. Seol, Youngsoo, 2019. "Limit theorems for an inverse Markovian Hawkes process," Statistics & Probability Letters, Elsevier, vol. 155(C), pages 1-1.
    2. Wang, Haixu, 2022. "Limit theorems for a discrete-time marked Hawkes process," Statistics & Probability Letters, Elsevier, vol. 184(C).
    3. Horst, Ulrich & Xu, Wei, 2021. "Functional limit theorems for marked Hawkes point measures," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 94-131.
    4. Li, Bo & Pang, Guodong, 2022. "Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regime," Stochastic Processes and their Applications, Elsevier, vol. 143(C), pages 285-339.
    5. Youngsoo Seol, 2023. "Large Deviations for Hawkes Processes with Randomized Baseline Intensity," Mathematics, MDPI, vol. 11(8), pages 1-10, April.
    6. Raviar Karim & Roger J. A. Laeven & Michel Mandjes, 2021. "Exact and Asymptotic Analysis of General Multivariate Hawkes Processes and Induced Population Processes," Papers 2106.03560, arXiv.org.
    7. Ulrich Horst & Wei Xu, 2024. "Functional Limit Theorems for Hawkes Processes," Papers 2401.11495, arXiv.org, revised Nov 2024.
    8. Youngsoo Seol, 2022. "Non-Markovian Inverse Hawkes Processes," Mathematics, MDPI, vol. 10(9), pages 1-12, April.

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