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Functional Limit Theorems for Marked Hawkes Point Measures

Author

Listed:
  • Ulrich Horst

    (Institut für Mathematik [Berlin] - TUB - Technical University of Berlin / Technische Universität Berlin)

  • Wei Xu

    (Institut für Mathematik [Berlin] - TUB - Technical University of Berlin / Technische Universität Berlin)

Abstract
This paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures and their corresponding shot noise processes. We prove that the normalized random measure can be approximated in distribution by the sum of a Gaussian wihte noise process plus an appropriate lifting map of a correlated one-dimensional Brownian motion. The Brownian results from the self-exiting arrivals of events. We apply our limit theorems for Hawkes point measures to analyze the population dynamics of budding microbes in a host.

Suggested Citation

  • Ulrich Horst & Wei Xu, 2019. "Functional Limit Theorems for Marked Hawkes Point Measures ," Working Papers hal-02443841, HAL.
    • Handle: RePEc:hal:wpaper:hal-02443841
    Note: View the original document on HAL open archive server: https://univ-lemans.hal.science/hal-02443841
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    File URL: https://univ-lemans.hal.science/hal-02443841/document
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    References listed on IDEAS

    as
    1. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
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    3. Thibault Jaisson & Mathieu Rosenbaum, 2013. "Limit theorems for nearly unstable Hawkes processes," Papers 1310.2033, arXiv.org, revised Mar 2015.
    4. Bacry, E. & Delattre, S. & Hoffmann, M. & Muzy, J.F., 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2475-2499.
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    6. Chavez-Demoulin, V. & McGill, J.A., 2012. "High-frequency financial data modeling using Hawkes processes," Journal of Banking & Finance, Elsevier, vol. 36(12), pages 3415-3426.
    7. Pakes, A. G., 1975. "Limit theorems for the integrals of some branching processes," Stochastic Processes and their Applications, Elsevier, vol. 3(1), pages 89-111, January.
    8. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Post-Print hal-01313994, HAL.
    9. Gabriele Stabile & Giovanni Luca Torrisi, 2010. "Risk Processes with Non-stationary Hawkes Claims Arrivals," Methodology and Computing in Applied Probability, Springer, vol. 12(3), pages 415-429, September.
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    Cited by:

    1. Ulrich Horst & Wei Xu, 2019. "The Microstructure of Stochastic Volatility Models with Self-Exciting Jump Dynamics," Papers 1911.12969, arXiv.org.

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