IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v153y2022icp57-78.html
   My bibliography  Save this article

Mean-field limits for non-linear Hawkes processes with excitation and inhibition

Author

Listed:
  • Pfaffelhuber, P.
  • Rotter, S.
  • Stiefel, J.

Abstract

We study a multivariate, non-linear Hawkes process ZN on the complete graph with N nodes. Each vertex is either excitatory (probability p) or inhibitory (probability 1−p). We take the mean-field limit of ZN, leading to a multivariate point process Z̄. If p≠12, we rescale the interaction intensity by N and find that the limit intensity process solves a deterministic convolution equation and all components of Z̄ are independent. In the critical case, p=12, we rescale by N1/2 and obtain a limit intensity, which solves a stochastic convolution equation and all components of Z̄ are conditionally independent.

Suggested Citation

  • Pfaffelhuber, P. & Rotter, S. & Stiefel, J., 2022. "Mean-field limits for non-linear Hawkes processes with excitation and inhibition," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 57-78.
    • Handle: RePEc:eee:spapps:v:153:y:2022:i:c:p:57-78
    DOI: 10.1016/j.spa.2022.07.006
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S030441492200165X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2022.07.006?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Ofer Yizhar & Lief E. Fenno & Matthias Prigge & Franziska Schneider & Thomas J. Davidson & Daniel J. O’Shea & Vikaas S. Sohal & Inbal Goshen & Joel Finkelstein & Jeanne T. Paz & Katja Stehfest & Roman, 2011. "Neocortical excitation/inhibition balance in information processing and social dysfunction," Nature, Nature, vol. 477(7363), pages 171-178, September.
    2. Chevallier, J. & Duarte, A. & Löcherbach, E. & Ost, G., 2019. "Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 1-27.
    3. Volker Pernice & Benjamin Staude & Stefano Cardanobile & Stefan Rotter, 2011. "How Structure Determines Correlations in Neuronal Networks," PLOS Computational Biology, Public Library of Science, vol. 7(5), pages 1-14, May.
    4. Chevallier, Julien, 2017. "Mean-field limit of generalized Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3870-3912.
    5. Chevallier, Julien & Ost, Guilherme, 2020. "Fluctuations for spatially extended Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5510-5542.
    6. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    7. Ditlevsen, Susanne & Löcherbach, Eva, 2017. "Multi-class oscillating systems of interacting neurons," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1840-1869.
    8. Felipe Gerhard & Moritz Deger & Wilson Truccolo, 2017. "On the stability and dynamics of stochastic spiking neuron models: Nonlinear Hawkes process and point process GLMs," PLOS Computational Biology, Public Library of Science, vol. 13(2), pages 1-31, February.
    9. Alan G. Hawkes, 2018. "Hawkes processes and their applications to finance: a review," Quantitative Finance, Taylor & Francis Journals, vol. 18(2), pages 193-198, February.
    10. Heesen, Sophie & Stannat, Wilhelm, 2021. "Fluctuation limits for mean-field interacting nonlinear Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 139(C), pages 280-297.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Duval, Céline & Luçon, Eric & Pouzat, Christophe, 2022. "Interacting Hawkes processes with multiplicative inhibition," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 180-226.
    2. 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.
    3. Schmutz, Valentin, 2022. "Mean-field limit of age and leaky memory dependent Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 39-59.
    4. Agathe-Nerine, Zoé, 2022. "Multivariate Hawkes processes on inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 86-148.
    5. Maxime Morariu-Patrichi & Mikko S. Pakkanen, 2017. "Hybrid marked point processes: characterisation, existence and uniqueness," Papers 1707.06970, arXiv.org, revised Oct 2018.
    6. Mercuri, Lorenzo & Perchiazzo, Andrea & Rroji, Edit, 2024. "A Hawkes model with CARMA(p,q) intensity," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 1-26.
    7. Kyungsub Lee, 2024. "Discrete Hawkes process with flexible residual distribution and filtered historical simulation," Papers 2401.13890, arXiv.org.
    8. Nikolaus Graf von Luckner & Rüdiger Kiesel, 2021. "Modeling Market Order Arrivals on the German Intraday Electricity Market with the Hawkes Process," JRFM, MDPI, vol. 14(4), pages 1-31, April.
    9. Heesen, Sophie & Stannat, Wilhelm, 2021. "Fluctuation limits for mean-field interacting nonlinear Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 139(C), pages 280-297.
    10. Simon Clinet & William T. M. Dunsmuir & Gareth W. Peters & Kylie-Anne Richards, 2021. "Asymptotic distribution of the score test for detecting marks in hawkes processes," Statistical Inference for Stochastic Processes, Springer, vol. 24(3), pages 635-668, October.
    11. Cao, Jingyi & Landriault, David & Li, Bin, 2020. "Optimal reinsurance-investment strategy for a dynamic contagion claim model," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 206-215.
    12. Pankaj Kumar, 2021. "Deep Hawkes Process for High-Frequency Market Making," Papers 2109.15110, arXiv.org.
    13. Fabio Vanni & David Lambert, 2023. "A detection analysis for temporal memory patterns at different time-scales," Papers 2309.12034, arXiv.org.
    14. Chevallier, J. & Duarte, A. & Löcherbach, E. & Ost, G., 2019. "Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 1-27.
    15. Yang Shen & Bin Zou, 2021. "Mean-Variance Portfolio Selection in Contagious Markets," Papers 2110.09417, arXiv.org.
    16. Kyungsub Lee, 2022. "Application of Hawkes volatility in the observation of filtered high-frequency price process in tick structures," Papers 2207.05939, arXiv.org, revised Sep 2024.
    17. Chevallier, Julien & Ost, Guilherme, 2020. "Fluctuations for spatially extended Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5510-5542.
    18. Bufalo, Michele & Ceci, Claudia & Orlando, Giuseppe, 2024. "Addressing the financial impact of natural disasters in the era of climate change," The North American Journal of Economics and Finance, Elsevier, vol. 73(C).
    19. Hui Qu & Tianyang Wang & Peng Shangguan & Mengying He, 2024. "Revisiting the puzzle of jumps in volatility forecasting: The new insights of high‐frequency jump intensity," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(2), pages 218-251, February.
    20. Hai-Chuan Xu & Wei-Xing Zhou, 2020. "Modeling aggressive market order placements with Hawkes factor models," PLOS ONE, Public Library of Science, vol. 15(1), pages 1-12, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:153:y:2022:i:c:p:57-78. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.