IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v549y2020ics0378437120301096.html
   My bibliography  Save this article

Fractional Hawkes processes

Author

Listed:
  • Hainaut, Donatien

Abstract

Hawkes processes have a self-excitation mechanism used for modeling the clustering of events observed in natural or social phenomena. In the first part of this article, we find the forward differential equations ruling the probability density function and the Laplace’s transform of the intensity of a Hawkes process, with an exponential decaying kernel. In the second part, we study the properties of the fractional version of this process. The fractional Hawkes process is obtained by subordinating the point process with the inverse of a α-stable Lévy process. This process is not Markov but the probability density function of its intensity is solution of a fractional Fokker–Planck equation. Finally, we find closed form expressions for moments and autocovariance of the fractional intensity.

Suggested Citation

  • Hainaut, Donatien, 2020. "Fractional Hawkes processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 549(C).
    • Handle: RePEc:eee:phsmap:v:549:y:2020:i:c:s0378437120301096
    DOI: 10.1016/j.physa.2020.124330
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437120301096
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2020.124330?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. Dassios, Angelos & Zhao, Hongbiao, 2013. "Exact simulation of Hawkes process with exponentially decaying intensity," LSE Research Online Documents on Economics 51370, London School of Economics and Political Science, LSE Library.
    2. Yosihiko Ogata, 1998. "Space-Time Point-Process Models for Earthquake Occurrences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(2), pages 379-402, June.
    3. BAUWENS, Luc & HAUTSCH, Nikolaus, 2006. "Modelling financial high frequency data using point processes," LIDAM Discussion Papers CORE 2006080, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Hainaut, Donatien, 2020. "Credit risk modelling with fractional self-excited processes," LIDAM Discussion Papers ISBA 2020002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Enrico Scalas, 2006. "Five Years of Continuous-time Random Walks in Econophysics," Lecture Notes in Economics and Mathematical Systems, in: Akira Namatame & Taisei Kaizouji & Yuuji Aruka (ed.), The Complex Networks of Economic Interactions, pages 3-16, Springer.
    6. Magdziarz, Marcin, 2009. "Stochastic representation of subdiffusion processes with time-dependent drift," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3238-3252, October.
    7. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    8. Hainaut, Donatien & Goutte, Stephane, 2018. "A switching microstructure model for stock prices," LIDAM Discussion Papers ISBA 2018014, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. Donatien Hainaut & Franck Moraux, 2019. "A switching self-exciting jump diffusion process for stock prices," Annals of Finance, Springer, vol. 15(2), pages 267-306, June.
    10. Piryatinska, A. & Saichev, A.I. & Woyczynski, W.A., 2005. "Models of anomalous diffusion: the subdiffusive case," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 349(3), pages 375-420.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Dupret, Jean-Loup & Hainaut, Donatien, 2023. "A fractional Hawkes process for illiquidity modeling," LIDAM Discussion Papers ISBA 2023001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Dupret, Jean-Loup & Hainaut, Donatien, 2022. "A subdiffusive stochastic volatility jump model," LIDAM Discussion Papers ISBA 2022001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Hainaut, Donatien, 2021. "A fractional multi-states model for insurance," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 120-132.
    4. Habyarimana, Cassien & Aduda, Jane A. & Scalas, Enrico & Chen, Jing & Hawkes, Alan G. & Polito, Federico, 2023. "A fractional Hawkes process II: Further characterization of the process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 615(C).
    5. Antonov, A. & Leonidov, A. & Semenov, A., 2021. "Self-excited Ising game," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 561(C).
    6. Ketelbuters, John-John & Hainaut, Donatien, 2022. "CDS pricing with fractional Hawkes processes," European Journal of Operational Research, Elsevier, vol. 297(3), pages 1139-1150.
    7. Hainaut, Donatien, 2021. "A fractional multi-states model for insurance," LIDAM Discussion Papers ISBA 2021019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

    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. Hainaut, Donatien, 2019. "Fractional Hawkes processes," LIDAM Discussion Papers ISBA 2019016, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Ketelbuters, John John & Hainaut, Donatien, 2021. "Time-Consistent Evaluation of Credit Risk with Contagion," LIDAM Discussion Papers ISBA 2021004, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. 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.
    4. Hainaut, Donatien & Leonenko, Nikolai, 2020. "Option pricing in illiquid markets: a fractional jump-diffusion approach," LIDAM Discussion Papers ISBA 2020003, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Ketelbuters, John-John & Hainaut, Donatien, 2022. "CDS pricing with fractional Hawkes processes," European Journal of Operational Research, Elsevier, vol. 297(3), pages 1139-1150.
    6. Steffen Volkenand & Günther Filler & Martin Odening, 2020. "Price Discovery and Market Reflexivity in Agricultural Futures Contracts with Different Maturities," Risks, MDPI, vol. 8(3), pages 1-17, July.
    7. Cavaliere, Giuseppe & Lu, Ye & Rahbek, Anders & Stærk-Østergaard, Jacob, 2023. "Bootstrap inference for Hawkes and general point processes," Journal of Econometrics, Elsevier, vol. 235(1), pages 133-165.
    8. Foad Shokrollahi, 2016. "Subdiffusive fractional Brownian motion regime for pricing currency options under transaction costs," Papers 1612.06665, arXiv.org, revised Aug 2017.
    9. Gajda, Janusz & Magdziarz, Marcin, 2014. "Large deviations for subordinated Brownian motion and applications," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 149-156.
    10. Lucio Maria Calcagnile & Giacomo Bormetti & Michele Treccani & Stefano Marmi & Fabrizio Lillo, 2015. "Collective synchronization and high frequency systemic instabilities in financial markets," Papers 1505.00704, arXiv.org.
    11. Francine Gresnigt & Erik Kole & Philip Hans Franses, 2017. "Specification Testing in Hawkes Models," Journal of Financial Econometrics, Oxford University Press, vol. 15(1), pages 139-171.
    12. Anatoliy Swishchuk & Aiden Huffman, 2018. "General Compound Hawkes Processes in Limit Order Books," Papers 1812.02298, arXiv.org.
    13. Gu, Hui & Liang, Jin-Rong & Zhang, Yun-Xiu, 2012. "Time-changed geometric fractional Brownian motion and option pricing with transaction costs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(15), pages 3971-3977.
    14. Giada Adelfio & Arianna Agosto & Marcello Chiodi & Paolo Giudici, 2021. "Financial contagion through space-time point processes," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(2), pages 665-688, June.
    15. Riccardo Brignone & Carlo Sgarra, 2020. "Asian options pricing in Hawkes-type jump-diffusion models," Annals of Finance, Springer, vol. 16(1), pages 101-119, March.
    16. Martin Magris, 2019. "On the simulation of the Hawkes process via Lambert-W functions," Papers 1907.09162, arXiv.org.
    17. Marcello Rambaldi & Emmanuel Bacry & Fabrizio Lillo, 2016. "The role of volume in order book dynamics: a multivariate Hawkes process analysis," Papers 1602.07663, arXiv.org.
    18. Roger Martins & Dieter Hendricks, 2016. "The statistical significance of multivariate Hawkes processes fitted to limit order book data," Papers 1604.01824, arXiv.org, revised Apr 2016.
    19. Anatoliy Swishchuk, 2021. "Modelling of Limit Order Books by General Compound Hawkes Processes with Implementations," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 399-428, March.
    20. Marius Pfeuffer & Goncalo dos Reis & Greig smith, 2018. "Capturing Model Risk and Rating Momentum in the Estimation of Probabilities of Default and Credit Rating Migrations," Papers 1809.09889, arXiv.org, revised Feb 2020.

    More about this item

    Statistics

    Access and download statistics

    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:phsmap:v:549:y:2020:i:c:s0378437120301096. 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.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    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.