IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v126y2016i6p1710-1743.html
   My bibliography  Save this article

Locally stationary Hawkes processes

Author

Listed:
  • Roueff, François
  • von Sachs, Rainer
  • Sansonnet, Laure

Abstract

This paper addresses the generalization of stationary Hawkes processes in order to allow for a time-evolving second-order analysis. Motivated by the concept of locally stationary autoregressive processes, we apply however inherently different techniques to describe the time-varying dynamics of self-exciting point processes. In particular we derive a stationary approximation of the Laplace functional of a locally stationary Hawkes process. This allows us to define a local mean density function and a local Bartlett spectrum which can be used to compute approximations of first and second order moments of the process. We complete the paper by some insightful simulation studies.

Suggested Citation

  • Roueff, François & von Sachs, Rainer & Sansonnet, Laure, 2016. "Locally stationary Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1710-1743.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:6:p:1710-1743
    DOI: 10.1016/j.spa.2015.12.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414915003075
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2015.12.003?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. Ioane Muni Toke & Fabrizio Pomponio, 2012. "Modelling Trades-Through in a Limit Order Book Using Hawkes Processes," Post-Print hal-00745554, HAL.
    2. Bowsher, Clive G., 2007. "Modelling security market events in continuous time: Intensity based, multivariate point process models," Journal of Econometrics, Elsevier, vol. 141(2), pages 876-912, December.
    3. Toke, Ioane Muni & Pomponio, Fabrizio, 2012. "Modelling trades-through in a limit order book using hawkes processes," Economics - The Open-Access, Open-Assessment E-Journal (2007-2020), Kiel Institute for the World Economy (IfW Kiel), vol. 6, pages 1-23.
    4. Luc, BAUWENS & Nikolaus, HAUTSCH, 2006. "Modelling Financial High Frequency Data Using Point Processes," Discussion Papers (ECON - Département des Sciences Economiques) 2006039, Université catholique de Louvain, Département des Sciences Economiques.
    5. E. Bacry & S. Delattre & M. Hoffmann & J. F. Muzy, 2013. "Modelling microstructure noise with mutually exciting point processes," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 65-77, January.
    6. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Modelling microstructure noise with mutually exciting point processes," Post-Print hal-01313995, HAL.
    7. 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.
    8. Zhu, Lingjiong, 2013. "Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 544-550.
    9. Dahlhaus, R., 1996. "On the Kullback-Leibler information divergence of locally stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 139-168, March.
    10. Møller, Jesper & Torrisi, Giovanni Luca, 2007. "The pair correlation function of spatial Hawkes processes," Statistics & Probability Letters, Elsevier, vol. 77(10), pages 995-1003, June.
    11. Dahlhaus, Rainer, 2009. "Local inference for locally stationary time series based on the empirical spectral measure," Journal of Econometrics, Elsevier, vol. 151(2), pages 101-112, August.
    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. Roueff, Francois & von Sachs, Rainer, 2017. "Time-frequency analysis of locally stationary Hawkes processes," LIDAM Discussion Papers ISBA 2017005, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Simon Clinet & Yoann Potiron, 2016. "Statistical inference for the doubly stochastic self-exciting process," Papers 1607.05831, arXiv.org, revised Jun 2017.
    3. von Sachs, Rainer, 2019. "Spectral Analysis of Multivariate Time Series," LIDAM Discussion Papers ISBA 2019008, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Stindl, Tom & Chen, Feng, 2018. "Likelihood based inference for the multivariate renewal Hawkes process," Computational Statistics & Data Analysis, Elsevier, vol. 123(C), pages 131-145.
    5. Xuefeng Gao & Xiang Zhou & Lingjiong Zhu, 2017. "Transform Analysis for Hawkes Processes with Applications in Dark Pool Trading," Papers 1710.01452, arXiv.org.
    6. E A K Cohen & A J Gibberd, 2022. "Wavelet spectra for multivariate point processes [The spectral analysis of point processes]," Biometrika, Biometrika Trust, vol. 109(3), pages 837-851.

    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. Roueff, Francois & von Sachs, Rainer & Sansonnet, Laure, 2015. "Time-frequency analysis of locally stationary Hawkes processes," LIDAM Discussion Papers ISBA 2015011, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. 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.
    3. Dassios, Angelos & Jang, Jiwook & Zhao, Hongbiao, 2019. "A generalised CIR process with externally-exciting and self-exciting jumps and its applications in insurance and finance," LSE Research Online Documents on Economics 102043, London School of Economics and Political Science, LSE Library.
    4. Giacomo Bormetti & Lucio Maria Calcagnile & Michele Treccani & Fulvio Corsi & Stefano Marmi & Fabrizio Lillo, 2013. "Modelling systemic price cojumps with Hawkes factor models," Papers 1301.6141, arXiv.org, revised Mar 2013.
    5. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    6. Ioane Muni Toke & Nakahiro Yoshida, 2020. "Marked point processes and intensity ratios for limit order book modeling," Papers 2001.08442, arXiv.org.
    7. Hainaut, Donatien, 2016. "A bivariate Hawkes process for interest rate modeling," Economic Modelling, Elsevier, vol. 57(C), pages 180-196.
    8. Ioane Muni Toke & Nakahiro Yoshida, 2020. "Analyzing order flows in limit order books with ratios of Cox-type intensities," Post-Print hal-01799398, HAL.
    9. Dassios, Angelos & Zhao, Hongbiao, 2017. "A generalised contagion process with an application to credit risk," LSE Research Online Documents on Economics 68558, London School of Economics and Political Science, LSE Library.
    10. Bilodeau, Yann, 2020. "Deep limit order book events dynamics," Working Papers 20-4, HEC Montreal, Canada Research Chair in Risk Management.
    11. Kyungsub Lee, 2022. "Application of Hawkes volatility in the observation of filtered high-frequency price process in tick structures," Papers 2207.05939, arXiv.org.
    12. Ulrich Horst & Wei Xu, 2024. "Functional Limit Theorems for Hawkes Processes," Papers 2401.11495, arXiv.org.
    13. Swishchuk, Anatoliy & Zagst, Rudi & Zeller, Gabriela, 2021. "Hawkes processes in insurance: Risk model, application to empirical data and optimal investment," Insurance: Mathematics and Economics, Elsevier, vol. 101(PA), pages 107-124.
    14. Anatoliy Swishchuk & Aiden Huffman, 2018. "General Compound Hawkes Processes in Limit Order Books," Papers 1812.02298, arXiv.org.
    15. Giacomo Bormetti & Lucio Maria Calcagnile & Michele Treccani & Fulvio Corsi & Stefano Marmi & Fabrizio Lillo, 2015. "Modelling systemic price cojumps with Hawkes factor models," Quantitative Finance, Taylor & Francis Journals, vol. 15(7), pages 1137-1156, July.
    16. Angelos Dassios & Jiwook Jang & Hongbiao Zhao, 2019. "A Generalised CIR Process with Externally-Exciting and Self-Exciting Jumps and Its Applications in Insurance and Finance," Risks, MDPI, vol. 7(4), pages 1-18, October.
    17. Lee, Kyungsub & Seo, Byoung Ki, 2017. "Modeling microstructure price dynamics with symmetric Hawkes and diffusion model using ultra-high-frequency stock data," Journal of Economic Dynamics and Control, Elsevier, vol. 79(C), pages 154-183.
    18. Angelos Dassios & Hongbiao Zhao, 2017. "A Generalized Contagion Process With An Application To Credit Risk," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 1-33, February.
    19. 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.
    20. 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.

    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:126:y:2016:i:6:p:1710-1743. 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.