IDEAS home Printed from https://ideas.repec.org/a/spr/jagbes/v30y2025i1d10.1007_s13253-024-00644-8.html
   My bibliography  Save this article

Expectations of Linear and Nonlinear Hawkes Processes Using a Field-Theoretical Approach

Author

Listed:
  • Lirong Cui

    (Qingdao University)

  • Didier Sornette

    (Southern University of Science and Technology)

Abstract

Moments play a crucial role for understanding the mathematical properties and practical applications of Hawkes processes. Here, we derive expectations of Hawkes processes and their intensity functions using a recently introduced Markovian embedding of (generally non-Markovian) linear and nonlinear Hawkes processes via a field-theoretical approach. The necessary and sufficient conditions for the stability of the Hawkes processes are also given by using the expectations of intensity functions directly via some matrix manipulations. Two kinds of Hawkes processes are considered, the standard linear Hawkes process with non-Markovian memory function expressed as a finite sum of exponentials, and the nonlinear Hawkes process with an intensity function that is quadratic as a function of an internal variable (“tension”) itself expressed as the sum over all past events with memory function given as a finite sum of exponentials and with zero mean random marks. All results obtained for the quadratic Hawkes processes are new contributions to the literature. The results obtained for linear Hawkes processes recover already known conclusions, while providing a novel alternative approach to existing methods. The matrix method presented in this paper gives a new way for finding the necessary and sufficient conditions for the stability of Hawkes processes.

Suggested Citation

  • Lirong Cui & Didier Sornette, 2025. "Expectations of Linear and Nonlinear Hawkes Processes Using a Field-Theoretical Approach," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 30(1), pages 63-88, March.
  • Handle: RePEc:spr:jagbes:v:30:y:2025:i:1:d:10.1007_s13253-024-00644-8
    DOI: 10.1007/s13253-024-00644-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13253-024-00644-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s13253-024-00644-8?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.

    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:spr:jagbes:v:30:y:2025:i:1:d:10.1007_s13253-024-00644-8. 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.

    We have no bibliographic references for this item. You can help adding them by using 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.