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Locally stationary Hawkes processes

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  • 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
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    References listed on IDEAS

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    9. 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.
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    Cited by:

    1. Xuefeng Gao & Xiang Zhou & Lingjiong Zhu, 2017. "Transform Analysis for Hawkes Processes with Applications in Dark Pool Trading," Papers 1710.01452, arXiv.org.
    2. 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.
    3. 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).
    4. Simon Clinet & Yoann Potiron, 2016. "Statistical inference for the doubly stochastic self-exciting process," Papers 1607.05831, arXiv.org, revised Jun 2017.
    5. 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).
    6. 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.

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