• Bernoulli
  • Volume 25, Number 2 (2019), 1355-1385.

Time-frequency analysis of locally stationary Hawkes processes

François Roueff and Rainer von Sachs

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Locally stationary Hawkes processes have been introduced in order to generalise classical Hawkes processes away from stationarity by allowing for a time-varying second-order structure. This class of self-exciting point processes has recently attracted a lot of interest in applications in the life sciences (seismology, genomics, neuro-science, …), but also in the modeling of high-frequency financial data. In this contribution, we provide a fully developed nonparametric estimation theory of both local mean density and local Bartlett spectra of a locally stationary Hawkes process. In particular, we apply our kernel estimation of the spectrum localised both in time and frequency to two data sets of transaction times revealing pertinent features in the data that had not been made visible by classical non-localised approaches based on models with constant fertility functions over time.

Article information

Bernoulli, Volume 25, Number 2 (2019), 1355-1385.

Received: April 2017
Revised: January 2018
First available in Project Euclid: 6 March 2019

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high frequency financial data locally stationary time series non-parametric kernel estimation self-exciting point processes time frequency analysis


Roueff, François; von Sachs, Rainer. Time-frequency analysis of locally stationary Hawkes processes. Bernoulli 25 (2019), no. 2, 1355--1385. doi:10.3150/18-BEJ1023.

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Supplemental materials

  • Supplement to “Time-frequency analysis of locally stationary Hawkes processes”. We provide some numerical experiments on transaction data illustrating our time frequency analysis for point processes. Moreover, we give detailed proofs of the uniform exponential moment bound for stable non-stationary Hawkes processes as stated in Proposition 6 and of the uniform approximation of the variance of a locally stationary Hawkes process as stated in Theorem 5.