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Second order statistics characterization of Hawkes processes and non-parametric estimation

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  • Emmanuel Bacry
  • Jean-Francois Muzy
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    Abstract

    We show that the jumps correlation matrix of a multivariate Hawkes process is related to the Hawkes kernel matrix by a system of Wiener-Hopf integral equations. A Wiener-Hopf argument allows one to prove that this system (in which the kernel matrix is the unknown) possesses a unique causal solution and consequently that the second-order properties fully characterize Hawkes processes. The numerical inversion of the system of integral equations allows us to propose a fast and efficient method to perform a non-parametric estimation of the Hawkes kernel matrix. We discuss the estimation error and provide some numerical examples. Applications to high frequency trading events in financial markets and to earthquakes occurrence dynamics are considered.

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    File URL: http://arxiv.org/pdf/1401.0903
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    Paper provided by arXiv.org in its series Papers with number 1401.0903.

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    Date of creation: Jan 2014
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    Handle: RePEc:arx:papers:1401.0903

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    Web page: http://arxiv.org/

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    1. E. Bacry & S. Delattre & M. Hoffmann & J. F. Muzy, 2011. "Modeling microstructure noise with mutually exciting point processes," Papers 1101.3422, arXiv.org.
    2. Mohler, G. O. & Short, M. B. & Brantingham, P. J. & Schoenberg, F. P. & Tita, G. E., 2011. "Self-Exciting Point Process Modeling of Crime," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 100-108.
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