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Measuring Discrepancies Between Poisson and Exponential Hawkes Processes

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  • Rachele Foschi

    (Università degli Studi di Pisa)

Abstract

Poisson processes are widely used to model the occurrence of similar and independent events. However they turn out to be an inadequate tool to describe a sequence of (possibly differently) interacting events. Many phenomena can be modelled instead by Hawkes processes. In this paper we aim at quantifying how much a Hawkes process departs from a Poisson one with respect to different aspects, namely, the behaviour of the stochastic intensity at jump times, the cumulative intensity and the interarrival times distribution. We show how the behaviour of Hawkes processes with respect to these three aspects may be very irregular. Therefore, we believe that developing a single measure describing them is not efficient, and that, instead, the departure from a Poisson process with respect to any different aspect should be separately quantified, by means of as many different measures. Key to defining these measures will be the stochastic intensity and the integrated intensity of a Hawkes process, whose properties are therefore analysed before introducing the measures. Such quantities can be also used to detect mistakes in parameters estimation.

Suggested Citation

  • Rachele Foschi, 2021. "Measuring Discrepancies Between Poisson and Exponential Hawkes Processes," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 219-239, March.
    • Handle: RePEc:spr:metcap:v:23:y:2021:i:1:d:10.1007_s11009-020-09833-7
    DOI: 10.1007/s11009-020-09833-7
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    References listed on IDEAS

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    1. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    2. Zhuang J. & Ogata Y. & Vere-Jones D., 2002. "Stochastic Declustering of Space-Time Earthquake Occurrences," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 369-380, June.
    3. Alan G. Hawkes, 2018. "Hawkes processes and their applications to finance: a review," Quantitative Finance, Taylor & Francis Journals, vol. 18(2), pages 193-198, February.
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