Measuring Discrepancies Between Poisson and Exponential Hawkes Processes

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.

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Acknowledgments

I would like to thank Cecilia Mancini and two anonymous Referees for valuable comments and suggestions, that have been very useful in improving the paper.

Funding

Open access funding provided by Università di Pisa within the CRUI-CARE Agreement.

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Correspondence to Rachele Foschi.

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Foschi, R. Measuring Discrepancies Between Poisson and Exponential Hawkes Processes. Methodol Comput Appl Probab 23, 219–239 (2021). https://doi.org/10.1007/s11009-020-09833-7

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Keywords

  • Stochastic intensity
  • Self-excitation
  • Inter-arrival times
  • Clusters

Mathematics Subject Classification (2010)

  • 60K99
  • 60E99
  • 65C50