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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15 May 2021
A Correction to this paper has been published: https://doi.org/10.1007/s11009-021-09868-4
References
Bacry E, Mastromatteo I, Muzy JF (2015) Hawkes processes in finance. Market Microstructure and Liquidity 1(1):59
Cha JH, Finkelstein M (2012) Information-based thinning of point processes and its application to shock models. J Stat Plan Inference 142:2345–2350
Daley DJ, Vere-Jones D (2008) An introduction to the theory of point processes, vol II: general theory and structure. Springer, New York. 2nd revised and extended ed edition
Delattre S, Fournier N, Hoffmannm M (2016) Hawkes processes on large networks. Ann Appl Probab 26(1):216–261
Foschi R, Lilla F, Mancini C (2019) Warnings about future jumps: properties of the exponential hawkes model. Available at SSRN: https://ssrn.com/abstract=3459443 or https://doi.org/10.2139/ssrn.3459443
Hawkes AG (1971) Spectra of some self-exciting and mutually exciting point processes. Biometrika 58(1):83–90
Hawkes AG (2018) Hawkes processes and their applications to finance: a review. Quant Finan 18(2):193–198
Karr A (1991) Point processes and their statistical inference. CRC Press
Rangan A, Grace RE (1988) A non-Markov model for the optimum replacement of self-repairing systems subject to shocks. J Appl Probab 25(2):375–382
Rasmussen JG (2011) Lecture notes: temporal point processes and the conditional intensity function. arXiv:1806.00221
Reynaud-Bouret P, Rivoirard V, Tuleau-Malot C (2013) Inference of functional connectivity in neurosciences via Hawkes processes. In: 2013 IEEE global conference on signal and information processing , pp 317–320
Zhuang J, Ogata Y, Vere-Jones D (2002) Stochastic declustering of space-time earthquake occurrences. J Am Stat Assoc 97(458):369–380
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.
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Open access funding provided by Università di Pisa within the CRUI-CARE Agreement.
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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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DOI: https://doi.org/10.1007/s11009-020-09833-7