Measuring Discrepancies Between Poisson and Exponential Hawkes Processes
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Measuring Discrepancies Between Poisson and Exponential Hawkes Processes Rachele Foschi1 Received: 15 March 2019 / Revised: 10 July 2020 / Accepted: 23 October 2020 / © The Author(s) 2020
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. Keywords Stochastic intensity · Self-excitation · Inter-arrival times · Clusters Mathematics Subject Classification (2010) 60K99 · 60E99 · 65C50
1 Introduction A Poisson process models, by means of the unordered vector of its jump times, the occurrence of similar and independent events. This means that the jumps of a Poisson process are in a sense unexpected and the Poisson process, despite its mathematical tractability, is an inadequate tool when we believe that some connection exists among events; in particular when some events are caused by some previous events and it becomes therefore possible to predict or even to prevent them.
Rachele Foschi
[email protected] 1
Department of Economics and Management, Universit`a degli Studi di Pisa, Via Cosimo Ridolfi 10, 56124 Pisa, Italy
Methodology and Computing in Applied Probability
A widespread tool for modelling this kind of phenomenon is through the use of Hawkes processes. The name Hawkes processes is due to the seminal work (Hawkes 1971), setting the theoretical basis for the study of self-exciting processes (see also Daley and Vere-Jones 2008; Bacry et al. 2015 and references therein), that were actually already used in engineering and reliability theory (see e.g. Rangan and Grace 1988 and references therein). Currently Hawkes processes are applied in a number of fields: in geology, to earthquakes or volcanic eruptions, in biology, to population growth, spread of infections, neuronal activity, in computer science or social sciences, to networks and social interactions, and in finance, to order book dynamics, defaults and so on (see e.g. Zhuang et al. 2002; Reynaud-Bouret et al. 2013; Delattre et al. 2016; Hawkes 2018). The main fe
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