Hawkes process

In probability theory and statistics, a Hawkes process, named after Alan G. Hawkes, is a kind of self-exciting point process.^[1] It has arrivals at times ${\textstyle 0<t_{1}<t_{2}<t_{3}<\cdots }$ where the infinitesimal probability of an arrival during the time interval ${\textstyle [t,t+dt)}$ is

\lambda _{t}\,dt=\left(\mu (t)+\sum _{t_{i}\,:\,t_{i}\,<\,t}\phi (t-t_{i})\right)\,dt.

The function ${\textstyle \mu }$ is the intensity of an underlying Poisson process. The first arrival occurs at time ${\textstyle t_{1}}$ and immediately after that, the intensity becomes ${\textstyle \mu (t)+\phi (t-t_{1})}$ , and at the time ${\textstyle t_{2}}$ of the second arrival the intensity jumps to ${\textstyle \mu (t)+\phi (t-t_{1})+\phi (t-t_{2})}$ and so on.^[2]

During the time interval ${\textstyle (t_{k},t_{k+1})}$ , the process is the sum of ${\textstyle k+1}$ independent processes with intensities ${\textstyle \mu (t),\phi (t-t_{1}),\ldots ,\phi (t-t_{k}).}$ The arrivals in the process whose intensity is ${\textstyle \phi (t-t_{k})}$ are the "daughters" of the arrival at time ${\textstyle t_{k}.}$ The integral $\int _{0}^{\infty }\phi (t)\,dt$ is the average number of daughters of each arrival and is called the branching ratio. Thus viewing some arrivals as descendants of earlier arrivals, we have a Galton–Watson branching process. The number of such descendants is finite with probability 1 if branching ratio is 1 or less. If the branching ratio is more than 1, then each arrival has positive probability of having infinitely many descendants.

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Applications

Hawkes processes are used for statistical modeling of events in mathematical finance,^[3] epidemiology,^[4] earthquake seismology,^[5] and other fields in which a random event exhibits self-exciting behavior.^[6]^[7]

References

^ Laub, Patrick J.; Lee, Young; Taimre, Thomas (2021). The Elements of Hawkes Processes. doi:10.1007/978-3-030-84639-8. ISBN 978-3-030-84638-1. S2CID 245682002.
^ Hawkes, Alan G. (1971). "Spectra of some self-exciting and mutually exciting point processes". Biometrika. 58 (1): 83–90. doi:10.1093/biomet/58.1.83. ISSN 0006-3444.
^ Hawkes, Alan G. (2018). "Hawkes processes and their applications to finance: a review". Quantitative Finance. 18 (2): 193–198. doi:10.1080/14697688.2017.1403131. ISSN 1469-7688. S2CID 158619662.
^ Rizoiu, Marian-Andrei; Mishra, Swapnil; Kong, Quyu; Carman, Mark; Xie, Lexing (2018). "SIR-Hawkes: Linking Epidemic Models and Hawkes Processes to Model Diffusions in Finite Populations". Proceedings of the 2018 World Wide Web Conference on World Wide Web - WWW '18. pp. 419–428. arXiv:1711.01679. doi:10.1145/3178876.3186108. S2CID 195346881.
^ Kwon, Junhyeon; Zheng, Yingcai; Jun, Mikyoung (2023). "Flexible spatio-temporal Hawkes process models for earthquake occurrences". Spatial Statistics. 54: 100728. arXiv:2210.08053. Bibcode:2023SpaSt..5400728K. doi:10.1016/j.spasta.2023.100728. S2CID 252917746.
^ Tench, Stephen; Fry, Hannah; Gill, Paul (2016). "Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army". European Journal of Applied Mathematics. 27 (3): 377–402. doi:10.1017/S0956792515000686. ISSN 0956-7925. S2CID 53692006.
^ Laub, Patrick J.; Taimre, Thomas; Pollett, Philip K. (2015). "Hawkes Processes". arXiv:1507.02822 [math.PR].

v t e Stochastic processes
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This page was last edited on 14 June 2024, at 08:59

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