Hawkes process

In probability theory and statistics, a Hawkes process, named after Alan G. Hawkes, is a kind of self-exciting point process.{{Cite book |last1=Laub |first1=Patrick J. |last2=Lee |first2=Young |last3=Taimre |first3=Thomas |date=2021 |title=The Elements of Hawkes Processes |url=https://link.springer.com/book/10.1007/978-3-030-84639-8 |language=en-gb |doi=10.1007/978-3-030-84639-8|isbn=978-3-030-84638-1 |s2cid=245682002 }} It has arrivals at times $0 < t_1 < t_2 < t_3 < \cdots$ where the infinitesimal probability of an arrival during the time interval $[t,t+dt)$ is

: $\lambda_t \, dt = \left( \mu(t) + \sum_{t_i\,:\, t_i\,<\,t} \phi(t-t_i) \right) \, dt.$

The function $\mu$ is the intensity of an underlying Poisson process. The first arrival occurs at time $t_1$ and immediately after that, the intensity becomes $\mu(t) + \phi(t-t_1)$ , and at the time $t_2$ of the second arrival the intensity jumps to $\mu(t) + \phi(t-t_1) + \phi(t-t_2)$ and so on.{{Cite journal |last=Hawkes |first=Alan G. |date=1971 |title=Spectra of some self-exciting and mutually exciting point processes |url=https://doi.org/10.1093/biomet/58.1.83 |journal=Biometrika |volume=58 |issue=1 |pages=83–90 |doi=10.1093/biomet/58.1.83 |issn=0006-3444|url-access=subscription }}

During the time interval $(t_k, t_{k+1})$ , the process is the sum of $k+1$ independent processes with intensities $\mu(t), \phi(t-t_1), \ldots, \phi(t-t_k).$ The arrivals in the process whose intensity is $\phi(t-t_k)$ are the "daughters" of the arrival at time $t_k.$ 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.

Applications

Hawkes processes are used for statistical modeling of events in mathematical finance,{{Cite journal |last=Hawkes |first=Alan G. |date=2018 |title=Hawkes processes and their applications to finance: a review |url=https://doi.org/10.1080/14697688.2017.1403131 |journal=Quantitative Finance |volume=18 |issue=2 |pages=193–198 |doi=10.1080/14697688.2017.1403131 |s2cid=158619662 |issn=1469-7688|url-access=subscription }} epidemiology,{{Cite book |last1=Rizoiu |first1=Marian-Andrei |last2=Mishra |first2=Swapnil |last3=Kong |first3=Quyu |last4=Carman |first4=Mark |last5=Xie |first5=Lexing |date=2018 |chapter=SIR-Hawkes: Linking Epidemic Models and Hawkes Processes to Model Diffusions in Finite Populations |arxiv=1711.01679 |title=Proceedings of the 2018 World Wide Web Conference on World Wide Web - WWW '18 |pages=419–428 |doi=10.1145/3178876.3186108|s2cid=195346881 }} earthquake seismology,{{Cite journal |last1=Kwon |first1= Junhyeon |last2= Zheng |first2= Yingcai |last3=Jun|first3=Mikyoung|date=2023 |title=Flexible spatio-temporal Hawkes process models for earthquake occurrences |url=https://doi.org/10.1016/j.spasta.2023.100728|journal=Spatial Statistics |volume=54 |pages= 100728 |doi=10.1016/j.spasta.2023.100728|arxiv=2210.08053 |bibcode= 2023SpaSt..5400728K |s2cid= 252917746 }} and other fields in which a random event exhibits self-exciting behavior.{{Cite journal |last1=Tench |first1=Stephen |last2=Fry |first2=Hannah |last3=Gill |first3=Paul |date=2016 |title=Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army |url=https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/article/abs/spatiotemporal-patterns-of-ied-usage-by-the-provisional-irish-republican-army/3BC0D8552313F1BC893D5F6519079BB8 |journal=European Journal of Applied Mathematics |language=en |volume=27 |issue=3 |pages=377–402 |doi=10.1017/S0956792515000686 |s2cid=53692006 |issn=0956-7925}}{{cite arXiv |last1=Laub |first1=Patrick J. |last2=Taimre |first2=Thomas |last3=Pollett |first3=Philip K. |date=2015 |title=Hawkes Processes |class=math.PR |eprint=1507.02822 }}

Hawkes process

Applications

See also

References

Further reading