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 [math]\displaystyle{ 0 \lt t_1 \lt t_2 \lt t_3 \lt \cdots }[/math] where the infinitesimal probability of an arrival during the time interval [math]\displaystyle{ [t,t+dt) }[/math] is

[math]\displaystyle{ \lambda_t \, dt = \left( \mu(t) + \sum_{t_i\,:\, t_i\,\lt \,t} \phi(t-t_i) \right) \, dt. }[/math]

The function [math]\displaystyle{ \mu }[/math] is the intensity of an underlying Poisson process. The first arrival occurs at time [math]\displaystyle{ t_1 }[/math] and immediately after that, the intensity becomes [math]\displaystyle{ \mu(t) + \phi(t-t_1) }[/math], and at the time [math]\displaystyle{ t_2 }[/math] of the second arrival the intensity jumps to [math]\displaystyle{ \mu(t) + \phi(t-t_1) + \phi(t-t_2) }[/math] and so on.^[2]

During the time interval [math]\displaystyle{ (t_k, t_{k+1}) }[/math], the process is the sum of [math]\displaystyle{ k+1 }[/math] independent processes with intensities [math]\displaystyle{ \mu(t), \phi(t-t_1), \ldots, \phi(t-t_k). }[/math] The arrivals in the process whose intensity is [math]\displaystyle{ \phi(t-t_k) }[/math] are the "daughters" of the arrival at time [math]\displaystyle{ t_k. }[/math] The integral [math]\displaystyle{ \int_0^\infty \phi(t)\,dt }[/math] 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,^[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) (in en-gb). The Elements of Hawkes Processes. doi:10.1007/978-3-030-84639-8. ISBN 978-3-030-84638-1. https://link.springer.com/book/10.1007/978-3-030-84639-8.
↑ 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. https://doi.org/10.1093/biomet/58.1.83.
↑ 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. https://doi.org/10.1080/14697688.2017.1403131.
↑ 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. doi:10.1145/3178876.3186108.
↑ Kwon, Junhyeon; Zheng, Yingcai; Jun, Mikyoung (2023). "Flexible spatio-temporal Hawkes process models for earthquake occurrences". Spatial Statistics 54: 100728. doi:10.1016/j.spasta.2023.100728. https://doi.org/10.1016/j.spasta.2023.100728.
↑ Tench, Stephen; Fry, Hannah; Gill, Paul (2016). "Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army" (in en). European Journal of Applied Mathematics 27 (3): 377–402. doi:10.1017/S0956792515000686. ISSN 0956-7925. 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.
↑ Laub, Patrick J.; Taimre, Thomas; Pollett, Philip K. (2015). "Hawkes Processes". arXiv:1507.02822 [math.PR].

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem
Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Karhunen–Loève_theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics System on Chip design Stochastic analysis Time series analysis Machine learning
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