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.

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. 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].

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	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Dyson Brownian motion Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes 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 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	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) 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
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Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
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 Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
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