Английская Википедия:Hawkes process

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In probability theory and statistics, a Hawkes process, named after Alan G. Hawkes, is a kind of self-exciting point process. It has arrivals at times <math display=inline> 0 < t_1 < t_2 < t_3 < \cdots </math> where the infinitesimal probability of an arrival during the time interval <math display=inline> [t,t+dt) </math> is

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

The function <math display=inline>\mu</math> is the intensity of an underlying Poisson process. The first arrival occurs at time <math display=inline> t_1</math> and immediately after that, the intensity becomes <math display=inline> \mu(t) + \phi(t-t_1) </math>, and at the time <math display=inline> t_2</math> of the second arrival the intensity jumps to <math display=inline> \mu(t) + \phi(t-t_1) + \phi(t-t_2) </math> and so on.

During the time interval <math display=inline> (t_k, t_{k+1}) </math>, the process is the sum of <math display=inline> k+1</math> independent processes with intensities <math display=inline> \mu(t), \phi(t-t_1), \ldots, \phi(t-t_k). </math> The arrivals in the process whose intensity is <math display=inline> \phi(t-t_k) </math> are the "daughters" of the arrival at time <math display=inline> t_k.</math> The integral <math> \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, epidemiology, earthquake seismology, and other fields in which a random event exhibits self-exciting behavior.

See also

• Point process
• Self-oscillation

Further reading

• Шаблон:Cite arXiv
• Шаблон:Cite arXiv