# Hawkes process

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Published: 2026-10-25T20:55:51.618Z

Updated: 2026-10-25T20:55:51.618Z

In probability theory and statistics, a Hawkes process is an age-dependent branching process driven by immigration from an inhomogeneous Poisson process.  The process, named after Alan G. Hawkes, is also called a self-exciting point process., i.e., the occurrence of an event increases the likelihood of another event occurring. It has arrivals at times

## Key facts

## Overview

In probability theory and statistics, a Hawkes process is an age-dependent branching process driven by immigration from an inhomogeneous Poisson process.  The process, named after Alan G. Hawkes, is also called a self-exciting point process., i.e., the occurrence of an event increases the likelihood of another event occurring. It has arrivals at times

0         \<

t

1

\<

t

2

\<

t

3

\<         ⋯

\{\\textstyle 0\<t\_\{1\}\<t\_\{2\}\<t\_\{3\}\<\\cdots \}

where the infinitesimal probability of an arrival during the time interval

\[         t         ,         t         +         d         t         )

\{\\textstyle \[t,t+dt)\}

is

λ

t

d         t         =

(

## Multivariate extension

In its seminal paper, Hawkes also considered mutually exciting processes which are now called multivariate Hawkes processes. The

n

\{\\textstyle n\}

point processes have arrival times denoted by

0         \<

t

1

i

\<

t

2

i

\<

t

3

i

\<         ⋯

\{\\textstyle 0\<t\_\{1\}^\{i\}\<t\_\{2\}^\{i\}\<t\_\{3\}^\{i\}\<\\cdots \}

for each type

i         =         1         ,         …         ,         n

## Density-dependent extension

Brémaud and Massoulié extended Hawkes processes to allow for a density-dependent birth-rate. They considered an infinitesimal probability of arrival of the following form:

λ

t

d         t         =         f

(

μ             (             t             )             +

∑

t

k

:

t

k

\<

t

ϕ             (             t             −

t

k

)

)

d         t         ,

## 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 Dynamic contagion process

## Further reading

Bacry, Emmanuel; Mastromatteo, Iacopo; Muzy, Jean-François (2015). "Hawkes processes in finance". arXiv:1502.04592 \[q-fin.TR\]. Rizoiu, Marian-Andrei; Lee, Young; Mishra, Swapnil; Xie, Lexing (2017). "A Tutorial on Hawkes Processes for Events in Social Media". arXiv:1708.06401 \[stat.ML\].

## Sources

- Hawkes process · Wikipedia: https://en.wikipedia.org/?curid=61323423

Source: Wikipedia contributors — https://en.wikipedia.org/?curid=61323423&oldid=1373523081

Licence: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/

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