Hawkes process

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].