# Cox process

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{{Short description|Poisson point process}}
In [probability theory](/source/probability_theory), a '''Cox process''', also known as a '''doubly stochastic Poisson process''' is a [point process](/source/point_process) which is a generalization of a [Poisson process](/source/Poisson_process) where the intensity that varies across the underlying mathematical space (often space or time) is itself a stochastic process. The process is named after the [statistician](/source/statistician) [David Cox](/source/David_Cox_(statistician)), who first published the model in 1955.<ref>{{Cite journal | last1 = Cox | first1 = D. R. | author-link = David Cox (statistician)| title = Some Statistical Methods Connected with Series of Events | journal = Journal of the Royal Statistical Society | volume = 17 | issue = 2 | pages = 129–164 | doi =  10.1111/j.2517-6161.1955.tb00188.x| year = 1955 }}</ref>

Cox processes are used to generate simulations of [spike train](/source/spike_train)s (the sequence of action potentials generated by a [neuron](/source/neuron)),<ref>{{Cite journal | last1 = Krumin | first1 = M. | last2 = Shoham | first2 = S. | doi = 10.1162/neco.2009.08-08-847 | title = Generation of Spike Trains with Controlled Auto- and Cross-Correlation Functions | journal = Neural Computation | volume = 21 | issue = 6 | pages = 1642–1664 | year = 2009 | pmid =  19191596}}</ref> and also in [financial mathematics](/source/financial_mathematics) where they produce a "useful framework for modeling prices of financial instruments in which [credit risk](/source/credit_risk) is a significant factor."<ref>{{Cite journal | last1 = Lando | first1 = David| title = On cox processes and credit risky securities | doi = 10.1007/BF01531332 | journal = Review of Derivatives Research | volume = 2 | issue = 2–3 | pages = 99–120| year = 1998 }}</ref>

== Definition ==
Let <math> \xi </math> be a [random measure](/source/random_measure).

A random measure <math> \eta </math> is called a Cox process directed by <math> \xi </math>, if <math> \mathcal L(\eta \mid \xi=\mu) </math> is a [Poisson process](/source/Poisson_process) with [intensity measure](/source/intensity_measure) <math> \mu </math>.

Here, <math> \mathcal L(\eta \mid \xi=\mu) </math> is the conditional distribution of <math> \eta </math>, given <math> \{ \xi=\mu\} </math>.

== Laplace transform ==
If <math> \eta </math> is a Cox process directed by <math> \xi </math>, then <math> \eta </math> has the [Laplace transform](/source/Laplace_transform)
:<math> \mathcal L_\eta(f)=\exp \left(- \int 1-\exp(-f(x))\; \xi(\mathrm dx)\right) </math>

for any positive, [measurable function](/source/measurable_function) <math> f </math>.

==See also==
* [Poisson hidden Markov model](/source/Poisson_hidden_Markov_model)
* [Doubly stochastic model](/source/Doubly_stochastic_model)
* [Inhomogeneous Poisson process](/source/Inhomogeneous_Poisson_process), where ''λ''(''t'') is restricted to a deterministic function
* [Ross's conjecture](/source/Ross's_conjecture)
* [Gaussian process](/source/Gaussian_process)
* [Mixed Poisson process](/source/Mixed_Poisson_process)
* [Intensity of counting processes](/source/Intensity_of_counting_processes)

==References==
;Notes
{{Reflist}}
;Bibliography
* [Cox, D. R.](/source/David_Cox_(statistician)) and [Isham, V.](/source/Valerie_Isham) ''[Point Processes](/source/Point_Processes)'', London: Chapman & Hall, 1980 {{ISBN|0-412-21910-7}}
* Donald L. Snyder and Michael I. Miller ''Random Point Processes in Time and Space'' Springer-Verlag, 1991 {{ISBN|0-387-97577-2}} (New York) {{ISBN|3-540-97577-2}} (Berlin)

{{Stochastic processes}}

Category:Poisson point processes

{{statistics-stub}}

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