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Cox process

Poisson point process

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated September 6, 2026

In probability theory, a Cox process, also known as a doubly stochastic Poisson process, is a point process which is a generalization of a 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 David Cox, who first published the model in 1955.

Cox processes are used to generate simulations of spike trains (the sequence of action potentials generated by a neuron), and also in financial mathematics where they produce a "useful framework for modeling prices of financial instruments in which credit risk is a significant factor."

01Definition

Let \xi be a random measure.

A random measure \eta is called a Cox process directed by \xi, if {\mathcal {L}}(\eta \mid \xi =\mu ) is a Poisson process with intensity measure \mu.

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

02Laplace transform

If \eta is a Cox process directed by \xi, then \eta has the Laplace transform

{\mathcal {L}}_{\eta }(f)=\exp \left(-\int 1-\exp(-f(x))\;\xi (\mathrm {d} x)\right)

for any positive, measurable function f.

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Sources and credits

This article is adapted from the Wikipedia article “Cox process”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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