Cox process

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 1955.¹

Cox processes are used to generate simulations of spike trains (the sequence of action potentials generated by a neuron),² 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."³

Let ${\displaystyle \xi }$ be a random measure.

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

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

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

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

for any positive, measurable function ${\displaystyle f}$.