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.

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."

Definition
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 $$ \{ \xi=\mu\} $$.

Laplace transform
If $$ \eta $$ is a Cox process directed by $$ \xi $$, then $$ \eta $$ has the Laplace transform
 * $$ \mathcal L_\eta(f)=\exp \left(- \int 1-\exp(-f(x))\; \xi(\mathrm dx)\right) $$

for any positive, measurable function $$ f $$.