Cox process

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