A Cox process also known as a doubly stochastic Poisson process or mixed Poisson process, is a stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

 which is a generalization of a Poisson process

Poisson process

A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...

. In the case of Cox processes, the time-dependent intensity is a stochastic process which is separated from the Poisson process.

An example would be a spike train

Action potential

In physiology, an action potential is a short-lasting event in which the electrical membrane potential of a cell rapidly rises and falls, following a consistent trajectory. Action potentials occur in several types of animal cells, called excitable cells, which include neurons, muscle cells, and...

 of a sensory neuron with external stimulation. If the stimulation is a stochastic process and it modulates the firing rate (intensity function) of the neuron, then the spike train can be thought of as a realization of a Cox process.

Another example of the use of Cox processes is in financial mathematics for modeling Credit risk

Credit risk

Credit risk is an investor's risk of loss arising from a borrower who does not make payments as promised. Such an event is called a default. Other terms for credit risk are default risk and counterparty risk....

.