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 be a random measure.
A random measure is called a Cox process directed by , if is a Poisson process with intensity measure .
Here, is the conditional distribution of , given .
If is a Cox process directed by , then has the Laplace transform
for any positive, measurable function .