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 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 .
Laplace transform
If is a Cox process directed by , then has the Laplace transform
for any positive, measurable function .
See also
Poisson hidden Markov model
Doubly stochastic model
Inhomogeneous Poisson process, where λ(t) is restricted to a deterministic function Ross's conjecture Gaussian process Mixed Poisson process
References
Notes
Bibliography
Cox, D. R. and Isham, V. Point Processes, London: Chapman & Hall, 1980
Donald L. Snyder and Michael I. Miller Random Point Processes in Time and Space Springer-Verlag, 1991 (New York) (Berlin)