Cox process
Poisson point process From Wikipedia, the free encyclopedia
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]
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
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