Cox process explained

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

\xi be a random measure.

A random measure

η is called a Cox process directed by \xi , if l L( η \mid \xi = \mu) is a Poisson process with intensity measure \mu .

Here,

l L( η \mid \xi = \mu) is the conditional distribution of η , given \{ \xi = \mu\} .

Laplace transform

If

η is a Cox process directed by \xi , then η has the Laplace transforml L _{η(f)=\exp} \left( - \int 1- \exp( - f(x)) \xi( dx)\right) f .

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)

Notes and References

1. Cox . D. R. . David Cox (statistician). Some Statistical Methods Connected with Series of Events . Journal of the Royal Statistical Society . 17 . 2 . 129–164 . 10.1111/j.2517-6161.1955.tb00188.x. 1955 .
2. Krumin . M. . Shoham . S. . 10.1162/neco.2009.08-08-847 . Generation of Spike Trains with Controlled Auto- and Cross-Correlation Functions . Neural Computation . 21 . 6 . 1642–1664 . 2009 . 19191596.
3. Lando . David. On cox processes and credit risky securities . 10.1007/BF01531332 . Review of Derivatives Research . 2 . 2–3 . 99–120. 1998 .