Cox process

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 ${\displaystyle \xi }$ be a random measure.

A random measure ${\displaystyle \eta }$ is called a Cox process directed by ${\displaystyle \xi }$, if ${\displaystyle {\mathcal {L}}(\eta \mid \xi =\mu )}$ is a Poisson process with intensity measure ${\displaystyle \mu }$.

Here, ${\displaystyle {\mathcal {L}}(\eta \mid \xi =\mu )}$ is the conditional distribution of ${\displaystyle \eta }$, given ${\displaystyle \{\xi =\mu \}}$.

Laplace transform

If ${\displaystyle \xi }$ is a Cox process directed by ${\displaystyle \eta }$, then ${\displaystyle \xi }$ has the Laplace transform

${\displaystyle {\mathcal {L}}_{\xi }(f)=\exp \left(-\int 1-\exp(-f(x))\;\eta (\mathrm {d} x)\right)}$

for any positive, measurable function ${\displaystyle f}$.

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

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

Bibliography

• Cox, D. R. and Isham, V. Point Processes, London: Chapman & Hall, 1980 ISBN 0-412-21910-7
• Donald L. Snyder and Michael I. Miller Random Point Processes in Time and Space Springer-Verlag, 1991 ISBN 0-387-97577-2 (New York) ISBN 3-540-97577-2 (Berlin)