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 time-dependent intensity 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]

1 Definition

2 Laplace transform

3 See also

4 References

Definition

Let $xi$ be a random measure.

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

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

Laplace transform

If $xi$ is a Cox process directed by $eta$ , then $xi$ has got 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 $f$ .

References

Notes

^ Cox, D. R. (1955). "Some Statistical Methods Connected with Series of Events". Journal of the Royal Statistical Society. 17 (2): 129–164. doi:10.2307/2983950.

^ 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.

^ 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)

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Cox process

Contents

Definition

Laplace transform

See also

References

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