Cox process - Misplaced Pages

(Redirected from Doubly stochastic Poisson process) Poisson point 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.

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 $\xi$ be a random measure.

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

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

Laplace transform

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

{\mathcal {L}}_{\eta }(f)=\exp \left(-\int 1-\exp(-f(x))\;\xi (\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:10.1111/j.2517-6161.1955.tb00188.x.
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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Definition

Laplace transform

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References