Cox process

Last updated January 26, 2022

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]

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 $f$ .

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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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[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.

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v t e Stochastic processes
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Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
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Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe
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List of topics Category

Cox process

Contents

Definition

Laplace transform

See also

Related Research Articles

References