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 $\xi$ 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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Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
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Financial models	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
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Cox process

Definition

Laplace transform

See also

References