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.{{Cite journal | last1 = Cox | first1 = D. R. | author-link = David Cox (statistician)| title = Some Statistical Methods Connected with Series of Events | journal = Journal of the Royal Statistical Society | volume = 17 | issue = 2 | pages = 129–164 | doi = 10.1111/j.2517-6161.1955.tb00188.x| year = 1955 }}

Cox processes are used to generate simulations of spike trains (the sequence of action potentials generated by a neuron),{{Cite journal | last1 = Krumin | first1 = M. | last2 = Shoham | first2 = S. | doi = 10.1162/neco.2009.08-08-847 | title = Generation of Spike Trains with Controlled Auto- and Cross-Correlation Functions | journal = Neural Computation | volume = 21 | issue = 6 | pages = 1642–1664 | year = 2009 | pmid = 19191596}} 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."{{Cite journal | last1 = Lando | first1 = David| title = On cox processes and credit risky securities | doi = 10.1007/BF01531332 | journal = Review of Derivatives Research | volume = 2 | issue = 2–3 | pages = 99–120| year = 1998 }}

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

: $\mathcal L_\eta(f)=\exp \left(- \int 1-\exp(-f(x))\; \xi(\mathrm dx)\right)$

for any positive, measurable function $f$ .

References

;Notes

;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)

Category:Poisson point processes

Cox process

Definition

Laplace transform

See also

References