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Cox process

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[edit]

Let ${\displaystyle \xi }$ be a random measure.

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

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

Laplace transform[edit]

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

${\displaystyle {\mathcal {L}}_{\eta }($f)=\exp \left(-\int 1-\exp(-f(x))\;\xi (\mathrm {d} x)\right)}

for any positive, measurable function ${\displaystyle f}$.

See also[edit]

• Poisson hidden Markov model
• Doubly stochastic model
• Inhomogeneous Poisson process, where λ(t) is restricted to a deterministic function
• Ross's conjecture
• Gaussian process
• Mixed Poisson process

References[edit]

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)

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