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.[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
be a random measure.
A random measure
is called a Cox process directed by
, if
is a Poisson process with intensity measure
.
Here,
is the conditional distribution of
, given
.
Laplace transform
If
is a Cox process directed by
, then
has the Laplace transform
![{\displaystyle {\mathcal {L}}_{\eta }(f)=\exp \left(-\int 1-\exp(-f(x))\;\xi (\mathrm {d} x)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f072e5176d68e85cfa254adee337d66ddc4e051)
for any positive, measurable function
.
See also
References
- Notes
- Bibliography