Poisson random measure

Let be some measure space with -sigma [finite measure|finite measure]. The Poisson random measure with intensity measure is a family of random variables defined on some probability space such that
i) is a Poisson random variable with rate.
ii) If sets don't intersect then the corresponding random variables from i) are mutually independent.
iii) is a measure on

Existence

If then satisfies the conditions i)–iii). Otherwise, in the case of finite measure, given, a Poisson random variable with rate, and, mutually independent random variables with distribution, define where is a degenerate measure located in. Then will be a Poisson random measure. In the case is not finite the measure can be obtained from the measures constructed above on parts of where is finite.

Applications

This kind of random measure is often used when describing jumps of stochastic processes, in particular in Lévy–Itō decomposition of the Lévy processes.

Generalizations

The Poisson random measure generalizes to the Poisson-type random measures, where members of the PT family are invariant under restriction to a subspace.