# Poisson random measure

Let be some measure space with -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.

## 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.

## References

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