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.

iii) is a measure on

Existence

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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

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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

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

References

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  • Sato, K. (2010). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. ISBN 978-0-521-55302-5.