Poisson random measure

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Let (E, \mathcal A, \mu) be some measurable space with σ-finite measure μ. The Poisson random measure with intensity measure μ is a family of random variables \{N_A\}_{A\in\mathcal{A}} defined on some probability space (\Omega, \mathcal F, \mathrm{P}) such that

i) \forall A\in\mathcal{A}\;N_A is a Poisson random variable with rate μ(A).

ii) If sets A_1,A_2,\ldots,A_n\in\mathcal{A} don't intersect then the corresponding random variables from i) are mutually independent.

iii) \forall\omega\in\Omega\;N_{\bullet}(\omega) is a measure on (E, \mathcal A)

[edit] Existence

If \mu\equiv 0 then N\equiv 0 satisfies the conditions i)-iii). Otherwise, in the case of finite measure μ given Z - Poisson random variable with rate μ(E) and X_1, X_2,\ldots - mutually independent random variables with distribution \frac{\mu}{\mu(E)} define N_{\bullet}(\omega) = \sum\limits_{i=1}^{Z(\omega)} \delta_{X_i(\omega)}(\bullet) where δc(A) is a degenerate measure located in c. Then N will be a Poisson random measure. In the case μ is not finite the measure N can be obtained from the measures constructed above on parts of E where μ is finite.

[edit] Applications

This kind of random measures ist often used when describing jumps of stochastic processes, in particular in Lévy-Itō decomposition of the Lévy processes.

[edit] References

  • Sato K. Lévy Processes and Infinitely Divisible Distributions Cambridge University Press, (1st ed.) ISBN 0-521-55302-4.