Poisson random measure

Poisson random measure

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

i) forall Ainmathcal{A};N_A is a Poisson random variable with rate mu(A).

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

iii) forallomegainOmega;N_{ullet}(omega) is a measure on (E, mathcal A)

Existence

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

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


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