Poisson random measure

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Let (E, \mathcal A, \mu) be some measure space with \sigma-finite measure \mu. The Poisson random measure with intensity measure \mu 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},\quad N_A is a Poisson random variable with rate \mu(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)

Existence[edit]

If \mu\equiv 0 then N\equiv 0 satisfies the conditions i)–iii). Otherwise, in the case of finite measure \mu, given Z, a 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_{\cdot}(\omega) = \sum\limits_{i=1}^{Z(\omega)} \delta_{X_i(\omega)}(\cdot) 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[edit]

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[edit]

  • Sato, K. (2010). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. ISBN 0-521-55302-4.