Poisson random measure

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

If $\mu\equiv 0$ then $N\equiv 0$ 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 $N_{\bullet}(\omega) = \sum\limits_{i=1}^{Z(\omega)} \delta_{X_i(\omega)}(\bullet)$ 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.