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) 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 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.