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# Poisson random measure

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

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

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

iii) $forallomegainOmega;N_\left\{bullet\right\}\left(omega\right)$ is a measure on $\left(E, mathcal A\right)$

## 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 $mu\left(E\right)$ and $X_1, X_2,ldots$ - mutually independent random variables with distribution $frac\left\{mu\right\}\left\{mu\left(E\right)\right\}$ define $N_\left\{bullet\right\}\left(omega\right) = sumlimits_\left\{i=1\right\}^\left\{Z\left(omega\right)\right\} delta_\left\{X_i\left(omega\right)\right\}\left(bullet\right)$ where $delta_c\left(A\right)$ 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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