Contents

Poisson random measure

ii) If sets ${\displaystyle A_{1},A_{2},\ldots ,A_{n}\in {\mathcal {A}}}$ don't intersect then the corresponding random variables from i) are mutually independent.

iii) ${\displaystyle \forall \omega \in \Omega \;N_{\bullet }(\omega )}$ is a measure on ${\displaystyle (E,{\mathcal {A}})}$

Existence[edit]

If${\displaystyle \mu \equiv 0}$ then ${\displaystyle N\equiv 0}$ satisfies the conditions i)–iii). Otherwise, in the case of finite measure ${\displaystyle \mu }$, given ${\displaystyle Z}$, a Poisson random variable with rate ${\displaystyle \mu ($E)} , and ${\displaystyle X_{1},X_{2},\ldots }$, mutually independent random variables with distribution ${\displaystyle {\frac {\mu }{\mu ($E)}}} , define ${\displaystyle N_{\cdot }(\omega )=\sum \limits _{i=1}^{Z(\omega )}\delta _{X_{i}(\omega )}(\cdot )}$ where ${\displaystyle \delta _{c}($A)} is a degenerate measure located in ${\displaystyle c}$. Then ${\displaystyle N}$ will be a Poisson random measure. In the case ${\displaystyle \mu }$ is not finite the measure ${\displaystyle N}$ can be obtained from the measures constructed above on parts of ${\displaystyle E}$ where ${\displaystyle \mu }$ 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.

Generalizations[edit]

The Poisson random measure generalizes to the Poisson-type random measures, where members of the PT family are invariant under restriction to a subspace.

References[edit]

• Sato, K. (2010). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. ISBN 978-0-521-55302-5.