Poisson random measure

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

i) ${\displaystyle \forall A\in {\mathcal {A}},\quad N_{A}}$ is a Poisson random variable with rate ${\displaystyle \mu (A)}$.

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[]

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[]

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[]

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[]

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