Poisson random measure

Let ( E , A , μ ) {\displaystyle (E,{\mathcal {A}},\mu )} be some measure space with σ {\displaystyle \sigma } -finite measure μ {\displaystyle \mu } . The Poisson random measure with intensity measure μ {\displaystyle \mu } is a family of random variables { N A } A ∈ A {\displaystyle \{N_{A}\}_{A\in {\mathcal {A}}}} defined on some probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathrm {P} )} such that i) ∀ A ∈ A , N A {\displaystyle \forall A\in {\mathcal {A}},\quad N_{A}} is a Poisson random variable with rate μ ( A ) {\displaystyle \mu (A)} . ii) If sets A 1 , A 2 , … , A n ∈ A {\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) ∀ ω ∈ Ω N ∙ ( ω ) {\displaystyle \forall \omega \in \Omega \;N_{\bullet }(\omega )} is a measure on ( E , A ) {\displaystyle (E,{\mathcal {A}})}

Existence If μ ≡ 0 {\displaystyle \mu \equiv 0} then N ≡ 0 {\displaystyle N\equiv 0} satisfies the conditions i)–iii). Otherwise, in the case of finite measure μ {\displaystyle \mu } , given Z {\displaystyle Z} , a Poisson random variable with rate μ ( E ) {\displaystyle \mu (E)} , and X 1 , X 2 , … {\displaystyle X_{1},X_{2},\ldots } , mutually independent random variables with distribution μ μ ( E ) {\displaystyle {\frac {\mu }{\mu (E)}}} , define N ⋅ ( ω ) = ∑ i = 1 Z ( ω ) δ X i ( ω ) ( ⋅ ) {\displaystyle N_{\cdot }(\omega )=\sum \limits _{i=1}^{Z(\omega )}\delta _{X_{i}(\omega )}(\cdot )} where δ c ( A ) {\displaystyle \delta _{c}(A)} is a degenerate measure located in c {\displaystyle c} . Then N {\displaystyle N} will be a Poisson random measure. In the case μ {\displaystyle \mu } is not finite the measure N {\displaystyle N} can be obtained from the measures constructed above on parts of E {\displaystyle E} where μ {\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 978-0-521-55302-5.