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}}_{Ain {mathcal {A}}}} defined on some probability space ( Ω , F , P ) {displaystyle (Omega ,{mathcal {F}},mathrm {P} )} such that 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}}_{Ain {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 Ain {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_{ullet }(omega )} is a measure on ( E , A ) {displaystyle (E,{mathcal {A}})} If μ ≡ 0 {displaystyle mu equiv 0} then N ≡ 0 {displaystyle Nequiv 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. This kind of random measure is often used when describing jumps of stochastic processes, in particular in Lévy–Itō decomposition of the Lévy processes.