Suppose is a filtration satisfying the usual conditions and are Poisson processes with respect to with parameters λ1, λ2, respectively. Suppose is a Poisson process with parameter λ1 + λ2. Prove...

Suppose

is a filtration satisfying the usual conditions and

are Poisson processes with respect to

with parameters λ₁, λ₂, respectively. Suppose

is a Poisson process with parameter λ₁
+ λ₂. Prove that P₁
and P₂
are independent processes.