Show that if P (1) and P (2) are independent Poisson processes with parameters λ 1 and λ 2 , If X is defined by (5.2), show that X is a Poisson process. Let X t be a stochastic process and let   be...

Show that if P⁽¹⁾
and P⁽²⁾
are independent Poisson processes with parameters λ₁
and λ₂,

If X is defined by (5.2), show that X is a Poisson process.

Let X_t
be a stochastic process and let
  be the filtration generated by X . Suppose X is a Poisson process with respect to the filtration
  Show that X is a Poisson process with respect to the minimal augmented filtration generated by X.