Under the setup of, verify whether or not has independent and homogeneous increments. Using (5.4.3), show by induction that for homogeneous Poisson processes, Consider the Poisson process and...

Under the setup of, verify whether or not

has independent and homogeneous increments.

Using (5.4.3), show by induction that for homogeneous Poisson processes,

Consider the Poisson process and show that for every

τ has the same distribution as X_t. Furthermore, show that

is independent of X_τ.