Strong Markov property explained.

Question

I have got 2 theorems,

Theorem 1 The increment $ (N_{t+u} - N_t)_{u\geq 0} $ of a Poisson process rate $\lambda$ is again a Poisson process rate $\lambda$ and is independent of $(N_s)_{0\leq s \leq t}$

Proof

$ P(N_{t+u} - N_t)= k|N_t=n, T_1=t_1,....T_n=t_n) $

$= P(N_{t+u} - N_t)= k) $

$= p_u (0,k) $

Theorem 2 The strong Markov property

$N = (N_t)_{t\geq0} $is a Poisson process rate $\lambda$

Let T be a stopping time

Define $N^T $ via $N^T_t = N_{T+t} - N_T $

Conditional on T being finite $(N^T_t)_{t\geq 0} $ is a Poisson process rate $\lambda$ which is independent of $ (N_t)_{0\leq t \leq T}$

Can anybody

A) explain the first proof why does it hold?

B) explain exactly how these 2 theorems are different?

Answer 1

The first proof is a direct consequence of the independent increment property of Poisson processes: the number of arrivals in $(t,t+u)$ is independent of $N_t$.

Strong Markov property is a result which states that the Markovian property of Theorem 1 not only holds for deterministic times, but also for some particular random times, which are called stopping times.

Strong Markov is a useful and strong extension of Markovity in Poisson processes, DTMCs and CTMCs.