# Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail.

Just to avoid misunderstanding, I point out that it is about Poisson point process and not about usual Poisson process.

Addition: here is what I mean exactly. Let's say we have a process $N$ on $\mathbb{R} _+\times \mathbb{R}^d$, $\mathbb{R} _+$ is time, with the intensity measure being the Lebesgue measure on $\mathbb{R} _+\times \mathbb{R}^d$. Let $\mathscr{F} _t$ be the minimal $\sigma$ -algebra containing all random variable $N(Q,U)$, where $Q \in \mathscr{B}([0;T])$, $U \in \mathscr{B}(\mathbb{R}^d)$. Or we may take minimal complete right-continuous $\sigma$ - algebra with this property. Let $\tau$ be a stopping time with respect to $(\mathscr{F} _t)$. It seems reasonable to conjecture, that the process $\bar N$ defined by

$$\bar N ([0;s],U) = N ([\tau;\tau + s],U) - N ([0;\tau],U), \ \ \ U \in \mathscr{B}(\mathbb{R}^d)$$

is a Poisson point process with the same intensity measure independent of $(\mathscr{F} _{\tau})$. I was not able however to find a reference to this statement.

In terms of random sets, it corresponds to the strong Markov property of the set $[0;\tau] \times \mathbb{R}^d$, which is not compact.

• math.chalmers.se/~zuev/files/smp.pdf
– Did
Commented May 24, 2014 at 15:26
• Thank you, that's very useful. Commented May 24, 2014 at 16:16
• Actually, in the article the strong Markov property is proven for stopping set, which are by definition compact. So it does not provide the answer. I was not able to find this statement in references, or in google either. Commented May 26, 2014 at 20:00
• Why does this make a difference?
– Did
Commented May 27, 2014 at 5:41
• The set $[0;\tau] \times \mathbb{R}^d$ is not compact when $\tau >0$. Probably one may still get desired property considering the sets $[0;\tau] \times K_n$, $K_n$ are compact subsets of $\mathbb{R}^d$, $\cup _n K_n = \mathbb{R}^d$. Yet it should be mentioned somewhere in literature, it seems to me. Commented May 27, 2014 at 8:24