Be $ \left \{ N(t):t \geq 0 \right \}$ a homogeneous Poisson Process with intensity λ.
Show that for arbitrary times $t_{1}< t_{2}<...t_{n}:$
$E[N(t_{n+1})|N(t_{n}),...,N(t_{1})]=N(t_{n})+\lambda (t_{n+1}-t_{n})$
and,is this result valid also in the non-homogeneous context?
I have really tried but I don't know how to approach the problem with the properties, if someone can help me
You can have a look here to check that a Poisson process is a Markov process:
Show the Markov Property of a Poisson Process
Therefore with this at hand, we start with this:
$E[N(t_{n+1})|N(t_{n}),...,N(t_{1})]=E[N(t_{n+1})|N(t_{n})]=$
And now:
$E[N(t_{n+1})|N(t_{n})=n]=$
$=E[N(t_{n+1})-N(t_{n})+N(t_{n})|N(t_{n})=n]=$
$=n+E[N(t_{n+1})-N(t_{n})|N(t_{n})=n]=$
$=n+E[N(t_{n+1})-N(t_{n})|N(t_{n})-N(0)=n]=$
(since $N(0)=0$). Now since the process has independent and Poisson distributed increments, we continue:
$=n+E[N(t_{n+1})-N(t_{n})]=$
$=n+\lambda(t_{n+1}-t_{n})$
This is equivalent to what you wrote.
UPDATE: for an inhomogeneous Poisson process:
1- The Markov property remains, since the proof in the link is based on the property of having independent increments, which is true also in the inhomogeneous case ( https://en.wikipedia.org/wiki/Poisson_point_process#Interpreted_as_a_counting_process )
2- The calculations above apply directly, but one has to substitute in the last line the integral of the intensity function, leading to:
$E[N(t_{n+1})|N(t_{n}),...,N(t_{1})]=N(t_{n})+\int_{t_n}^{t_{n+1}}\lambda(t) dt$