Poisson Process : Markov Process

Can anyone please give an example of how the poisson process is a Markov process? I got a bit confused with the poisson process rate, the independent increments, the state space of the process.
N(t) records the number of occurrences of an event within the time interval 0 & t; and the events occur singly and at a rate lambda. So when an event occurs between 0 & t it adds up 1 to N(t). So I was wondering what is the Markov process here and what is the state space ?
Moreover having read the 'Poisson Process revisited' (page - 24 Chapter - 5) I was wondering that with the process jumping from 0 to 1, 1 to 2, and so on does the state 2 includes the state 1 (+1 increment) and state 3 includes state 2(+1 increment) and so on ???
I am really mixed up, a clear explanation would help me a lot. Thank You.

 

Because, Poisson Process does not depend on its previous than current state(even Poisson process is does not depends on current state)

Here, states are number of occurrence, so you can say 2=1+1 etc

 

Thank you for the reply mate , but I need a more detailed explanation on the query.

 

Hi, kindly ask specific query about any concepts... It's tough to provide full info at once.

 

Poisson process is basically a counting process, like no. of claims arrived until time t, So it's state space is {0, 1, 2, 3 ......}

It's not only markov but trivially markov, because the distribution of future events doesn't even depend on the current state of the process. The distribution of no. of events occurring in time t is Poi(λt) regardless of the current state of the process.

For example if I want to calculate the Prob. that the next claim arrives after 1hr given the process is in state

i) 3
ii) 4

then the required probability is same for both the cases i.e. exp(-λ) (assuming claims are arriving at the rate of λ per hour)