Is an AR(p)-process a martingale? Is an AR(p)-process a martingale? 
I think it is not, but I don't know how to explain this. The expected value of the martingale must be zero. In the case of an AR(p)-process it isn't but in the case of AR(1)-process it is. So an AR(1)-process would be a martingale. 

An AR(1) process is not a martingale in general. Suppose our AR(1) model is given by 
$$X_{t + 1} = c + \phi X_t + \epsilon_t$$
where the $\epsilon_t$ have mean 0 and $\phi$ and $c$ are constants. Then we see that
$$E[X_{t+1}| X_{t}, X_{t-1} \dots, ] = c + \phi X_t$$
The martingale property requires that 
$$E[X_{t+1}| X_{t}, X_{t-1} \dots, ] = X_t$$
so the above process is not a martingale unless $c = 0$ and $\phi = 1$.
