When is Markov a Martingale

I have two questions and I am very confused about the concepts

1. Can a Markov process of order one also be a a Martingale?
2. Is any Markov process of order one also a Martingale?

For 1. I would say yes, since a Markov of order one is also an AR(1) it would be a martingale if c=0 and phi=1. However, I am very confused about the second. My answer for 2 would be no, but I cannot explain it properly.

Can anyone maybe help me?

• Hi: I don't follow your question and it seems that others may not have either. markov processes are general so they are not just time series models. also, what do you mean by "order ?". the order of a time series usually refers to how many times you need to difference it for it to be stationary. I'm not familar with other meanings ? – mark leeds Jan 22 at 17:28
• hello mark, by order I meant the dependence in history. – gkc Jan 22 at 17:33
• Hi: gkc: A martingale satisfies $E(X_{t+1}| X_{t}, X_{t-1}, \ldots) = X_{t}$ so it needs to depend on only the current history by definition.. But clearly there can be cases where the process depends on the current history and is not a martingale. For example, $E(X_{t+1}| X_{t}, X_{t-1}, \ldots ) = 2 \times X_{t}$. – mark leeds Jan 23 at 14:09