Markov process which is not martingale

Question

I have seen the examples of a discrete time martingale that is not a Markov Process. Can you construct me an example of discrete time Markov Process that is not a martingale?

Answer 1

In order for a discrete time Markov process to be a martingale the transition probabilities would have to lead to a "fair" probability distribution in the long run. So just take a two state Markov chain where $P(0|0) = 3/4$ and $P(1|0) = 1/4$ and $P(0|1) = P(1|1) = 1/2$ for the transition probabilities. This will not be a martingale, as can be easily verified by computing conditional expectations.

Answer 2

The process $X_{t}:=t+B_{t}$ where $B_{t}$ is BM is Markov but not Martingale.

Not Martingale b/c

$E[X_{t}|F_{s}]=t+B_{s}=t-s+X_{s}$

and Markov b/c

$E[e^{u(t+s+B_{t+s})}|F_{t}]=e^{u(t+s+B_{t})}E[e^{u(B_{t+s}-B_{t})}|F_{t}]=E[e^{u(t+s+B_{t+s})}|B_{t}].$