Stochastic process that is Martingale but not Markov? Can you please help me by giving an example of a stochastic process that is Martingale but not Markov process for discrete case?
 A: Markov chains have a finite memory, Martingales can have an infinite one.
Pick a random value for $X_0$. Let the sequence of random variables $\{\epsilon_n,\,n>0\}$ be i.i.d. with mean$=E[\epsilon_{n}]=0$ and independent of $X_0$.
The process governed by $X_{n+1}=X_n+\epsilon_{n+1}X_0$


*

*is a martingale as 
\begin{align}
&E[X_{n+1}|X_0,\dots,X_n]
=E[X_{n}|X_0,\dots,X_n]+E[\epsilon_{n+1}X_0|X_0,\dots,X_n]\\
&=X_n+E[\epsilon_{n+1}|X_0,\dots,X_n]E[X_0|X_0,\dots,X_n]\\
&= X_n+E[\epsilon_{n+1}]X_0\\
&= X_n+0X_0\\&=X_n
\end{align}
Here, it is key that $\epsilon_n$ is independent of the $\{X_i, 0\leq i\leq n\}$.

*is not Markov as it is clear that $Pr[X_n+\epsilon_{n+1}X_0|X_n]\neq Pr[X_n+\epsilon_{n+1}X_0|X_0,\dots,X_n]$.
To determine $X_{n+1}$ not only the value of $X_n$ but the entire path to it (at which value did the path start at $X_0$) is needed.
Martingale are about expectation and the Markov property about probability, which of course is also an expectation, but that's stuff for another post.
Inspitration was drawn from http://djalil.chafai.net/blog/2012/01/20/martingales-which-are-not-markov-chains/
