Can you please help me by giving an example of a stochastic process that is Martingale but not Markov process for discrete case?
2026-04-07 05:53:04.1775541184
Stochastic process that is Martingale but not Markov?
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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/