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Let $(\Omega, \mathcal{F}, (\mathcal{F}_n)_{n\in\{0,1,2,\ldots,N\}}, P)$ be a stochastic basis, carrying an adapted and integrable stochastic process $X=X_n$. Show that X is a martingale iff $E[X_T]=E[X_S]$ for all bounded stopping times $S ≤T$.

I am stuck at this problem and don't know where to start. Could anyone give a help ?

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Optionnal stoppping time theorem gives you one implication.

For the other one, suppose $A \in \mathcal{F}_n$. One wants to prove that $E(1_A X_n) = E(1_A X_{n+1})$. Define $T$ by $T(\omega) = n$ if $\omega \in A$ and $T(\omega) = n+1$ otherwise, and let $S$ be constant equal to $n+1$.

What does the condition $E(X_T) = E(X_S)$ say ?

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