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This is a series problem and I'm struggling with the last part. I assume that the last part has nothing to do with previous ones, so i won't put up the other parts. Question is :

Let $\tau$ be a stopping time and $X=(X_n,{\cal F_n})$ be a martingale. Show that stopped sequence $X_{n}^{\tau}=( X_{\tau_{n}},{\cal F_n})$ where $\tau_{n}=\min\{\tau,n\}$ is a martingale.

I started with $E(X_{\tau_{n}}|F_{n-1})=E(X_{\min\{\tau,n\}}|F_{n-1})$ which then would have two cases, but all ends up with $E(X_{n-1})$, or should that be $E(X_{\tau_{n-1}})$?

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Neither. The result should be $X_{\tau_{n-1}}$. // You could decompose the conditional expectation along the events $[\tau\leqslant n-1]$ and $[\tau\geqslant n]$ and compute the two terms separately and you might first try to show that $[\tau\geqslant n]$ belongs to $F_{n-1}$. –  Did Nov 4 '11 at 8:26
kkk: Any luck with these hints? –  Did Nov 5 '11 at 11:24

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