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Here is the scenario I am presented with. I am supposed to show it is a martingale:

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Here is what I have done so far. I am fairly sure it is correct, I just don't know how to simplify it.

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Thanks for the help.

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It looks like you are on the right track. Here's a hint, what is $E( S_n )$ for any $n$? More importantly, if you have two r.v. X and Y and X is F_n measurable, then E(XY|F_n) = X E(Y|F_n). Notice that $S_m$ is independent of $S_{m+n}-S_m$. –  Alex R. Mar 31 '12 at 23:40
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