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I'm stuck with an exercise here. So there's a random walk on the integers with probability 0.5 of going left or right, i.e. $P[X_i=1]=P[X_i=-1]=1/2$ and $S_n=X_1+...+X_n$. for a real number $c$, set

$M_n=e^{cS_n}(\frac{2}{e^c+e^{-c}})$ and show that $M_n$ is a martingale.

What does that mean? Do I have to show that we have a martingale measure, i.e. some risk-neutral probabilities that sum up to 1? Would that be $\sum_{n \in \mathbb{Z}} M_n=1$? I don't see how to apply the definition on this, if I'm even taking the right definition here. Can someone please explain me what I exactly have to do?

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    No. You have to show that $M_n$ is a martingale, so it satisfies three properties: $M_n$ is $\mathcal{F}_n$-measurable; it is integrable and $E[M_{n+1}|\mathcal{F}_n]=M_n$, where (I suppose) $\mathcal{F}_n$ is the natural filtration.2012-03-21
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    Try [this](http://en.wikipedia.org/wiki/Martingale_(probability_theory)#Definitions). (And you are probably asked to show that $(M_n)_{n\geqslant0}$, not $M_n$, is a martingale.)2012-03-21
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    Ah okay, then I think I know what to do. Cool!2012-03-21

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