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Let $Y_{i}$ be IID with $P(Y_{i}=1)$= $\frac{1}{2}=P(Y_{i}=-1)$. Define $K_{n}=\sum_{i=1}^{n}{Y_{i}}$.

Find the constant $r$ so that so that $$\Lambda_{n}=exp(K_{n} + rn)$$ is a Martingale.

I'm not sure that I have the right answer on this one. Any help will be appreciated.

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    I guess this is a follow-up to this [question](http://math.stackexchange.com/q/2154311/36150) or, better to say, my answer... note that the constant $r$ is **given** in the first part of my answer. – 2017-02-21
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    @saz I wish you told me 5 minutes ago - I almost completed my answer! OP do you know the definition of a Martingale $$ \mathbb{E}\left[\Lambda_{n+1}|\mathcal{F}_{n}\right] = \Lambda_{n} $$ – 2017-02-21
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    @Chinny84 Sorry about that... but I believe that your answer will be helpful for the OP anyway; so just go ahead and post your answer. – 2017-02-21
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    Thank you for your answer. But the problem that I have is deriving it through $E[\Lambda_{n+1}|F_{n}]=\Lambda_{n}$. Is there something obvious that I'm missing? – 2017-02-21
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    @saz I hope my joke didn't miss the target :)! – 2017-02-21
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    Hi @saz I deleted my previous question. I'm not sure how to use my $r$ to find the moment generating function. I'm really puzzled. Can you give me the final answer? – 2017-02-26

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The definition of a Martingale $$ \mathbb{E}\left[\Lambda_{n+1}|\mathcal{F}_{n}\right] = \Lambda_{n} $$ we have $$ \mathbb{E}\left[\mathrm{e}^{K_{n+1} + r(n+1)}|\mathcal{F}_{n}\right] = \mathbb{E}\left[\mathrm{e}^{Y_{n+1} + K_{n} + r(n+1)}|\mathcal{F}_{n}\right] $$ we can re-write the argument as $$ K_{n} + rn + Y_{n+1} + r $$ or $$ \mathbb{E}\left[\mathrm{e}^{K_{n+1} + r(n+1)}|\mathcal{F}_{n}\right] = \mathbb{E}\left[\Lambda_{n}\mathrm{e}^{Y_{n+1} + r}|\mathcal{F}_{n}\right] = \Lambda_{n}\mathbb{E}\left[\mathrm{e}^{Y_{n+1} + r}|\mathcal{F}_{n}\right] $$ we can extract the $Y_{n}$ since we know this up to the filtration time $n$. so we need to compute $$ \mathbb{E}\left[\mathrm{e}^{Y_{n+1} + r}|\mathcal{F}_{n}\right] $$ For our condition to hold true we need $$ \mathbb{E}\left[\mathrm{e}^{Y_{n+1} + r}|\mathcal{F}_{n}\right] = 1 $$ can you compute the expectation of the above? You have a simple calculation since we have discrete binary system!