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let $S(k), k\geq 0$ a discrete random process. Suppose $S(N)$ is with probability one either 100 or 0 and that $S(0)=50$. Suppose further there is at least a sixty percent probability that the price will at some point dip below 40 and then subsequently rise above 60 before time $N$. How do you prove that $S(k)$ cannot be a martingale?

By advance, thank you very much for your help.

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If this is homework, you should add the "homework" tag. –  Nate Eldredge Dec 8 '10 at 21:30
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1 Answer 1

Let $T_1$ be the first time it dips below $40$ and $T_2$ be the first time it rises to $60$, then $T_1<N$ and $T_2<N$ almost surely.

By the Optional Sampling Theorem,

$E S(0)=ES(T_1)=ES(T_2)$

but

none of $40, 50, 60$ equals to each other. You have actually given more conditions than that is required, one of dipping below 40 or rising above 60 would have been sufficient.

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