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I was trying to calculate the expectation of $T^2$ using some martingale and got that I needed the expectation of $TS_T$. Any idea?

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up vote 8 down vote accepted

If $(S_n)$ is a simple symmetric random walk on the integers, then $S_n^3-3nS_n$ is a martingale. Applying the optional stopping theorem for martingales, we get $$ \mathbb{E}_x(S_T^3-3TS_T)=\mathbb{E}_x(S_0^3-0)=x^3.$$ Here $-a\leq x\leq a$ is the starting point of the random walk and $T$ is the hitting time of the boundary $\{-a,a\}$. Standard results tell us that $$\mathbb{P}_x(S_T=a)={a+x\over 2a}\quad\mbox{ and }\quad\mathbb{P}_x(S_T=-a)={a-x\over 2a}$$ so that $\mathbb{E}_x(S_T^3)=a^2 x$. It follows that $\mathbb{E}_x(T\ S_T)=(a^2x-x^3)/3$.


To find $\mathbb{E}_x(T^2)$, you could use the martingale $S_n^4-6nS_n^2+2n+3n^2$.


Update: The non-symmetric case.

Here is an infinite family of exponential martingales. For any real $\theta$ you can check that $\exp(\theta S_n)/[m_X(\theta)]^n$ is a martingale. Here, $m_X$ is the moment generating function of the increment $X$. For a non-symmetric random walk this will be $m_X(t)=q \exp(-\theta)+p \exp(\theta)$.

Starting with the exponential martingales, you can generate other martingales by differentiating with respect to $\theta$, then setting $\theta=0$. The first two martingales you get this way are $S_n-n\mu$ and $(S_n-n\mu)^2-n\sigma^2$ where $\mu,\sigma^2$ are the mean and variance of the increment $X$.

For more on this topic, take a look at Example M.1.3 (page 3) of Professor Glynn's notes on martingales.

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This MO answer may help understand why these martingales @Byron is using, and many others, exist and how they may be computed. –  Did Oct 26 '11 at 18:28
    
@Didier That is a very impressive MO answer. I hadn't noticed it before, so thanks for the reference. –  Byron Schmuland Oct 26 '11 at 18:33
    
Wow thanks a lot for your answer Byron, and for the awesome link Didier! –  wircho Oct 26 '11 at 22:07
    
I just realized the original question from the source does not specify it must be symmetric. Where can I find a list of martingales for the asymmetric case? –  wircho Oct 27 '11 at 21:26
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