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I'm doing an exercice about the second equality of Wald.

Let $(X_i)_{i\ge 1}$ be a sequence of integrable random variables. Let $F = (F_i)$ be a filtration such as $X$ is adapted. We suppose that $X_i$ and $F_{i-1}$ are independant for all $i\ge 2$ and that $\mathbb{E}[X_1^2] < \infty$. We put $S_n := X_1 + ... + X_n$. Show that if $T$ is an integrable stopping time such as $T \ge 1$, then $\mathbb{E}[S_T ^2 - T\mathbb{E}[X_1 ^2]]^2 = \sigma ^2 (X_1)\mathbb{E}[T]$.

I would like to do it this way : Show that $Y_n := Z_n^2 - n\sigma (X_1)$ is a martingale, where $Z_n := X_1 + ... + X_n - n\mathbb{E}[X_1]$ is a martingale (this result as been shown in an other exercice).

But I can't show that. We have $\mathbb{E}[Y_{n+1} | F_n] = \mathbb{E}[Z_{n+1} ^2 - (n+1)\sigma ^2 (X_1) | F_n] = \mathbb{E}[Z_{n+1} ^2 | F_n] - (n+1)\sigma ^2 (X_1)$. And then, I tried to develop the square, but I don't get any good result.... Maybe is there a mistake in the beginning ?

Thanks for your help.

P.S. I suppose there is another way to prove that equality, but I want to do it this way.

P.P.S As you can see, English is not my mother tongue, so if you see any mistake, I would be glad to learn how to write it correctly.

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closed as off topic by Valerio Capraro, Igor Rivin, Did, Anthony Quas, Louigi Addario-Berry Jun 4 '12 at 8:07

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

Also on math.SE: – Theo Buehler Jun 3 '12 at 12:47
This forum is not for exercises. You can certainly get helped in – Valerio Capraro Jun 3 '12 at 12:48
No the right place for the question, voting to close. – Igor Rivin Jun 3 '12 at 12:48
I didn't know that this forum was inappropriate for exercices. Thanks for the information – Merli Jun 3 '12 at 12:56
Is there a way to delete the post by myself ? – Merli Jun 3 '12 at 12:58

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