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Let $(S_n)$ a martingale refer to $(X_n)$. Show that for all integer $k\leq l\leq m$ $$\mathbb E[(S_m-S_l)S_k]=0.$$

I don't understand the to following equality: $$\mathbb E[(S_m-S_l)S_k]=\mathbb E\big[\mathbb E[(S_m-S_l)S_k\mid X_k,...,X_1]\big]=\mathbb E\big[S_k\mathbb E[(S_m-S_l)\mid X_k,...,X_1]\big].$$

Which properties is used ?

idm
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1 Answers1

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The first equality is always true. The second is the definition of the conditional expectation, plus the fact that $S_k$ is a deterministic function of $$ X_k \dots X_1 $$

mookid
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  • Thanks for your answer. I don't understand why the first equality is always true, neither the argument for the second equality. Could you tell me more please ? tkanks, – idm May 05 '15 at 13:53
  • both are based of the definition of the conditional expectation: $E(XY) = E(E(X|F) Y) $ when $Y$ is measurable with respect to $F$. in the first equality, take $Y = 1$ and in the second, $Y = S_k$, and $F = \sigma(X_k\dots X_1)$. – mookid May 05 '15 at 16:00