Expected value of product of dependent variables.

Question

I've just made a quick search but can't seem to find a satisfying explanation for the following:

Let $X_i,...,X_n$ be $\{-1,1\}$-variables that are not necessarily independent, and $E[X_i]=0$. Then: $$E[e^{\sum_i X_i}]= \prod_i E[e^{X_i}|\{ X_j: j < i \}]$$.

Can I have a derivation and/or the intuitive idea behind this?

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Edit: This is in the context of a discussion of martingales. Please see the screenshot below. Also if my question could be rephrased better, please help me do so.

Answer 1

You can derive this using the Law of Total Expectation, which states that $E[X] = E[E[X \mid Y]]$. With that, just note that: $$ E[e^{X_1+...+X_n}] = E[E[e^{X_1+...+X_n}\mid X_1,...,X_{n-1}]]= $$ $$= E[e^{X_1+...+X_{n-1}}E[e^{X_n} \mid X_1,...,X_{n-1}]]=$$

$$ = E [E[e^{X_1+...+X_{n-1}}\mid X_1,...,X_{n-2}]\cdot E[e^{X_n} \mid X_1,...,X_{n-1}]]= $$

$$= E [e^{X_1+...+X_{n-2}} E[e^{X_{n-1}}\mid \{X_j: j<n-1\}]\cdot E[e^{X_n} \mid \{X_j: j < n \}]] = $$ $$ =...=\prod_iE[e^{X_i}\mid \{X_j : j<i\}] $$

Which proves the identity.