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Let $(X_n)$ be a sequence of IID random variables with $P(X_i = 1) = P(X_i = -1) = 1/2$. Let $(\mathcal{F}_n)$ be the natural filtration. Define $(S_n)$ by $S_0 = 0$ and $S_n = S_{n-1} + X_N \sqrt{1+S_{n-1}^2}$ if $n \geq 1$.

I am trying to show that $S_n$ is a martingale:

$E(S_n | \mathcal{F}_{n-1}) = E(S_{n-1} | \mathcal{F}_{n-1}) + E(X_n\sqrt{1+S_{n-1}^2} | \mathcal{F}_{n-1})$ now, I don't know what to do, as we don't know anything about $S_n$.

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Start from your equation, we could have $E(S_n |F_{n-1})$ = $E(S_{n-1} |F_{n-1})$ + $1/2 \sqrt{1 + S_{n-1}^2} - 1/2 \sqrt{1 + S_{n-1}^2}$.

This is because of the probability distribution of $X_i$.

Then we have $E(S_n |F_{n-1})$ = $E(S_{n-1} |F_{n-1}) = S_{n-1}$ which is a martingale.

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