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 $S_n$.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.