Suppose $X_i$ are independent $N(0,1)$ distributed random variables. Let $S_n=X_1+\cdots+X_n$ and let $Y_n=e^{S_n-\frac{n}{2}}$ and let $F^{Y_n}$ be the algebra generated by $Y_1, \ldots, Y_n$. Show that $Y_n$ is a martingale in this algebra.
The problem basically comes down to finding the conditional expectation for each $Y_n$ but I'm now sure how to proceed. Any help would be appreciated.
$Y_n$ is integrable since it is lognormally distributed. It is also $F^{Y_n}$-measurable. Then \begin{align*}\forall n\in\mathbb{N}^* : \quad \mathbb{E}\left[Y_{n+1}\mid F^{Y_n}\right]&=\mathbb{E}\left[\exp\left(S_n-\frac{n+1}{2}\right)\exp\left(Y_{n+1}\right)\mid F^{Y_n}\right] \\ &=\exp\left(S_n-\frac{n+1}{2}\right)\mathbb{E}\exp Y_{n+1} \\ &=Y_n\end{align*} where I use the fact that $Y_n$ is $F^{Y_n}$-measurable and that $Y_{n+1}$ is independent of $F^{Y_n}$ in the second equality, and the fact that $\mathbb{E}e^{Y_{n+1}}=e^{1/2}$ in the third equality.