Let $\newcommand{\F}{\mathcal F} S_n=S_{n-1} +X_n $ where $S_0=0$ , and $X_k$ are iid, and let $\phi(t)=\mathbb{E}e^{itX_1}$ be the characteristic function of $X_k$.
Consider a process $Y_n=e^{itS_n-n\log(\phi(t))}$. Show that the process $(Y_n, \F_n)$ is a martingale, where $\F_n=\sigma(X_1,...,X_n)$.
I am not too sure about how to calculate the conditional expectation $\mathbb{E}[Y_n \mid \F_{n-1}]$.
$\mathbb{E}[Y_n \mid \F_{n-1}]=\mathbb{E}[e^{itS_n-n\log\phi(t)}]=\mathbb{E}[e^{it(S_{n-1}+X_n)-n\log\phi(t)} \mid \F_{n-1}]=e^{itS_{n-1}}\mathbb{E}[e^{itX_n-n\log\phi(t)} \mid \F_{n-1}]$ $=e^{itS_{n-1}}\mathbb{E} [e^{itX_n} - \phi (t)^n \mid \F_{n-1}]$
At this point, I am stuck.