Let $X_k$ be i.i.d. random variables such that $E[X_1]=m<\infty$. Consider $S_n = \sum_{k=1}^{n} X_k$. Let $\tau$ be a stopping time independent of $X_k$ with respect to the filtration $\{F_n\}_{n \geq 1}$ generated by the random variables $X_k$, $(k=1,2,...,n)$. Compute $E[e^{\sum_{k=1}^{\tau} X_k}]$. My professor gave me a hint: Compute the Laplace transform $E[e^{-u S_{\tau}}], u>0$.
** I am looking for constructive hints. I don't know how to start. **
What I am thinking: Let $g(s) = E[e^{-sX_1}],s>0$ and $f(z) = E[z^{\tau}], |z| \leq 1$.