Assume that $S_n=\sum_{i=1}^n X_i$ where $X_i$ are iid r.v. with finite mean $E(X_1)=\mu$. Assume that $\tau$ is a stopping time with finite expectation. What's the expectation of $\tau S_\tau$? Is it $\mu E(\tau^2)$? How can we derive it?
I tried it in this way: $\mathbb{E}(\tau S_\tau)=\sum_{n=1}^\infty \mathbb{E}\tau S_\tau I(\tau=n)=\sum_{n=1}^\infty \mathbb{E}n S_n I(\tau=n)=\sum_{n=1}^\infty \mathbb{E}n \sum_{k=1}^n X_k I(\tau=n)$, then I got stuck with the term $\mathbb{E} X_i I(\tau=j)$ for $i