 Exponential martingale of a random walk

For a random walk with independent and identically distributed random variables as increments and finite moment-generating function at $\theta$, put $\kappa (\theta )=\log \mathbb{E}{e}^{\theta X_1}$. Then $e^{\theta S_n-n\kappa (\theta )}$ is a martingale by conditioning on the next independent increment. In particular, if $r>0$ and $\mathbb{E}{r}^{X_1}=1$, then $r^{S_n}$ is a martingale. Stopping at a finite interval exit can make this process bounded, allowing the dominated convergence theorem to justify passage from bounded stopping times to the exit time.