= Exponential martingale of a random walk
{title2=$e^{\theta S_n-n\kappa(\theta)}$}
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.
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