= Stopped likelihood ratio under exponential tilting
{title2=$\mathbb P(T\leq n)=\mathbb E^\lambda[\phi(\lambda)^T e^{-\lambda S_T}\mathbf1_{\{T\leq n\}}]$}
For a <random walk> with <independent and identically distributed random variables> as increments, let $M_n=e^{\lambda S_n}/\phi(\lambda)^n$ define its finite-time <exponential tilting>. Under the tilted law, $L_n=M_n^{-1}$ is a <martingale>: the conditional mean of $\phi(\lambda)e^{-\lambda X_{n+1}}$ is one. For a <stopping time> $T$, partition $\{T\leq n\}$ into $\{T=j\}$ and apply the <tower property of conditional expectation> to obtain the displayed identity. Only finite-time <absolute continuity of measures> is needed; there need not be an infinite-horizon likelihood-ratio density.
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