Stopped likelihood ratio under exponential tilting ($\mathbb{P}(T\le n)={\mathbb{E}}^{\lambda }[\varphi (\lambda {)}^T{e}^{-\lambda S_T}{1}_{\{T\le n\}}]$)

For a random walk with independent and identically distributed random variables as increments, let $M_n=e^{\lambda S_n}/\varphi (\lambda {)}^n$ define its finite-time exponential tilting. Under the tilted law, $L_n=M_n^{-1}$ is a martingale: the conditional mean of $\varphi (\lambda )e^{-\lambda X_{n+1}}$ is one. For a stopping time $T$, partition $\{T\le n\}$ 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.