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Choosing a stopping time to maximize an expected reward. For an adapted integrable reward process over a finite discrete horizon, the Snell envelope computes the conditional optimal reward by comparing immediate exercise with continuation.

An optimal stopping time attains the supremum of the expected reward over an admissible class of stopping times. For a finite-horizon Snell envelope, the first time at which its value equals the current reward is optimal.

For a Markov process with reward $f$, the value starting at time $t$ in state $s$ is the supremum of expected rewards over admissible stopping times. In finite discrete time it obeys $V(T,s)=f(s)$ and $V(t,s)=\max \{f(s),PV(t+1,\cdot )(s)\}$ for the transition operator $P$, wherever expectations are well-defined. Statewise absolute integrability of every remaining-horizon reward makes these values finite.

Let $M$ be a continuous local martingale and let $({\tau }_s)$ be an increasing continuous family of finite stopping times. Under the usual compatibility conditions, $M_{{\tau }_s}$ is a continuous local martingale for the time-changed filtration ${\mathcal{F}}_{{\tau }_s}$. A finite-variation process remains of finite variation after the same time change.