The finite-horizon Snell envelope of rewards is defined backward by and . It is the least supermartingale dominating .
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.
With independent uniform distributions on as sequential offers and a finite number of rounds, the continuation value with offers remaining is . Accept the current offer when it reaches that threshold. Integrating the maximum of the offer and continuation value gives the recursion, by the Snell envelope principle.
At a regular exercise boundary for a diffusion optimal stopping problem, value matching is commonly accompanied by matching first derivatives. This makes the value across the boundary and eliminates a derivative-jump local time of a semimartingale term in the generalized Itô formula. It does not assert that second derivatives agree, or that smooth fit holds for every possible stopping problem.
For a Markov process with reward , the value starting at time in state is the supremum of expected rewards over admissible stopping times. In finite discrete time it obeys and for the transition operator , wherever expectations are well-defined. Statewise absolute integrability of every remaining-horizon reward makes these values finite.
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.
The first time the Snell-envelope compensator is about to become positive is an optimal stopping time; before it the martingale part equals both value and reward at stopping.
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The Snell envelope is a concept used primarily in the fields of stochastic control and optimal stopping theory. It provides a way to characterize the value of optimal stopping problems, particularly in scenarios where a decision-maker can stop a stochastic process at various times to maximize their expected payoff. Mathematically, the Snell envelope is defined as the least upper bound of the expected values of stopping times given a stochastic process. Formally, if \( X_t \) is a stochastic process (e.g.