Optimal stopping and Snell’s envelope

For a finite discrete horizon and integrable adapted rewards, define the envelope at the final time to equal the reward. Move backward by taking the maximum of stopping now and the conditional expected envelope next period. This process dominates the reward and is a supermartingale. If another supermartingale dominates the reward, backward induction shows it also dominates this envelope. Thus the recursion constructs the smallest supermartingale with the required domination property.

All stopping times here take values in the finite decision grid. The first time the envelope equals the reward is a stopping time, since equality is observable at that date. Before contact, the recursion chooses the conditional continuation value, so the stopped envelope is a martingale. Optional sampling and contact then give expected reward equal to the initial envelope, while any other stopping rule is bounded by it.

A dominating supermartingale bounds every allowed stopped reward through optional sampling. To attain that bound, a rule must collect the envelope at contact and avoid losing expected envelope value before it stops. Hence contact together with the martingale property of the stopped envelope certifies optimality. A rule that contacts only after the envelope has already fallen in expectation may be suboptimal. The martingale condition keeps track of that otherwise invisible loss.

In continuous time, define the value family using an essential supremum over stopping times, rather than a pointwise supremum over sample paths. Its aggregation into a right-continuous Snell process and the existence of an optimal first-contact rule require regularity. For example, a bounded continuous adapted reward on a finite horizon with the usual Brownian filtration has the standard optimal-contact result. Boundedness ensures uniform integrability of stopped rewards; more general theorems replace it by class D and appropriate regularity along stopping times.

An infinite horizon requires additional terminal or transversality control when limits of finite-horizon envelopes and stopping rules are taken. Uniform integrability alone does not force an optimal finite stopping time to exist. Consider deterministic rewards increasing to one while remaining strictly below it at every finite date. Waiting longer approaches the supremum, but every finite stopping rule misses it. If stopping at infinity is allowed, its payoff must be defined explicitly, producing a different problem.

Hindsight creates a separate issue even on a bounded finite tree. A pathwise rule that chooses whichever earlier date later turns out best need not be a stopping time: its decision to stop now can depend on a future branch. Its expected payoff is a useful upper bound, often called a prophet value, but it is not automatically attainable. The filtration-based recursion computes what can actually be achieved under the allowed information.