the Snell envelope
/ snel /
The Snell envelope is the abstract, model-free solution to an optimal stopping problem. Given a reward process Z_t (the payoff if you stop at time t), the Snell envelope U_t is defined as the smallest supermartingale that dominates Z, i.e. U_t >= Z_t for all t, U is a supermartingale, and any other supermartingale dominating Z is at least U. Equivalently, in discrete time it is built by backward induction: U_T = Z_T at the terminal time, and U_t = max( Z_t, E[ U_{t+1} given F_t ] ). Read that recursion in words: the value of holding the option to stop at time t is the larger of 'take the reward now' (Z_t) and 'wait one step and continue optimally' (the conditional expectation of next period's value).
Its defining property answers the optimal stopping question exactly: U_0 = sup_tau E[ Z_tau ] is the optimal expected reward, and the optimal stopping time is tau* = inf{ t : U_t = Z_t }, the first time the envelope touches the reward. The mechanism is the Doob-Meyer decomposition: U_t = M_t - A_t splits the envelope into a martingale M and a nondecreasing predictable process A, and tau* is exactly the last time before A starts to increase — up to tau* the envelope is a martingale (waiting is fair), and once A begins to grow strictly, continuing strictly loses value, so you must already have stopped. This makes precise the intuition 'stop as soon as the value of waiting equals the value of stopping'.
The Snell envelope is the bridge between the probabilistic and the analytic pictures of optimal stopping: in a Markovian diffusion model it equals V(X_t) for the value function V solving the free-boundary problem, but the envelope itself needs no Markov structure or PDE — it works for any adapted reward process satisfying mild integrability (e.g. E[ sup_t Z_t^- ] < infinity, or the family { Z_tau } uniformly integrable). An honest caveat: in continuous time, existence and the regularity needed to write U_t = M_t - A_t require the reward process to be of class D (uniformly integrable over stopping times) and right-continuous; without such conditions the supremum over stopping times may not be attained by any single tau.
Pricing a 3-period American put on a binomial tree: at expiry set U = (K - S)^+. Step back one period: at each node U = max( (K - S)^+, e^(-r dt) E*[ U_next ] ) under the risk-neutral measure. Whenever the immediate payoff (K-S)^+ wins the max, that node is an early-exercise node. The root value U_0 is the option price, and the set of nodes where U = payoff is the discrete stopping region.
Backward induction on a binomial tree IS the Snell envelope; early-exercise nodes are where it touches the payoff.
The Snell envelope is not the discounted reward — it is the smallest supermartingale ABOVE it; the optimal stopping time is the first time the two meet, not before. In continuous time you need a class-D, right-continuous reward for the supremum to be attained.