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Optimal stopping asks: given a stochastic process X(t) and a payoff function g(X(t)), when should you stop to maximize the expected reward? The solution is characterized by the Snell envelope — the smallest supermartingale dominating the payoff process — and the optimal stopping time is the first time the process hits the boundary of the "continuation region." In continuous time, this leads to a free-boundary problem where the stopping boundary itself must be determined as part of the solution.
Optimal stopping is the mathematical theory of deciding when to act. Given a stochastic process X(t) and a payoff g(X(t)) received upon stopping, the problem is to choose the stopping time τ that maximizes E[g(X(τ))]. The classic examples are selling an asset (stop when the price is "high enough"), exercising an option (stop when the intrinsic value justifies giving up future optionality), and the secretary problem (stop when the current candidate is likely the best). The theory draws on martingales, dynamic programming, and free-boundary problems.
In discrete time with finite horizon, the solution is given by backward induction. Define V_N(x) = g(x) (at the terminal time, you must stop). For earlier times, V_n(x) = max{g(x), E[V_{n+1}(X_{n+1}) | X_n = x]} — the maximum of stopping now versus the expected value of continuing optimally. The optimal stopping time is τ* = min{n : V_n(X_n) = g(X_n)} — the first time the value function equals the immediate payoff, meaning there is nothing to gain from waiting. The value process V_n(X_n) is the Snell envelope — the smallest supermartingale that dominates the payoff process g(X_n).
In continuous time, optimal stopping for diffusions leads to free-boundary problems. For the process dX = μ dt + σ dW with payoff g(x) and discount rate r, the value function V(x) satisfies the equation LV - rV = 0 in the continuation region C = {x : V(x) > g(x)}, where L is the generator of X, and V(x) = g(x) in the stopping region S = {x : V(x) = g(x)}. The boundary ∂C between the two regions is free — it must be determined as part of the solution. The smooth-pasting condition V'(x*) = g'(x*) at the free boundary x* is the additional equation that pins down the boundary location.
The most important financial application is American option pricing. An American put on a stock following GBM with strike K has value V(S,t) = sup_τ E_Q[e-r(τ-t)(K-S_τ)⁺]. This is a free-boundary problem: in the continuation region {S > S*(t)}, V satisfies the Black-Scholes PDE; in the stopping region {S ≤ S*(t)}, V = K - S. The exercise boundary S*(t) is a decreasing function of time (as maturity approaches, the threshold for exercising drops because there is less future optionality). Unlike European options, no closed-form formula exists for American options — they are computed by binomial trees, finite difference methods, or least-squares Monte Carlo (the Longstaff-Schwartz algorithm). The optimal stopping framework unifies these computational approaches with a rigorous mathematical foundation.