optimal stopping and the free-boundary problem
An optimal stopping problem asks: I watch a random process X_t evolve, and at any moment of my choosing I may STOP and collect a reward g(X_tau) (possibly discounted, possibly minus a running cost). When should I pull the trigger? The decision must be made without foresight — the stopping time tau must be a stopping time, depending only on what I have seen so far, not on the future. This is the prototype of every 'act now or wait' decision under uncertainty: when to exercise an option, when to sell an asset, when to stop sampling and announce a verdict, when to maintain or scrap a machine. The value function V(x) = sup_tau E_x[ e^(-r tau) g(X_tau) ] is the best expected reward starting from X_0 = x, where the supremum is over all stopping times.
The state space splits into two regions. The continuation region C = { x : V(x) > g(x) } is where waiting is strictly better than stopping; the stopping region D = { x : V(x) = g(x) } is where you should stop immediately. The boundary between them is the free boundary — 'free' because its location is not given in advance but is part of the unknown, determined by the problem itself. On C the value function solves the homogeneous PDE (L - r) V = 0, where L is the generator of X (for a diffusion, a second-order elliptic operator); on D it equals g. Gluing these together gives a free-boundary problem, equivalently the variational inequality min{ rV - L V, V - g } = 0 on the whole space, with the smooth-fit condition that V and its first derivative match g across the boundary. Probabilistically, the discounted value process e^(-rt) V(X_t) is a supermartingale that becomes a martingale up to the optimal stopping time tau* = inf{ t : X_t in D }, the first entry into the stopping region.
This is where probability meets PDE most cleanly: the Snell envelope (the smallest supermartingale dominating the reward) is the abstract solution, while the variational inequality is its analytic face, and the optimal rule is always a threshold/region rule of the form 'stop the first time X leaves C'. An honest caveat: smooth fit (continuity of V') is a theorem under regularity (e.g. for non-degenerate diffusions, when g is smooth enough and the boundary is regular), not a free lunch — for problems with jumps or degenerate diffusion it can fail, and one must verify it rather than assume it. Also the supremum need not be attained if the horizon is infinite and rewards do not decay; integrability/transversality conditions are part of the honest statement.
Take X_t a Brownian motion with drift and the reward g(x) = (x - K)^+ (a perpetual call). The continuation region is an interval (-infinity, b*) and you exercise the first time X reaches the threshold b*. The threshold b* is found by the smooth-fit condition: V(b*) = b* - K and V'(b*) = 1. Below b* you wait (V > g); at b* the line g and the curve V touch tangentially — that tangency IS smooth fit.
The perpetual call: stop the first time the state hits a threshold fixed by smooth fit (value and slope match).
The optimal rule is 'stop on first exit from the continuation region' — but smooth fit (continuity of V') is a theorem requiring regularity, not an axiom; for jump or degenerate models it can fail and must be checked.