Optimal Stopping Problems

An optimal stopping problem seeks to find a stopping time $\tau$ that maximizes $\mathbb{E}[f(X_\tau)]$ subject to constraints. These problems arise in finance (American options), statistics (sequential testing), and operations research.

Problem formulation

Given a process $(X_t)$ and payoff function $g(t, X_t)$ , find:

$V_0 = \sup_{\tau} \mathbb{E}[e^{-r\tau} g(\tau, X_\tau)],$

where the supremum is over all stopping times $\tau \leq T$ .

Theorem8.1Optimal stopping theorem

The optimal stopping time is $\tau^* = \inf\{t : V_t = g(t, X_t)\}$ , the first time the value function $V_t$ equals the immediate payoff $g(t, X_t)$ . The value function satisfies the free boundary problem.

Example: American put option

An American put on a stock $S_t$ gives the right to sell at strike $K$ at any time up to maturity $T$ . The payoff is $(K - S_\tau)^+$ . The optimal exercise boundary is a critical level $S^*(t)$ : exercise when $S_t \leq S^*(t)$ .

For perpetual options ( $T = \infty$ ), explicit solutions exist via the smooth pasting principle.

ExampleSelling a house

You receive offers $X_n$ i.i.d. from distribution $F$ . Each day costs $c$ . When should you accept? The optimal rule: accept the first offer exceeding threshold $x^*$ , determined by $x^* = c + \mathbb{E}[\max(X_1, x^*)]$ .

Summary

Optimal stopping problems combine martingale theory, dynamic programming, and variational inequalities to determine when to take an action that maximizes expected reward.