Optimal Stopping

Optimal stopping is the problem of choosing a stopping time to maximize the expected value of a stochastic reward. For a reward process adapted to a filtration , the finite-horizon value process is

where is the set of admissible stopping times taking values in . The operator denotes the essential supremum, which is the least almost-sure upper bound for the conditional expectations, ignoring changes on events of probability zero. Under standard integrability conditions, is the Snell envelope of , and an optimal rule is often the first time at which .

Valuing an American option is an optimal stopping problem whose reward is the discounted exercise payoff. When the reward depends on a Markov process, the same problem can be described by a variational inequality or a free boundary problem.