Stopping Times and Optional Stopping

A stopping time is a random time whose arrival is recognizable from the information so far, and the optional stopping theorem says that a fair game stopped at such a time stays fair, a principle of surprising reach.

Definition

A stopping time is a random time at which the decision to stop depends only on the information available up to that time, and the optional stopping theorem states that, under suitable conditions, the expected value of a martingale evaluated at a stopping time equals its initial expected value.

Scope

The topic covers the definition of a stopping time relative to a filtration and the sigma-algebra of events known by a stopping time, the stopped process, the optional stopping and optional sampling theorems with the integrability and boundedness conditions they require, Wald's identities for sums stopped at a random time, and applications to gambler's ruin, hitting probabilities, and expected hitting times.

Core questions

• What makes a random time a stopping time, and why does that distinction matter?
• Under what conditions does stopping a martingale preserve its expected value?
• Why can the optional stopping theorem fail without integrability or boundedness assumptions?
• How do stopping times yield hitting probabilities and expected durations?

Key concepts

• stopping time
• stopped process
• optional sampling
• Wald's identities
• gambler's ruin

Key theories

Optional stopping theorem

If a stopping time is bounded, or the stopped martingale is uniformly integrable, or the time has finite mean with bounded increments, then the expected value of the martingale at the stopping time equals its initial value, the precise sense in which a fair game cannot be exploited by clever quitting rules.

Wald's identities

For a sum of independent identically distributed variables stopped at a stopping time of finite mean, the expected sum equals the mean times the expected stopping time, and a corresponding identity holds for the variance, results obtained by martingale optional stopping.

Clinical relevance

Optional stopping is the analytic engine for computing ruin probabilities and expected playing times in gambling and insurance, for the error probabilities and expected sample sizes of Wald's sequential probability ratio test, and for first-passage calculations in queueing, reliability, and the pricing of American-style financial options.

History

Doob formulated the optional sampling theorems for martingales, and Wald, working on sequential analysis during the 1940s, derived the identities for randomly stopped sums that the martingale framework later unified and explained.

Key figures

• Joseph L. Doob
• Abraham Wald
• David Williams

Seminal works

• williams1991

Frequently asked questions

Why must a stopping time be recognizable from past information?

If one could stop based on the future, one could quit a fair game exactly at favorable moments and win systematically; the requirement that the stopping decision use only information up to the present is precisely what keeps optional stopping honest.

When does the optional stopping theorem fail?

It can fail when the stopping time is unbounded and the martingale is not uniformly integrable, as in an unrestricted simple random walk where stopping at the first visit to a positive level gives an expected value different from the start.