In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value.

The concept can be understood through the following key principles:

• Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem implies that, on average, nothing can be gained by stopping play based on the information obtainable so far (i.e., without looking into the future).

• Certain conditions are necessary for this result to hold true. In particular, the theorem applies to doubling strategies and illustrates mathematically why such strategies cannot guarantee a profit with finite resources.

• The optional stopping theorem is an important tool of mathematical finance in the context of the fundamental theorem of asset pricing, helping to evaluate the expected returns of various pricing models.

Statement

denoting the set of natural numbers, including zero.

be a discrete-time martingale a stopping time, both with respect to a filtration. Assume that one of the following three conditions holds:

1. ( is almost surely bounded, i.e., there exists a constant almost surely.

2. ( has finite expectation and the conditional expectations of the absolute value.

3. (.

.

is a submartingale or a supermartingale for a supermartingale.

Remark

Under condition(. See the proof below for details.

Applications

The optional stopping theorem has widespread applications in probability, paradox resolution, and random walk theory:

Impossibility of perfect betting strategies: The theorem can be used to prove the impossibility of successful betting strategies for a gambler with a finite lifetime (giving condition()) or a house limit on bets (condition()).

• dollars on a fair coin flip at times 1, 2, 3, etc., winning their wager if the coin comes up heads and losing it if the coin comes up tails.

• Suppose further that the gambler can quit whenever they like, but cannot predict the outcome of gambles that have not happened yet.

• . In other words, the gambler leaves with the same amount of money on average as when they started.

• The same result holds if the gambler has a finite limit on their line of credit or how far in debt they may go, rather than a house limit on individual bets.

Expected stop position of a random walk: Suppose a random walk goes up or down by one with equal probability on each step.

• ; the time at which this first occurs is a stopping time.

• If it is known that the expected time at which the walk ends is finite (e.g., from Markov chain.

• .

  Expected time of a random walk: martingale from .

Violations of the theorem (Counterexamples): Care must be taken to ensure that one of the conditions of the theorem holds.

• .

• .

• . The failure of the optional stopping theorem in this case shows that all three of the conditions fail for a one-sided stopping time with an unrestricted state space.

Proof

denote the stopped process, which is also a martingale (or a submartingale or supermartingale, respectively). The framework of the proof relies on analyzing this process under the provided conditions:

• Under condition() or( is well defined.

• Under condition( is bounded, hence by Doob's martingale convergence theorem.

If condition(

By the monotone convergence theorem

If condition( is integrable.

If condition() holds, then we continue by inserting a conditional expectation where a representation of the expected value of non-negative integer-valued random variables is used for the last equality.

, the dominated convergence theorem

is a submartingale or supermartingale, respectively, change the equality in the last two formulas to the appropriate inequality.

References

External links

• Doob's Optional Stopping Theorem