Optional stopping theorem

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. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says 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.

The optional stopping theorem is an important tool of mathematical finance in the context of the fundamental theorem of asset pricing.

Statement

A discrete-time version of the theorem is given below, with denoting the set of natural numbers, including zero.

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

() The stopping time is almost surely bounded, i.e., there exists a constant such that almost surely

() The stopping time has finite expectation and the conditional expectations of the absolute value of the martingale increments are almost surely bounded, more precisely, and there exists a constant such that almost surely on the event for all .

() There exists a constant such that almost surely for all .

Then is an almost surely well defined random variable and .

Similarly, if the stochastic process is a submartingale or a supermartingale and one of the above conditions holds, then

for a submartingale, and

for a supermartingale.

Remark

Under condition () it is possible that happens with positive probability. On this event is defined as the almost surely existing pointwise limit of . See the proof below for details.

Applications

• The optional stopping theorem can be used to prove the impossibility of successful betting strategies for a gambler with a finite lifetime (which gives condition ()) or a house limit on bets (condition ()). Suppose that the gambler can wager up to 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. Then the gambler's fortune over time is a martingale, and the time at which they decide to quit (or go broke and are forced to quit) is a stopping time. So the theorem says that . 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, instead of having a house limit on individual bets, has a finite limit on their line of credit or how far in debt they may go, though this is easier to show with another version of the theorem.)
• Suppose a random walk starting at that goes up or down by one with equal probability on each step. Suppose further that the walk stops if it reaches 0 or ; 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 (say, from Markov chain theory), the optional stopping theorem predicts that the expected stop position is equal to the initial position . Solving for the probability that the walk reaches before 0 gives .
• Now consider a random walk that starts at 0 and stops if it reaches or , and use the martingale from . If is the time at which first reaches , then . This gives .
• Care must be taken, however, to ensure that one of the conditions of the theorem hold. For example, suppose the last example had instead used a 'one-sided' stopping time, so that stopping only occurred at , not at . The value of at this stopping time would therefore be . Therefore, the expectation value must also be , seemingly in violation of the theorem which would give . The failure of the optional stopping theorem shows that all three of the conditions fail.

Proof

Let denote the stopped process, it is also a martingale (or a submartingale or supermartingale, respectively). Under condition () or (), the random variable is well defined. Under condition () the stopped process is bounded, hence by Doob's martingale convergence theorem it converges almost surely pointwise to a random variable which we call .

If condition () holds, then the stopped process is bounded by the constant random variable . Otherwise, writing the stopped process as

gives for all , where

By the monotone convergence theorem

If condition () holds, then this series only has a finite number of non-zero terms, hence is integrable.

If condition () holds, then we continue by inserting a conditional expectation and using that the event is known at time (note that is assumed to be a stopping time with respect to the filtration), hence

where a representation of the expected value of non-negative integer-valued random variables is used for the last equality.

Therefore, under any one of the three conditions in the theorem, the stopped process is dominated by an integrable random variable . Since the stopped process converges almost surely to , the dominated convergence theorem implies

By the martingale property of the stopped process,

hence

Similarly, if 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