In probability theory, the optional stopping theorem 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. 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: 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: 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 ) or a house limit on bets ). Suppose that the gambler can wager up to c dollars on a fair coin flip at times 1, 2, 3, etc., winning his wager if the coin comes up heads and losing it if the coin comes up tails. Suppose further that he can quit whenever he likes, but cannot predict the outcome of gambles that haven't happened yet. Then the gambler's fortune over time is a martingale, and the time at which he decides 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 he started.
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 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, 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 gives.
Now consider a random walk that starts at and stops if it reaches or, and use the martingale from the examples section. 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. 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 a.s. 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 , 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.
