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 \mathbb{N}0 denoting the set of natural integers, including zero. Let be a discrete-time martingale and τ a stopping time with values in \mathbb{N}0 ∪ {∞} , both with respect to a filtration (t)t∈\mathbb{N} 0 . Assume that one of the following three conditions holds: τ is almost surely bounded, i.e., there exists a constant c ∈ \mathbb{N} such that τ ≤ c a.s. τ 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 c such that almost surely on the event {τ > t} for all t ∈ \mathbb{N}0 . c such that a.s. for all t ∈ \mathbb{N}0 where ∧ denotes the minimum operator. Then Xτ 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 Xτ is defined as the almost surely existing pointwise limit of (Xt)t∈\mathbb{N} 0 , see the proof below for details.

Applications

τ at which he decides to quit (or goes broke and is forced to quit) is a stopping time. So the theorem says that E[Xτ] = E[X0] . In other words, the gambler leaves with the same amount of money on average as when he started. (The same result holds if the gambler, instead of having a house limit on individual bets, has a finite limit on his line of credit or how far in debt he may go, though this is easier to show with another version of the theorem.) a ≥ 0 that goes up or down by one with equal probability on each step. Suppose further that the walk stops if it reaches 0 or m ≥ a a . Solving for the probability p that the walk reaches m before 0 gives . X that starts at 0 and stops if it reaches –m or +m , and use the martingale from the examples section. If τ is the time at which X first reaches ±m , then . This gives E[τ] = m2 . +m , not at −m . The value of X at this stopping time would therefore be m . Therefore, the expectation value E[Xτ] must also be m , seemingly in violation of the theorem which would give E[Xτ] = 0 . The failure of the optional stopping theorem shows that all three of the conditions fail.

Proof

Let Xτ denote the stopped process, it is also a martingale (or a submartingale or supermartingale, respectively). Under condition or, the random variable Xτ is well defined. Under condition the stopped process Xτ is bounded, hence by Doob's martingale convergence theorem it converges a.s. pointwise to a random variable which we call Xτ . If condition holds, then the stopped process Xτ is bounded by the constant random variable M := c . Otherwise, writing the stopped process as gives for all t ∈ \mathbb{N}0 , where By the monotone convergence theorem If condition holds, then this series only has a finite number of non-zero terms, hence M is integrable. If condition holds, then we continue by inserting a conditional expectation and using that the event {τ > s} is known at time s (note that τ 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 M . Since the stopped process Xτ converges almost surely to Xτ , the dominated convergence theorem implies By the martingale property of the stopped process, hence Similarly, if X is a submartingale or supermartingale, respectively, change the equality in the last two formulas to the appropriate inequality.