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Inprobability theory, theoptional stopping theorem (or sometimesDoob's optional sampling theorem, for American probabilistJoseph Doob) says that, under certain conditions, theexpected value of amartingale at astopping 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 todoubling strategies.

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

A discrete-time version of the theorem is given below, with${\displaystyle {\mathbb{N}}_0}$ denoting the set ofnatural numbers, including zero.

Let${\displaystyle X=(X_t{)}_{t\in {\mathbb{N}}_0}}$ be a discrete-timemartingale and${\displaystyle \tau }$ astopping time with values in${\displaystyle {\mathbb{N}}_0\cup \{\mathrm{\infty }\}}$, both with respect to afiltration${\displaystyle ({\mathcal{F}}_t{)}_{t\in {\mathbb{N}}_0}}$. Assume that one of the following three conditions holds:

(a) The stopping time${\displaystyle \tau }$ isalmost surely bounded, i.e., there exists aconstant${\displaystyle c\in \mathbb{N}}$ such that${\displaystyle \tau \le c}$ almost surely

(b) The stopping time${\displaystyle \tau }$ has finite expectation and the conditional expectations of theabsolute value of the martingale increments are almost surely bounded, more precisely, and there exists a constant${\displaystyle c}$ such that${\displaystyle \mathbb{E}[|X_{t+1}-X_t||{\mathcal{F}}_t]\le c}$ almost surely on the event${\displaystyle \{\tau >t\}}$ for all${\displaystyle t\in {\mathbb{N}}_0}$.

(c) There exists a constant${\displaystyle c}$ such that${\displaystyle |X_{min\{t,\tau \}}|\le c}$ almost surely for all${\displaystyle t\in {\mathbb{N}}_0}$.

Then${\displaystyle X_{\tau }}$ is an almost surely well defined random variable and${\displaystyle \mathbb{E}[X_{\tau }]=\mathbb{E}[X_0]}$.

Similarly, if the stochastic process${\displaystyle X=(X_t{)}_{t\in {\mathbb{N}}_0}}$ is asubmartingale or asupermartingale and one of the above conditions holds, then$${\displaystyle \mathbb{E}[X_{\tau }]\ge \mathbb{E}[X_0]}$$for a submartingale, and$${\displaystyle \mathbb{E}[X_{\tau }]\le \mathbb{E}[X_0]}$$for a supermartingale.

Under condition (c) it is possible that${\displaystyle \tau =\mathrm{\infty }}$ happens with positive probability. On this event${\displaystyle X_{\tau }}$ is defined as the almost surely existing pointwise limit of${\displaystyle X=(X_t{)}_{t\in {\mathbb{N}}_0}}$. See the proof below for details.

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• 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 (a)) or a house limit on bets (condition (b)). Suppose that the gambler can wager up to${\displaystyle c}$ dollars on afair 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${\displaystyle \tau }$ at which they decide to quit (or go broke and are forced to quit) is a stopping time. So the theorem says that${\displaystyle \mathbb{E}[X_{\tau }]=\mathbb{E}[X_0]}$. In other words, the gambler leaves with the same amount of moneyon 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 arandom walk starting at${\displaystyle a\ge 0}$ that goes up or down by one with equal probability on each step. Suppose further that the walk stops if it reaches 0 or${\displaystyle m\ge a}$; 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, fromMarkov chain theory), the optional stopping theorem predicts that the expected stop position is equal to the initial position${\displaystyle a}$. Solving${\displaystyle a=pm+(1-p)0}$ for the probability${\displaystyle p}$ that the walk reaches${\displaystyle m}$ before 0 gives${\displaystyle p=a/m}$.
• Now consider a random walk${\displaystyle X}$ that starts at 0 and stops if it reaches${\displaystyle -m}$ or${\displaystyle +m}$, and use the${\displaystyle Y_n=X_n^2-n}$ martingale fromMartingale (probability theory) § Examples of martingales. If${\displaystyle \tau }$ is the time at which${\displaystyle X}$ first reaches${\displaystyle \pm m}$, then${\displaystyle 0=\mathbb{E}[Y_0]=\mathbb{E}[Y_{\tau }]=m^2-\mathbb{E}[\tau ]}$. This gives${\displaystyle \mathbb{E}[\tau ]=m^2}$.
• 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${\displaystyle +m}$, not at${\displaystyle -m}$. The value of${\displaystyle X}$ at this stopping time would therefore be${\displaystyle m}$. Therefore, the expectation value${\displaystyle \mathbb{E}[\tau ]}$ must also be${\displaystyle m}$, seemingly in violation of the theorem which would give${\displaystyle \mathbb{E}[\tau ]=0}$. The failure of the optional stopping theorem shows that all three of the conditions fail.

Let${\displaystyle X^{\tau }}$ denote thestopped process, it is also a martingale (or a submartingale or supermartingale, respectively). Under condition (a) or (b), the random variable${\displaystyle X^{\tau }}$ is well defined. Under condition (c) the stopped process${\displaystyle X^{\tau }}$ is bounded, hence by Doob'smartingale convergence theorem it converges almost surely pointwise to a random variable which we call${\displaystyle X_{\tau }}$.

If condition (c) holds, then the stopped process${\displaystyle X^{\tau }}$ is bounded by the constant random variable${\displaystyle M:=c}$. Otherwise, writing the stopped process as$${\displaystyle X_t^{\tau }=X_0+\sum _{s=0}^{\tau -1\wedge t-1}(X_{s+1}-X_s),t\in {\mathbb{N}}_0}$$gives${\displaystyle X_t^{\tau }\le M}$ for all${\displaystyle t\in {\mathbb{N}}_0}$, where$${\displaystyle M:=|X_0|+\sum _{s=0}^{\tau -1}|X_{s+1}-X_s|=|X_0|+\sum _{s=0}^{\mathrm{\infty }}|X_{s+1}-X_s|\cdot {\mathbf{1}}_{\{\tau >s\}}.}$$

By themonotone convergence theorem$${\displaystyle \mathbb{E}[M]=\mathbb{E}[|X_0|]+\sum _{s=0}^{\mathrm{\infty }}\mathbb{E}[|X_{s+1}-X_s|\cdot {\mathbf{1}}_{\{\tau >s\}}].}$$

If condition (a) holds, then this series only has a finite number of non-zero terms, hence${\displaystyle M}$ is integrable.

If condition (b) holds, then we continue by inserting aconditional expectation and using that the event${\displaystyle \{\tau >s\}}$ is known at time${\displaystyle s}$ (note that${\displaystyle \tau }$ is assumed to be a stopping time with respect to the filtration), hencewhere arepresentation 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${\displaystyle M}$. Since the stopped process${\displaystyle X^{\tau }}$ converges almost surely to${\displaystyle X_{\tau }}$, thedominated convergence theorem implies$${\displaystyle \mathbb{E}[X_{\tau }]=\underset{t\to \mathrm{\infty }}{lim}\mathbb{E}[X_t^{\tau }].}$$

By the martingale property of the stopped process,$${\displaystyle \mathbb{E}[X_t^{\tau }]=\mathbb{E}[X_0],t\in {\mathbb{N}}_0,}$$hence$${\displaystyle \mathbb{E}[X_{\tau }]=\mathbb{E}[X_0].}$$

Similarly, if${\displaystyle X}$ is a submartingale or supermartingale, respectively, change the equality in the last two formulas to the appropriate inequality.

1. Grimmett, Geoffrey R.; Stirzaker, David R. (2001).Probability and Random Processes (3rd ed.). Oxford University Press. pp. 491–495.ISBN 9780198572220.
2. Bhattacharya, Rabi; Waymire, Edward C. (2007).A Basic Course in Probability Theory. Springer. pp. 43–45.ISBN 978-0-387-71939-9.

• Doob's Optional Stopping Theorem

• Theorems in probability theory
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• Martingale theory

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