In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph L. Doob, Joseph Doob) says that, under certain conditions, the expected value of a martingale (probability theory), 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 Martingale (betting system), 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 integers, including zero. Let be a discrete-time Martingale (probability theory), martingale and a stopping time with values in , both with respect to a Filtration (probability theory), filtration . Assume that one of the following three conditions holds: :() The stopping time is almost surely bounded, i.e., there exists a mathematical constant, constant such that a.s. :() The stopping time has finite expectation and the conditional expectations of the absolute value of the martingale increments are almost surely bounded, more precisely,

\mathbb[\tau]<\infty

and there exists a constant such that

\mathbb\bigl[, X_-X_t, \,\big\vert\,_t\bigr]\le c

almost surely on the event for all . :() There exists a constant such that a.s. for all where denotes the join and meet, minimum operator. Then is an almost surely well defined random variable and

\mathbb[X_]=\mathbb[X_0].

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

\mathbb[X_\tau]\ge\mathbb[X_0],

for a submartingale, and :

\mathbb[X_\tau]\le\mathbb[X_0],

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 ''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 (or goes broke and is 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 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.) *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 (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 gives . *Now consider a random walk that starts at and stops if it reaches or , and use the martingale from the Martingale (probability theory)#Examples of martingales, 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 (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 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 :

X_t^\tau=X_0+\sum_^(X_-X_s),\quad t\in_0,

gives for all , where :

M:=, X_0, +\sum_^, X_-X_s, =, X_0, +\sum_^\infty, X_-X_s, \cdot\mathbf_

. By the monotone convergence theorem :

\mathbb[M]=\mathbb[, X_0, ]+\sum_^\infty \mathbb\bigl[, X_-X_s, \cdot\mathbf_\bigr]

. 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 :

\begin\mathbb[M]
&=\mathbb[, X_0, ]+\sum_^\infty \mathbb\bigl[\underbrace_\bigr]\\
&\le\mathbb[, X_0, ]+c\sum_^\infty\mathbb(\tau>s)\\
&=\mathbb[, X_0, ]+c\,\mathbb[\tau]<\infty,\\
\end

where a Expected value#Discrete distribution taking only non-negative integer values, 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 :

\mathbb[X_\tau]=\lim_\mathbb[X_t^\tau].

By the martingale property of the stopped process, :

\mathbb[X_t^\tau]=\mathbb[X_0],\quad t\in_0,

hence :

\mathbb[X_\tau]=\mathbb[X_0].

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

References

# # {{cite book , last1=Bhattacharya , first1=Rabi , last2=Waymire , first2=Edward C. , title=A Basic Course in Probability Theory , publisher=Springer , year=2007 , pages=43–45 , isbn=978-0-387-71939-9, url=https://books.google.com/books?id=clQQVdUNBmcC&q=%22Optional+stopping%22

External links

Doob's Optional Stopping Theorem
Theorems in probability theory Theorems in statistics Articles containing proofs Martingale theory