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Originally,
martingale referred to a class of
betting strategies popular in 18th century France. The simplest of
these strategies was designed for a game in which the gambler wins
his stake if a coin comes up heads and loses it if the coin comes
up tails. The strategy had the gambler double his bet after every
loss, so tha
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can use it to prove the impossibility of successful betting
strategies for a gambler with a finite lifetime (which gives
conditions (a) and (b)) and a house limit on bets (condition (c)).
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 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[X1].
In other words, the gambler leaves with the same amount of money
on average as when he started.
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Suppose we have a random walk 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; the time at which this first
occurs is a stopping time. If we happen to know that the expected
time that the walk ends is finite (say, from Markov chain theory),
the optional stopping theorem tells us that the expected position
when we stop is equal to the initial position a. Solving
a = pm + (1−p)0 for the probability p
that we reach m before 0 gives p = a/m.
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