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Math 489/Math 889

Stochastic Processes and

Advanced Mathematical Finance

Dunbar, Fall 2010

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Problem Statement

Using the preceding two problems prove the Theorem The number of visits to the point
a > 0 that take place prior
to the first return to the origin has expectation (p/q)^{a} when p <
q and 1 when p = q.

The truly amazing implications of the theorem are best
explained in the language of fair games. A perfect coin is
tossed until the first equalization of the accumulated
numbers of heads and tails. The gambler receives one penny
for every time the accumulated number of heads exceeds the
accumulated number of tails by m. The ”fair entrance fee” for
playing this game equals 1, independently of m!.

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Solution

This problem is adapted from W. Feller, in Introduction to Probability Theory and Applications, Volume I, Chapter
XIV, Section 9, problem 5, page 367.

The number of visits to a without first returning to the
origin is a geometric random variable with ratio (2 * a -
1)/(2 * a).
Therefore the expected value is

Using a standard series summation formula (easily derived by
differentiating both sides of the geometric sum formula

with respect to r) namely

we see that the desired sum is

The interpretation in terms of the fair game is seen by
associating a heads with a +1 and a tails with -1 in the random walk, or
equivalently the sum of independent, identically distributed,
Bernoulli random variables. Then the accumulated number of
heads over the number of tails after n throws of the coin is just

or equivalently the position in the random walk. This is
astonishing since the entrance fee is the same whether m = 5, meaning that a
”penny” is earned every time the number of heads
exceeds the number of tails by 5 before the number of heads
and tails is equal, ending the game, (not unreasonable in our
experience) or if m = 1, 000,
000, which does seem unreasonable by comparison!

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