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Stochastic Processes and

Advanced Mathematical Finance

Dunbar, Fall 2010

Prove: In a random walk starting at a
> 0 the probability to reach the origin before
returning to the starting point equals qq_{a-1}.

This problem is taken from W. Feller, in Introduction to Probability Theory and Applications, Volume I, Chapter XIV, Section 9, problem 2 (b), page 367.

If the walker starts at a and
goes to a + 1 at the first step,
then the walk must return to a again before possibly reaching
0. Hence we need only consider the possibility of the walk
starting from the point a - 1 at the first step, and then
reaching the value 0 before return to the starting point
a. The probability of going to
a - 1 is q, and then from a - 1
the subsequent independent probability of reaching 0 before
returning to a is the same as
the probability of the gambler being ruined, which is q_{a-1}. Therefore the joint probability of
the two events in succession is qq_{a-1}.

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