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Math 489/Math 889 Stochastic Processes and Advanced Mathematical Finance Dunbar, Fall 2010

Problem Statement

In a random walk starting at the origin find the probability that the point a ¿ 0 will be reached before the point -b ¡ 0.

Solution

This problem is taken from W. Feller, in Introduction to Probability Theory and Applications, Volume I, Chapter XIV, Section 9, problem 1, page 367.

Let the probability that Xi = +1 be p, that is, a step to the right occurs with probability p. The stated random walk problem is equivalent to the ruin probability ruin problem where a gambler starts with initial fortune b, and succeeds by reaching the level a + b before being ruined by reaching 0. This is the complementary probability to the ruin probability and so may be expressed as

This can be simplified to

Alternatively, we can view this as the ruin of the gambler’s adversary, and using the idea in the first corollary, express this as:

One can verify that the expressions are equal.

In the case p = 1/2 = q, these simplify to

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