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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

 (q/p)a+b---(q/p)b pb = 1- qb = 1 - (q/p)a+b- 1

This can be simplified to

 b -(q/p)---1--. (q/p)a+b- 1

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

(p/q)a+b---(p/q)a (p/q)a+b- 1 .

One can verify that the expressions are equal.

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

pb = 1 - qb = 1- (1- b/(a + b)) = b/(a + b)

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