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


Problem Statement

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

Solution

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 qa-1. Therefore the joint probability of the two events in succession is qqa-1.


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