Hi - I am trying to find the probability of hitting one of two boundaries in a simple random walk (I describe the problem precisely below). Actually, my main concern is to find the probability distribution over time to hit either one of two boundaries. I think that this is a very standard problem, i.e. time to ruin in the Gambler's ruin problem, and while I am able to find material describing the probability of hitting one, or other, boundary, I have not been successful in finding any material describing the probability distribution over hitting times. Could anyone help?(adsbygoogle = window.adsbygoogle || []).push({});

many thanks,

Mark

The problem is as follows:

A particle x begins at time t=0, with a value of 0. At each time interval, t=1,2,... it is incremented by 1 with probability p, and decremented by 1 with probability q=1-p. There are two boundaries a>0 and b<0, such that when the particle hits either one it stops. I would like to know 1) the probability that the particle hits a before b and 2) assuming it has hit a (or b), the probability distribution over the time taken to hit it.

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# Random walk on integers with two absorbing boundaries

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