On a gambler's ruin problem (Q2702759)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On a gambler's ruin problem |
scientific article |
Statements
13 March 2001
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classical ruin problem
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simple random walk
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absorption probabilities
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On a gambler's ruin problem (English)
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A particle moves in independent steps on the real line. Its initial position is 0. The \(u\)th step is described by a random variable \(X_u\) such that \(X_u=1\) w.p. \(p\), \(X_u=-1\) w.p. \(q\), and \(X_u=0\) w.p. \(r\), where \(p+q+r=1\). Let \(a\) and \(b\) be positive integers. Consider barriers at points \(-b\) and \(a\). If the particle reaches \(-b\), it is immediately returned to the given point \(j\), \(-b<j<a\). The barrier \(a\) is absorbing. The authors give an explicit result for the absorption probabilities. The derivation is based on a generating function.
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