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On a gambler’s ruin problem. (English) Zbl 0965.60043
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.

60G40 Stopping times; optimal stopping problems; gambling theory
60G50 Sums of independent random variables; random walks
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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