Hypergeometric solution to a gambler's ruin problem with a nonzero halting probability

TL;DR

This paper analyzes a modified gambler's ruin problem incorporating a halting probability, deriving exact solutions using hypergeometric functions, and explores the long-term behavior of the ruin probability.

Contribution

It introduces a novel gambler's ruin model with a halt state and provides exact solutions using hypergeometric functions, including moment calculations and long-time asymptotics.

Findings

Exact solution for ruin probability using hypergeometric functions.

Derived the moment generating function and moments of the duration.

Identified power-law decay of ruin probability in symmetric hopping cases.

Abstract

This paper treats of a kind of a gambler's ruin problem, which seeks the probability that a random walker first hits the origin at a certain time. In addition to a usual random walk which hops either rightwards or leftwards, the present paper introduces the `halt' that the walker does not hop with a certain probability. The solution to the problem can be obtained exactly using a Gauss hypergeometric function. The moment generating function of the duration is also calculated, and a calculation technique of the moments is developed. The author derives the long-time behavior of the ruin probability, which exhibits power-law behavior if the walker hops to the right and left with equal probability.