Extended gambler's ruin problem

TL;DR

This paper extends the classical gambler's ruin problem by incorporating delayed actions and absorption states, analyzing absorption probabilities, extremal values, and asymptotic behaviors of the process.

Contribution

It introduces a generalized gambler's ruin model with new features like staying in place and absorption, providing analytical results and asymptotic analysis.

Findings

Derived absorption probabilities and extremal value distributions.

Analyzed asymptotic behavior of absorption probabilities.

Introduced a conjugate version of the random walk.

Abstract

In the extended gambler's ruin problem we can move one step forward or backward (classical gambler's ruin problem), we can stay where we are for a time unit (delayed action) or there can be absorption in the current state (game is terminated without reaching an absorbing barrier). We obtain absorption probabilities, probabilities for maximum and minimum values of the ruin problem, expected time until absorption and the value of the game. We also investigate asymptotic behavior of absorption probabilities and expected time until absorption. We introduce a conjugate version of our random walk.