Increasing Gambler's Ruin duration and Brownian Motion exit times

TL;DR

This paper demonstrates that increasing fairness in Gambler's Ruin prolongs game duration and that Brownian motion's exit times from symmetric intervals increase as drift approaches zero, extending existing theoretical results.

Contribution

It provides new stochastic comparisons for game duration and Brownian motion exit times, extending prior results with two different proof techniques.

Findings

Gambler's Ruin duration increases with game fairness

Brownian motion exit times grow as drift nears zero

Results extend previous theoretical understanding

Abstract

In Gambler's Ruin when both players start with the same amount of money, we show the playing time stochastically increases when the games are made more fair. We give two different arguments for this fact that extend results from \cite{Pek2021}. We then use this to show that the exit time from a symmetric interval for Brownian motion with drift stochastically increases as the drift moves closer to zero; this result is not easily obtainable from available explicit formulas for the density.