Increasing Gambler's Ruin duration and Brownian Motion exit times

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.