Stochastic Domination of Exit Times for Random Walks and Brownian Motion with Drift

TL;DR

This paper demonstrates that the exit times for biased random walks and drifted Brownian motions on symmetric intervals increase monotonically with the drift parameter, using elementary probabilistic transforms and comparison theorems.

Contribution

It provides a unified, elementary proof of the stochastic monotonicity of exit times for both discrete and continuous processes, extending recent results and offering new insights.

Findings

Exit times are stochastically monotone with respect to drift.

Elementary Girsanov transform simplifies the proof.

Parallel arguments for discrete and continuous cases.

Abstract

In this note, by an elementary use of Girsanov's transform we show that the exit time for either a biased random walk or a drifted Brownian motion on a symmetric interval is stochastically monotone with respect to the drift parameter. In the random walk case, this gives an alternative proof of a recent result of E. Pek\"oz and R. Righter in 2024. Our arguments in both discrete and continuous cases are parallel to each other. We also outline a simple SDE proof for the Brownian case based on a standard comparison theorem.