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

2024/08/01 by Xi Geng, Geng, Xi, Greg Markowsky +1
Business, Management and Accounting · Mathematics · Physics and Astronomy · #60G50 #60J65 #Advanced Queuing Theory Analysis #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2408.00277

openalex publication_date 2024/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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. Peköz 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.

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