Stochastic domination of exit times for random walks and Brownian motion with drift

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, while the Brownian motion case is the continuous analogue as discussed in the same paper. 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.