Optimal Multistage Sampling in a Boundary-Crossing Problem

Abstract

Brownian motion with known positive drift is sampled in stages until it crosses a positive boundary $a$. A family of multistage samplers that control the expected overshoot over the boundary by varying the stage size at each stage is shown to be optimal for large $a$, minimizing a linear combination of overshoot and number of stages. Applications to hypothesis testing are discussed.