Optimal Multistage Sampling in a Boundary-Crossing Problem

TL;DR

This paper introduces an optimal multistage sampling method for Brownian motion with positive drift crossing a boundary, minimizing overshoot and stages, with applications to hypothesis testing.

Contribution

It develops a family of multistage samplers that optimize overshoot control and stage count for large boundaries, advancing boundary-crossing problem solutions.

Findings

The proposed samplers are asymptotically optimal for large boundaries.

They effectively minimize a linear combination of overshoot and number of stages.

Applications to hypothesis testing demonstrate practical utility.

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.