Minimax Optimality of Shiryaev-Roberts Procedure for Quickest Drift Change Detection of a Brownian motion

TL;DR

This paper proves that a specially initialized Shiryaev-Roberts procedure is exactly optimal for quickest detection of drift changes in Brownian motion under a min-max framework with uncertain prior parameters.

Contribution

It establishes the minimax optimality of a modified Shiryaev-Roberts procedure for drift change detection in Brownian motion with unknown prior parameters.

Findings

Shiryaev-Roberts procedure with a specific starting point is exactly optimal.

The analysis covers a worst-case scenario with respect to prior parameters.

Both analytical and numerical evidence support the optimality claim.

Abstract

The problem of detecting a change in the drift of a Brownian motion is considered. The change point is assumed to have a modified exponential prior distribution with unknown parameters. A worst-case analysis with respect to these parameters is adopted leading to a min-max problem formulation. Analytical and numerical justifications are provided towards establishing that the Shiryaev-Roberts procedure with a specially designed starting point is exactly optimal for the proposed mathematical setup.