Quickest Search over Brownian Channels

TL;DR

This paper presents an optimal strategy for quickly identifying a Brownian motion with nonzero drift among a sequence of such processes with unknown drift, by reducing the problem to a solvable optimal stopping problem.

Contribution

It introduces a novel reduction of the quickest search problem over Brownian channels to an explicit optimal stopping problem for reflected diffusion.

Findings

Explicit solution for the optimal stopping rule.

Reduction of the search problem to a single filtration impulse control.

Provides a practical procedure for quickest detection in Brownian channels.

Abstract

In this paper we resolve an open problem proposed by Lai, Poor, Xin, and Georgiadis (2011, IEEE Transactions on Information Theory). Consider a sequence of Brownian Motions with unknown drift equal to one or zero, which we may be observed one at a time. We give a procedure for finding, as quickly as possible, a process which is a Brownian Motion with nonzero drift. This original quickest search problem, in which the filtration itself is dependent on the observation strategy, is reduced to a single filtration impulse control and optimal stopping problem, which is in turn reduced to an optimal stopping problem for a reflected diffusion, which can be explicitly solved.