Sequential change-point detection in continuous time when the post-change drift is unknown

Abstract

Let W_t(0≤t<∞) denote a Brownian motion process which has zero drift during the time interval [0,ν) and drift θ during the time interval [ν,∞), where θ and ν are unknown. The process W is observed sequentially. The general goal is to find a stopping time T of W that 'detects' the unknown time point ν as soon and as reliably as possible on the basis of this information. Here stopping always means deciding that a change in the drift has already occurred. We discuss two particular loss structures in a Bayesian framework. Our first Bayes risk is closely connected to that of the Bayes tests of power one of Lerche. The second Bayes risk generalizes the disruption problem of Shiryayev to the case of unknown θ.