Asymptotic Optimality of Change-Point Detection Schemes in General Continuous-Time Models

Abstract

In the early 1960s, Shiryaev obtained the structure of Bayesian stopping rules for detecting abrupt changes in independent and identically distributed sequences as well as in a constant drift of the Brownian motion. Since then, the methodology of optimal change-point detection has concentrated on the search for stopping rules that achieve the best balance of the mean detection delay and the rate of false alarms or minimize the mean delay under a fixed false alarm probability. In this respect, analysis of the performance of the Shiryaev procedure has been an open problem. Recently, Tartakovsky and Veeravalli (2005) investigated asymptotic performance of the Shiryaev Bayesian change detection procedure, the Page procedure, and the Shiryaev–Roberts procedure when the false alarm probability goes to zero for general discrete-time models. In this article, we investigate the asymptotic performance of Shiryaev and Shiryaev–Roberts procedures for general continuous-time stochastic models for a small false alarm probability and small cost of detection delay. We show that the Shiryaev procedure has asymptotic optimality properties under mild conditions, while the Shiryaev–Roberts procedure may or may not be asymptotically optimal depending on the type of the prior distribution. The presented asymptotic Bayesian detection theory substantially generalizes previous work in the field of change-point detection for continuous-time processes.