Optimality properties of the shiryaev-roberts procedure

In 1961, for detecting a change in the drift of a Brownian motion, Shiryaev introduced what is now usually referred to as the Shiryaev-Roberts procedure. This procedure has a number of optimality and asymptotic optimality properties in various settings. Shiryaev (1961, 1963), and more recently Feinberg and Shiryaev (2006), established exact optimality properties in the context of monitoring a Brownian motion for a (known) change of drift. Their method of proof relies on techniques particular to Brownian motion that are not applicable in discrete time. Here we establish analogous results in a general discrete time setting, where surveillance is not relegated to a change of mean or to normal observations only. Our method of proof relies on asymptotic Bayesian analysis and on renewal theory.