On the Best 2-CUSUM Stopping Rule for Quickest Detection of Two-Sided Alternatives in a Brownian Motion Model

This work examines the problem of sequential detection of a change in the drift of a Brownian motion in the case of two-sided alternatives. Traditionally, 2-CUSUM stopping rules have been used for this problem due to their asymptotically optimal character as the mean time between false alarms tends to $\infty$. In particular, attention has focused on 2-CUSUM harmonic mean rules due to the simplicity of calculating their first moments. In this paper, expressions for the first moment of a general 2-CUSUM stopping rule and its rate of change are derived. These expressions are used to obtain explicit upper and lower bounds for it and its rate of change as one of the threshold parameters changes. Using these expressions we prove not only the existence but also the uniqueness of the best classical 2-CUSUM stopping rule with respect to the extended Lorden criterion suggested in [O. Hadjiliadis and G. V. Moustakides, Theory Probab. Appl., 50 (2006), pp. 75–85]. In particular, in both the symmetric and the nonsymmetric case we identify the class of the best 2-CUSUM stopping rule.