Abstract
This work examines the problem of sequential change detection in the constant drift of a Brownian motion in the case of multiple alternatives. As a performance measure an extended Lorden's criterion is proposed. When the possible drifts, assumed after the change, have the same sign, the \hbox{CUSUM} rule, designed to detect the smallest in absolute value drift, is proven to be the optimum. If the drifts have opposite signs, then a specific 2-\hbox{CUSUM} rule is shown to be asymptotically optimal as the frequency of false alarms tends to infinity.
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