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 Lordenas criterion is proposed. When the possible drifts, assumed after the change, have the same sign, the 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 specific 2-CUSUM rule is shown to be asymptotically optimal as the frequency of false alarms tends to infinity.