  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. Applications
to real life situations in which two-sided changes can occur are discussed.
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 in calculating their first moments. In this
paper, we derive closed-form expressions for the first moment of a general
2-CUSUM stopping rule. We use these expressions to obtain explicit upper and
lower bounds for it. Moreover, we derive an expression for the rate of change
of this first moment as one of the threshold parameters changes. Based on these
expressions we obtain explicit upper and lower bounds to this rate of change.
Using these expressions we are able to find the best 2-CUSUM stopping rule with
respect to the extended Lorden criterion. In fact, we demonstrate not only the
existence but also the uniqueness of the best 2-CUSUM stopping both in the case
of a symmetric change and in the case of a non-symmetric case. Furthermore, we
discuss the existence of a modification of the 2-CUSUM stopping rule that has a
strictly better performance than its classical 2-CUSUM counterpart for small
values of the mean time between false alarms. We conclude with a discussion on
the open problem of strict optimality in the case of two-sided alternatives.
