The problem of detecting a change in the probability distribution of
a random sequence is considered. Stopping times are derived that optimize the
tradeoff between detection delay and false alarms within two criteria. In both
cases, the detection delay is penalized exponentially rather than linearly, as
has been the case in previous formulations of this problem. The first of these
two criteria is to minimize a worst-case measure of the exponential detection
delay within a lower-bound constraint on the mean time between false alarms.
Expressions for the performance of the optimal detection rule are also
developed for this case. It is seen, for example, that the classical Page CUSUM
test can be arbitrarily unfavorable relative to the optimal test under
exponential delay penalty. The second criterion considered is a Bayesian one,
in which the unknown change point is assumed to obey a geometric prior
distribution. In this case, the optimal stopping time effects an optimal
trade-off between the expected exponential detection delay and the probability
of false alarm. Finally, generalizations of these results to problems in which
the penalties for delay may be path dependent are also considered.