Detection problem with post-change drift uncertainty

We consider the problem of detection of abrupt changes when there is uncertainty about the post-change distribution. In particular we examine this problem in the prototypical model of continuous time in which the drift of a Wiener process changes at an unknown time from zero to a random value. It is assumed that the change time is an unknown constant while the drift assumed after the change has a Bernoulli distribution with all values of the same sign independent of the process observed. We set up the problem as a stochastic optimization in which the objective is to minimize a measure of detection delay subject to a frequency of false alarm constraint. As a measure of detection delay we consider that of a worst detection delay weighed by the probabilities of the different possible drift values assumed after the change point to which we are able to compute a lower bound amongst the class of all stopping times. Our objective is to then construct low complexity, easy to implement decision rules, that achieve this lower bound exactly, while maintaining the same frequency of false alarms as the family of stopping times. In this effort, we consider a special class of decision rules that are delayed versions of CUSUM algorithm. In this enlarged collection, we are able to construct a family of computationally efficient decision rules that achieve the lower bound with equality, and then choose a best one whose performance is as close to the performance of a stopping time as possible.