The decentralized quickest detection problem

We consider two hypotheses, H0 and H1, and two detectors. Initially hypothesis H0 is true with some probability P0, but at some random time θ a jump occurs and hypothesis H1 becomes true and remains true thereafter. The two detectors do not communicate. Each detector has to detect the time of the jump as accurately as possible based on his own measurements. After a detector declares that the jump has occurred he stops. Let τi(i=1,2) be the time that the detector i declares that the jump has occurred. The problem is to determine the decision rules of the two detectors to minimize a cost of the form EJ (τ1, τ2, θ), where false alarms are uniformly penalized and delays in detection are penalized linearly. We show that the member by member optimal strategies of the detectors are thresholds. The thresholds of the detectors are coupled and can be determined by the solution of non-linear algebraic equations.