A Decentralized Detection Problem

Two detectors make independent observations of a Markov chain xt which jumps from state 0 to state 1 at the random time ¿. Based on its own observation detector i declares that the jump occurred at time ¿i, 1= 2. The problem is to find the stopping times ¿i that minimize the cost EJ(¿1, ¿2,¿) where J = 1(¿1≪¿)1(¿2≪¿)+C1(¿1-¿)1(¿1¿¿)+C2(¿2-¿)1(¿2¿¿) This cost puts a constant penalty for false alarms and, for each detector, a penalty proportional to the delay in detecting the jump. The major result is that the optimal stopping time ¿*i is characterized by a deterministic threshold P*i(t): ¿*i= min{t/P(¿≫t/Yⁱt)≪ P*i(t)}. where Yⁱt denotes the observations of i up to t. The thresholds P*1(t) are solutions of a pair of coupled dynamic programming equations.