Application of a Functional Equation to a Class of Stochastic Detection Models

In a proposed class of abstract stochastic detection models, the occurrence of detection is related to the time-dependent behavior of a detection functional defined on the signal-to-noise process sample paths. We describe a potential operations-analysis application in certain sonar detection problems. The method developed for computing detection probabilities for these models requires the solution of a functional equation involving the detection functional and the weak infinitesimal operator of the signal-to-noise process, assumed Markovian with stationary transition probabilities. Two hypothetical signal-to-noise processes considered in detail are the random sampling process and the stationary Gauss-Markov (Ornstein-Uhlenbeck) process in the special case when the detection functional measures time above a constant threshold level. We compare expressions for the mean time to detect, obtained for these processes, numerically.