On Optimum Methods in Quickest Detection Problems

In this paper optimum methods are developed for observing a process (1), in which the moment when a "disorder" $\theta$ appears is not known. The basic quantity characterizing the quality of this observation method is the mean time delay $\tau$ for detection of a disorder.
After making assumption (4) it is shown that for a set false alarm probability $\omega$ or for a set $\mathbf{N}$ – mathematical expectation of false alarm numbers occurring up till the moment the disorder appears the observation method minimizing $\tau=\tau(\omega)$ or $\tau=\tau(\mathbf{N})$ is based on an observation of aposteriori probability (23).
In § 3 a case is considered, wherein the disorder appears on the background of steadystate conditions arising when the disordes is absent. A method is found for minimizing $\tau=\tau(\mathbf{T})$ for a set $\mathbf{T}$ – mathematical expectation of the time between two false alarms. The dependency $\tau=\tau(\mathbf{T})$ is given by formula (36).