Quickest Detection of a Minimum of Disorder Times

A multi-source quickest detection problem is considered. Assume there are two independent Poisson processes X¹and X²with disorder times θ1and θ2, respectively: that is the intensities of X¹and X²change at random unobservable times θ1and θ2, respectively. θ1and θ2are independent of each other and are exponentially distributed. Define θ#25B5\\θ1#22C0\\θ2=min {θ1, θ2}. For any stopping time τ that is measurable with respect to the filtration generated by the observations define a penalty function of the form Rτ=#2219(τ<θ)+c#1D404[(τ-θ)⁺], where c > 0 and (τ - 0)⁺is the positive part of τ - 0. It is of interest to find a stopping time τ that minimizes the above performance index. Since both observations X¹and X²reveal information about the disorder time θ, even this simple problem is more involved than solving the disorder problems for X¹and X²separately. This problem is formulated in terms of a two dimensional sufficient statistic, and the corresponding optimal stopping problem is examined. Using a suitable single jump operator, this problem is solved explicitly.