Quickest Detection of a Minimum of Disorder Times

A multi-source quickest detection problem is considered. Assume there are two independent Poisson processes X
¹
and X
²
with disorder times θ
1
and θ
2
, respectively: that is the intensities of X
¹
and X
²
change at random unobservable times θ
1
and θ
2
, respectively. θ
1
and θ
2
are independent of each other and are exponentially distributed. Define θ\#25B5\θ
1
\#22C0\θ
2
=min {θ
1
, θ
2
}. For any stopping time τ that is measurable with respect to the filtration generated by the observations define a penalty function of the form R
τ
=\#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 X
¹
and X
²
reveal information about the disorder time θ, even this simple problem is more involved than solving the disorder problems for X
¹
and X
²
separately. 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.