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 $\theta_{1}$ and $\theta_{2}$, respectively; that is, the intensities of $X^1$ and $X^2$ change at random unobservable times $\theta_1$ and $\theta_2$, respectively. $\theta_1$ and $\theta_2$ are independent of each other and are exponentially distributed. Define $\theta \triangleq \theta_1 \wedge \theta_2=\min\{\theta_{1},\theta_{2}\}$ . For any stopping time $\tau$ that is measurable with respect to the filtration generated by the observations define a penalty function of the form $$R_{\tau}=\mathbb{P}(\tau<\theta)+c \mathbb{E}[(\tau-\theta)^{+}],$$ where $c>0$ and $(\tau-\theta)^{+}$ is the positive part of $\tau-\theta$. It is of interest to find a stopping time $\tau$ that minimizes the above performance index. Since both observations $X^{1}$ and $X^{2}$ reveal information about the disorder time $\theta$, 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 three dimensional sufficient statistic, and the corresponding optimal stopping problem is examined. A two dimensional optimal stopping problem whose optimal stopping time turns out to coincide with the optimal stopping time of the original problem for some range of parameters is also solved. The value function of this problem serves as a tight upper bound for the original problem's value function. The two solutions are characterized by iterating suitable functional operators.