Quickest detection of a minimum of disorder times

TL;DR

This paper addresses a multi-source quickest detection problem involving two Poisson processes with unknown disorder times, formulating an optimal stopping problem to minimize a combined penalty of false alarms and delay, and providing solutions through iterative operators.

Contribution

It introduces a novel formulation for detecting the earliest disorder among multiple Poisson sources and solves the associated optimal stopping problem using a reduced two-dimensional approach.

Findings

Derived a three-dimensional sufficient statistic for the problem

Solved a two-dimensional optimal stopping problem with matching solutions

Provided a tight upper bound for the original problem's value function

Abstract

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.