A quickest detection problem with false negatives

TL;DR

This paper addresses a quickest detection problem involving false negatives, formulating it as an optimal multiple stopping problem, and deriving explicit strategies and costs using free boundary methods, extending classical detection models.

Contribution

It introduces a novel quickest detection model with false negatives, providing explicit solutions and connecting it to classical detection problems.

Findings

Explicit formulas for expected detection cost.

Optimal detection strategy characterized by free boundary methods.

Extension of classical models to include false negatives.

Abstract

We formulate and solve a variant of the quickest detection problem which features false negatives. A standard Brownian motion acquires a drift at an independent exponential random time which is not directly observable. Based on the observation in continuous time of the sample path of the process, an optimizer must detect the drift as quickly as possible after it has appeared. The optimizer can inspect the system multiple times upon payment of a fixed cost per inspection. If a test is performed on the system before the drift has appeared then, naturally, the test will return a negative outcome. However, if a test is performed after the drift has appeared, then the test may fail to detect it and return a false negative with probability $\epsilon\in(0,1)$. The optimisation ends when the drift is eventually detected. The problem is formulated mathematically as an optimal multiple stopping problem, and it is shown to be equivalent to a recursive optimal stopping problem. Exploiting such connection and free boundary methods we find explicit formulae for the expected cost and the optimal strategy. We also show that when $\epsilon = 0$ our expected cost is an affine transformation of the one in Shiryaev's classical optimal detection problem with a rescaled model parameter.