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.

翻译：我们提出并解决了一种包含假阴性的最快检测问题变体。一个标准布朗运动在独立的指数随机时间获得漂移，该时间无法直接观测。基于对过程样本路径的连续时间观测，优化者必须在漂移出现后尽快检测到它。优化者可通过每次支付固定检测成本对系统进行多次检测。若在漂移出现前对系统进行测试，测试结果自然为阴性。然而，若在漂移出现后进行测试，测试可能以$\epsilon\in(0,1)$的概率未能检测到漂移而返回假阴性结果。当漂移最终被检测到时优化过程终止。该问题在数学上被表述为最优多重停止问题，并证明其等价于递归最优停止问题。利用这种关联性和自由边界方法，我们得到了期望成本和最优策略的显式公式。我们还证明当$\epsilon = 0$时，我们的期望成本是Shiryaev经典最优检测问题中期望成本经过模型参数重新标度后的仿射变换。