论文标题
部分可观测时空混沌系统的无模型预测
A quickest detection problem with false negatives
论文作者
论文摘要
我们制定并解决了具有假阴性的最快检测问题的变体。标准的布朗运动在独立的指数随机时间中获得漂移，这是无法直接观察的。基于过程样品路径的连续观察，优化器必须在出现后尽快检测到漂移。优化器可以在每次检查支付固定成本后多次检查系统。如果在漂移出现之前对系统进行测试，则自然，测试将返回负面结果。但是，如果在漂移出现后执行测试，则测试可能无法检测到它，并以$ε\ in（0,1）$返回假阴性。当最终检测到漂移时，优化结束。该问题在数学上是作为最佳多重停止问题而提出的，并且证明与递归的最佳停止问题相当。利用这种连接和自由边界方法，我们找到了预期成本和最佳策略的明确公式。我们还表明，当$ε= 0 $时，我们的预期成本是Shiryaev的经典最佳检测问题的仿射转换，并具有重新制定的模型参数。
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 $ε\in(0,1)$. 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 $ε= 0$ our expected cost is an affine transformation of the one in Shiryaev's classical optimal detection problem with a rescaled model parameter.
