论文标题
最快的布朗坐标漂移的实时检测
Quickest Real-Time Detection of a Brownian Coordinate Drift
论文作者
论文摘要
考虑两个或多个维度的布朗粒子的运动，其坐标过程是标准的布朗尼运动，最初是零漂移的零，然后在某些随机/无法观察的时间下，其中一个坐标过程将永久地（已知）非零漂移。鉴于布朗粒子的位置正在实时观察到，问题是要检测坐标过程尽可能准确地漂移的时间。当随机/无法观察的时间为（i）指数分布，并且（ii）独立于初始运动而无需漂移时，我们将在最不确定的情况下解决此问题。该解决方案是用停止时间来表示的，该时间最大程度地减少了虚假的早期检测的可能性和预期的遗漏晚期检测的延迟。据我们所知，这是第一次在文献中完全解决了这样的问题。
Consider the motion of a Brownian particle in two or more dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, one of the coordinate processes gets a (known) non-zero drift permanently. Given that the position of the Brownian particle is being observed in real time, the problem is to detect the time at which a coordinate process gets the drift as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the initial motion without drift. The solution is expressed in terms of a stopping time that minimises the probability of a false early detection and the expected delay of a missed late detection. To our knowledge this is the first time that such a problem has been solved exactly in the literature.
