Wiener Disorder Problem with Observations at Fixed Discrete Time Epochs

Suppose that a Wiener process gains a known drift rate at some unobservable disorder time with some zero-modified exponential distribution. The process is observed only at known fixed discrete time epochs, which may not always be spaced in equal distances. The problem is to detect the disorder time as quickly as possible by means of an alarm that depends only on the observations of Wiener process at those discrete time epochs. We show that Bayes optimal alarm times, which minimize expected total cost of frequent false alarms and detection delay time, always exist. Optimal alarms may in general sound between observation times and when the space-time process of the odds that disorder happened in the past hits a set with a nontrivial boundary. The optimal stopping boundary is piecewise-continuous and explodes as time approaches from left to each observation time. On each observation interval, if the boundary is not strictly increasing everywhere, then it first decreases and then increases. It is strictly monotone wherever it does not vanish. Its decreasing portion always coincides with some explicit function. We develop numerical algorithms to calculate nearly optimal detection algorithms and their Bayes risks, and we illustrate their use on numerical examples. The solution of Wiener disorder problem with discretely spaced observation times will help reduce risks and costs associated with disease outbreak and production quality control, where the observations are often collected and/or inspected periodically.