Wiener disorder problem with observations at fixed discrete time epochs

Date
2010
Authors
Dayanik, S.
Advisor
Instructor
Source Title
Mathematics of Operations Research
Print ISSN
0364-765X
Electronic ISSN
1526-5471
Publisher
Institute for Operations Research and the Management Sciences (I N F O R M S)
Volume
35
Issue
4
Pages
756 - 785
Language
English
Type
Article
Journal Title
Journal ISSN
Volume Title
Abstract

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 irst 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.

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Keywords
Optimal stopping, Sequential change detection, Wiener disorder problem, Discrete time epochs, Disease outbreaks, Exponential distributions, Numerical algorithms, Numerical example, Observation interval, Optimal detection algorithm, Optimal stopping, Piecewise-continuous, Production quality, Sequential change detection, Wiener disorder problem, Alarm systems, Algorithms, Disease control, Signal detection, Optimization
Citation
Published Version (Please cite this version)