Abstract
We study the sequential hypothesis testing and quickest change-point (or disorder) detection problems with linear delay penalty costs for observable Wiener processes under (constantly) delayed detection times. The method of proof consists of the reduction of the associated delayed optimal stopping problems for one-dimensional diffusion processes to the equivalent free-boundary problems and solution of the latter problems by means of the smooth-fit conditions. We derive closed-form expressions for the Bayesian risk functions and optimal stopping boundaries for the associated weighted likelihood ratio processes in the original problems of sequential analysis.
Highlights
The problem of sequential testing for two simple hypotheses about the drift rate of an observable Wiener process is to detect the form of its constant drift rate from one of the two given alternatives
In the Bayesian formulation of this problem, it is assumed that these alternatives have an a priori given distribution
Gapeev change-point θ at which the local drift rate of the process changes from one constant value to another
Summary
The problem of sequential testing for two simple hypotheses about the drift rate of an observable Wiener process (or Brownian motion) is to detect the form of its constant drift rate from one of the two given alternatives. The sequential testing and quickest change-point detection problems in the distributional properties of certain observable time-homogeneous diffusions processes were studied in Gapeev and Shiryaev (2011, 2013) on infinite time intervals. These two classical problems of sequential analysis for the case of observable compound Poisson processes, in which the unknown probabilitstic characteristics were the intensities and distributions of jumps, were investigated by Dayanik and Sezer (2006a, b).
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