Bayesian Switching Multiple Disorder Problems

Abstract

This document is the author's final accepted version of the journal article. There may be differences between this version and the published version. You are advised to consult the publisher's version if you wish to cite from it. Bayesian switching multiple disorder problemsPavel V. Gapeev * To appear in Mathematics of Operations ResearchThe switching multiple disorder problem seeks to determine an ordered infinite sequence of times of alarms which are as close as possible to the unknown times of disorders, or change-points, at which the observable process changes its probability characteristics. We study a Bayesian formulation of this problem for an observable Brownian motion with switching constant drift rates. The method of proof is based on the reduction of the initial problem to an associated optimal switching problem for a three-dimensional diffusion posterior probability process and the analysis of the equivalent coupled parabolic-type free-boundary problem. We derive analytic-form estimates for the Bayesian risk function and the optimal switching boundaries for the components of the the posterior probability process.

Sequential tracking of an unobservable two-state Markov process under Brownian noise

“…This provides a restatement of the optimisation problem (4) in terms of the process M , which is introduced directly as a (unique strong) solution to (8).…”

“…Let us briefly mention other results in the literature related to our paper. A similar multiple changepoint detection problem was studied by Gapeev [8]. His optimality criterion is somewhat different from ours, and he considers general (non-symmetric) two-state Markov processes.…”

“…His optimality criterion is somewhat different from ours, and he considers general (non-symmetric) two-state Markov processes. The main results of [8] consist in reduction of the changepoint detection problem to coupled optimal stopping problems and, further, to coupled free-boundary problems. Then the optimal boundaries at which switching occurs are identified as unique functions satisfying the smooth fit conditions.…”

A change of variable formula with applications to multi-dimensional optimal stopping problems

“…A general study of optimal stopping boundaries for multi-dimensional diffusions can be found in [8]. In the context of quickest detection problems, multi-dimensional situations arise for example in [27], [25] and [16]. In problems of singular control (that can be linked to optimal stopping) solved via free boundary methods we find the contributions [10], [3], [21], among others.…”

A sequential test for the drift of a Brownian motion with a possibility to change a decision

“…In the present paper, we deal with a similar tracking procedure and a penalty function, but the difference is that the unobservable drift coefficient does not change. Among other results on multiple changepoint detection, one can mention the paper [3], where a tracking problem for a general two-state Markov process with a Brownian noise was considered, and the paper [2], which studied a tracking problem for a compound Poisson process.…”