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Bayesian Quickest Detection Problems for Some Diffusion Processes

Part of: Stochastic processes Sequential methods Functional-differential and differential-difference equations Inference from stochastic processes Markov processes Decision theory

Published online by Cambridge University Press:  04 January 2016

Pavel V. Gapeev  and
Albert N. Shiryaev
Pavel V. Gapeev*
Affiliation:
London School of Economics
Albert N. Shiryaev*
Affiliation:
Steklov Institute of Mathematics
*
∗ Postal address: Department of Mathematics, London School of Economics, Houghton Street, London, WC2A 2AE, UK. Email address: p.v.gapeev@lse.ac.uk
∗∗ Postal address: Steklov Institute of Mathematics, Russian Academy of Sciences, Gubkina Street 8, Moscow 119991, Russia. Email address: albertsh@mi.ras.ru
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Abstract

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We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process with linear and exponential penalty costs for a detection delay. The optimal times of alarms are found as the first times at which the weighted likelihood ratios hit stochastic boundaries depending on the current observations. The proof is based on the reduction of the initial problems into appropriate three-dimensional optimal stopping problems and the analysis of the associated parabolic-type free-boundary problems. We provide closed-form estimates for the value functions and the boundaries, under certain nontrivial relations between the coefficients of the observable diffusion.

Keywords

disorder detectiondiffusion processmultidimensional optimal stoppingstochastic boundarya change-of-variable formula with local time on surfacesparabolic-type free-boundary problem

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 62M20: Prediction 34K10: Boundary value problems
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 62L15: Optimal stopping 60J60: Diffusion processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 1 , March 2013 , pp. 164 - 185
Copyright
© Applied Probability Trust 

Footnotes

Supported by the Alexander von Humboldt Fellowship for Experienced Researchers.

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