Advances in Applied Probability

Bayesian quickest detection problems for some diffusion processes

Pavel V. Gapeev and Albert N. Shiryaev

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Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 45, Number 1 (2013), 164-185.

Dates
First available in Project Euclid: 15 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aap/1363354107

Digital Object Identifier
doi:10.1239/aap/1363354107

Mathematical Reviews number (MathSciNet)
MR3077545

Zentralblatt MATH identifier
1261.62077

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 34K10: Boundary value problems
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 62L15: Optimal stopping [See also 60G40, 91A60] 60J60: Diffusion processes [See also 58J65]

Keywords
disorder detection diffusion process multidimensional optimal stopping stochastic boundary a change-of-variable formula with local time on surfaces parabolic-type free-boundary problem

Citation

Gapeev, Pavel V.; Shiryaev, Albert N. Bayesian quickest detection problems for some diffusion processes. Adv. in Appl. Probab. 45 (2013), no. 1, 164--185. doi:10.1239/aap/1363354107. https://projecteuclid.org/euclid.aap/1363354107


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