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On the Disorder Problem for a Negative Binomial Process

Part of: Miscellaneous topics - Partial differential equations Decision theory Stochastic processes Sequential methods

Published online by Cambridge University Press:  30 January 2018

Bruno Buonaguidi  and
Pietro Muliere
Bruno Buonaguidi*
Affiliation:
Bocconi University
Pietro Muliere*
Affiliation:
Bocconi University
*
∗ Postal address: Department of Decision Sciences, Bocconi University, Via Roentgen 1, 20136 Milan, Italy.
∗ Postal address: Department of Decision Sciences, Bocconi University, Via Roentgen 1, 20136 Milan, Italy.
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Abstract

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We study the Bayesian disorder problem for a negative binomial process. The aim is to determine a stopping time which is as close as possible to the random and unknown moment at which a sequentially observed negative binomial process changes the value of its characterizing parameter p ∈ (0, 1). The solution to this problem is explicitly derived through the reduction of the original optimal stopping problem to an integro-differential free-boundary problem. A careful analysis of the free-boundary equation and of the probabilistic nature of the boundary point allows us to specify when the smooth fit principle holds and when it breaks down in favour of the continuous fit principle.

Keywords

Disorder problemfree-boundary problemnegative binomial processoptimal stoppingprinciples of continuous and smooth fitregular boundarysequential detection

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 62L10: Sequential analysis 35R35: Free boundary problems
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 167 - 179
Copyright
© Applied Probability Trust 

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