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Disorder detection with costly observations

Part of: Parametric inference Sequential methods

Published online by Cambridge University Press:  25 April 2022

Erhan Bayraktar ,
Erik Ekström  and
Jia Guo
Erhan Bayraktar*
Affiliation:
University of Michigan
Erik Ekström*
Affiliation:
Uppsala University
Jia Guo*
Affiliation:
University of Michigan
*
*Postal address: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48104, USA
***Postal address: Department of Mathematics, Uppsala University, Box 256, 75105 Uppsala, Sweden. Email: [email protected]
*Postal address: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48104, USA
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Abstract

We study the Wiener disorder detection problem where each observation is associated with a positive cost. In this setting, a strategy is a pair consisting of a sequence of observation times and a stopping time corresponding to the declaration of disorder. We characterize the minimal cost of the disorder problem with costly observations as the unique fixed point of a certain jump operator, and we determine the optimal strategy.

Keywords

Wiener disorder detectiondiscrete cost of observationoptimal stopping

MSC classification

Primary: 62L10: Sequential analysis
Secondary: 62F15: Bayesian inference 62L15: Optimal stopping
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 2 , June 2022 , pp. 338 - 349
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Balmer, D. W. (1975). On a quickest detection problem with costly information. J. Appl. Prob. 12, 87–97.10.2307/3212410CrossRefGoogle Scholar
Bayraktar, E., Dayanik, S. and Karatzas, I. (2005). The standard Poisson disorder problem revisited. Stoch. Process. Appl. 115, 1437–1450.10.1016/j.spa.2005.04.011CrossRefGoogle Scholar
Bayraktar, E., Dayanik, S. and Karatzas, I. (2006). Adaptive Poisson disorder problem, Ann. Appl. Prob. 16, 1190–1261.10.1214/105051606000000312CrossRefGoogle Scholar
Bayraktar, E. and Kravitz, R. (2015). Quickest detection with discretely controlled observations. Sequent. Anal. 34, 77–133.10.1080/07474946.2015.995993CrossRefGoogle Scholar
Bouchard, B. and Touzi, N. (2011). Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49, 948–962.10.1137/090752328CrossRefGoogle Scholar
Çınlar, E. (2011). Probability and Stochastics (Graduate Texts in Math. 261). Springer, New York.Google Scholar
Dalang, R. C. and Shiryaev, A. N. (2015). A quickest detection problem with an observation cost. Ann. Appl. Prob. 25, 1475–1512.10.1214/14-AAP1028CrossRefGoogle Scholar
Davis, M. H. A. (1993), Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman & Hall, London.CrossRefGoogle Scholar
Dayanik, S. (2010). Wiener disorder problem with observations at fixed discrete time epochs. Math. Operat. Res. 35, 756–785.10.1287/moor.1100.0471CrossRefGoogle Scholar
Dyrssen, H. and Ekström, E. (2018). Sequential testing of a Wiener process with costly observations. Sequent. Anal. 37, 47–58.10.1080/07474946.2018.1427973CrossRefGoogle Scholar
Gapeev, P. V. and Shiryaev, A. N. (2013). Bayesian quickest detection problems for some diffusion processes. Adv. Appl. Prob. 45, 164–185.10.1239/aap/1363354107CrossRefGoogle Scholar
Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
Poor, H. V. and Hadjiliadis, O. (2009). Quickest Detection. Cambridge University Press.Google Scholar
Shiryaev, A. N. (1969). Two problems of sequential analysis. Cybernetics 3, 63–69.10.1007/BF01078755CrossRefGoogle Scholar
Shiryaev, A. N. (2004). A remark on the quickest detection problems. Statist. Decisions 22, 79–82.10.1524/stnd.22.1.79.32716CrossRefGoogle Scholar
Shiryaev, A. N. (2008). Optimal Stopping Rules (Stochastic Model. Appl. Prob. 8). Springer, Berlin.Google Scholar