Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-25T17:36:31.253Z Has data issue: false hasContentIssue false
  • English
  • Français

Disorder detection with costly observations

Part of: Sequential methods Parametric inference

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: ekstrom@math.uu.se
*Postal address: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48104, USA
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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