Skip to main content
SpringerLink
Log in

Finite Horizon Sequential Detection with Exponential Penalty for the Delay

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The problem of the sequential detection of a change in the drift of a one-dimensional Brownian motion is considered under the assumptions that the detection must eventually occur within a finite horizon and the detection delay is exponentially penalized. Our results extend those obtained by Beibel for the infinite horizon sequential detection with exponential penalty (Ann Stat 28:1696–1701, 2000) and by Gapeev and Peskir for the finite horizon sequential detection with linear penalty (Stoch Process Appl 116:1770–1791, 2006).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Bayraktar, E., Dayanik, S.: Poisson disorder problem with exponential penalty for delay. Math. Oper. Res. 31, 217–233 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bayraktar, E., Dayanik, S., Karatzas, I.: The standard Poisson disorder problem revisited. Stoch. Process. Appl. 115, 1437–1450 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bayraktar, E., Dayanik, S., Karatzas, I.: Adaptive Poisson disorder problem. Ann. Appl. Probab. 16, 1190–1261 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beibel, M.: A note on sequential detection with exponential penalty for the delay. Ann. Stat. 28, 1696–1701 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernhart, G., Mai, J.F.: A note on the numerical evaluation of the Hartman-Watson density and distribution function. In: Glau, K., Scherer, M., Zagst, R. (eds.) Innovations in Quantitative Risk Mangement, pp. 337–345. Springer, Berlin (2015)

    Google Scholar 

  6. Buonaguidi, B.: The disorder problem for purely jump Lévy processes with completely monotone jumps. J. Stat. Plan. Inference 205, 203–218 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buonaguidi, B.: On the dimension reduction in the quickest detection problem for diffusion processes with exponential penalty for the delay. Electron. Commun. Probab. 26, 1–12 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  8. Buonaguidi, B., Muliere, P.: On the disorder problem for a negative Binomial process. J. Appl. Probab. 52, 167–179 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, F., Guo, X., Liao, ZW.: Optimal stopping time on semi-Markov processes with finite horizon. J. Optim. Theory Appl. https://doi.org/10.1007/s10957-022-02026-x

  10. Dayanik, S., Poor, H.V., Sezer, S.O.: Multisource Bayesian sequential change detection. Ann. Appl. Probab. 18, 552–590 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dayanik, S., Sezer, S.O.: Compound Poisson disorder problem. Math. Oper. Res. 31, 649–672 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. De Angelis, T., Peskir, G.: Global \(C^{1}\) regularity of the value function in optimal stopping problems. Ann. Appl. Probab. 30, 1007–1031 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Du Toit, J., Peskir, G.: The trap of complacency in predicting the maximum. Ann. Probab. 35, 340–365 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Du Toit, J., Peskir, G.: Selling a stock at the ultimate maximum. Ann. Probab. 19, 983–1014 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Ernst, P.A., Peskir, G.: Quickest real-time detection of a Brownian coordinate drift. Ann. Appl. Probab 32, 2652–2670 (2022)

  16. Gapeev, P.V.: The disorder problem for compound Poisson processes with exponential jumps. Ann. Appl. Probab. 15, 487–499 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gapeev, P.V., Peskir, G.: The Wiener disorder problem with finite horizon. Stoch. Process. Appl. 116, 1770–1791 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gapeev, P.V., Shiryaev, A.N.: Bayesian quickest detection problems for some diffusion processes. Adv. Appl. Probab. 45, 164–185 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gapeev, P.V., Stoev, Y.I.: On the sequential testing and quickest change-point detection problems for Gaussian processes. Stochastics 89, 1143–1165 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gerhold, S.: The Hartman-Watson distribution revisited: asymptotics for pricing Asian options. J. Appl. Probab. 48, 892–899 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Johnson, P., Peskir, G.: Quickest detection problems for Bessel processes. Ann. Appl. Probab. 27, 1003–1056 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1991)

    MATH  Google Scholar 

  23. Klein, M.: Comment on “Investment timing under incomplete information.” Math. Oper. Res. 34, 249–254 (2009)

  24. Ludkovski, M., Sezer, S.O.: Finite horizon decision timing with partially observable Poisson processes. Stoch. Model. 28, 207–247 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Peskir, G.: A change-of-variable formula with local time on curves. J. Theor. Probab. 18, 499–535 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Peskir, G.: On the American option problem. Math. Financ. 15, 169–181 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Peskir, G.: The Russian option: finite horizon. Finance Stochast. 9, 251–267 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Peskir, G.: On the fundamental solution of the Kolmogorov–Shiryaev equation. In: Kabanov, Y., Liptser, R., Stoyanov, J. (eds.) The Shiryaev Festschrift. From Stochastic Calculus to Mathematical Finance, pp. 535–546. Springer, Berlin (2006)

    Chapter  Google Scholar 

  29. Peskir, G.: Continuity of the optimal stopping boundary for two-dimensional diffusions. Ann. Appl. Probab. 29, 505–530 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Peskir, G., Shiryaev, A.N.: Solving the Poisson disorder problem. In: Sandmann, K., Schönbucher, P. (eds.) Advances in Finance and Stochastics, Essays in Honour of Dieter Sondermann, pp. 295–312. Springer, Berlin (2002)

    Chapter  Google Scholar 

  31. Peskir, G., Shiryaev, A.N.: Optimal Stopping and Free-Boundary Problems, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2006)

    MATH  Google Scholar 

  32. Peskir, G., Uys, N.: On Asian options of American type. In: Kyprianou, A., Schoutens, W., Wilmott, P. (eds.) Exotic Option Pricing and Advanced Lévy Models, pp. 217–235. Wiley, New York (2005)

    Google Scholar 

  33. Pirjol, D.: Asymptotic expansion for the Hartman–Watson distribution. arXiv:2001.09579v2 (2020)

  34. Poor, H.V.: Quickest detection with exponential penalty for delay. Ann. Stat. 26, 2179–2205 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shiryaev, A.N.: Optimal Stopping Rules. Springer, Berlin (1978)

    MATH  Google Scholar 

  36. Shreve, S.E.: Stochastic Calculus for Finance II. Continuous-Time Models. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The author wishes to thank the Editor, the Associate Editor and the referees for their insightful comments, which improved the presentation of the paper. Support from UCSC (D1 research grant) is also acknowledged.

Author information

Authors and Affiliations

  1. Department of Statistical Sciences, Università Cattolica, 20123, Milan, Italy

    Bruno Buonaguidi

Authors
  1. Bruno Buonaguidi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bruno Buonaguidi.

Additional information

Communicated by Xiaolu Tan.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Buonaguidi, B. Finite Horizon Sequential Detection with Exponential Penalty for the Delay. J Optim Theory Appl 198, 224–238 (2023). https://doi.org/10.1007/s10957-023-02239-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-023-02239-8

Keywords

Mathematics Subject Classification

Search

Navigation