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The Wiener disorder problem with finite horizon

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  • Gapeev, P.V.
  • Peskir, G.

Abstract

The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of 'disorder' when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is finite. The method of proof is based on reducing the initial problem to a parabolic free-boundary problem where the continuation region is determined by a continuous curved boundary. By means of the change-of-variable formula containing the local time of a diffusion process on curves we show that the optimal boundary can be characterized as a unique solution of the nonlinear integral equation.

Suggested Citation

  • Gapeev, P.V. & Peskir, G., 2006. "The Wiener disorder problem with finite horizon," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1770-1791, December.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:12:p:1770-1791
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    References listed on IDEAS

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    1. Goran Peskir, 2005. "On The American Option Problem," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 169-181, January.
    2. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14, April.
    3. Goran Peskir, 2005. "The Russian option: Finite horizon," Finance and Stochastics, Springer, vol. 9(2), pages 251-267, April.
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    Cited by:

    1. Buonaguidi, B., 2022. "The disorder problem for diffusion processes with the ϵ-linear and expected total miss criteria," Statistics & Probability Letters, Elsevier, vol. 189(C).
    2. Ameur, Hachmi Ben & Han, Xuyuan & Liu, Zhenya & Peillex, Jonathan, 2022. "When did global warming start? A new baseline for carbon budgeting," Economic Modelling, Elsevier, vol. 116(C).
    3. Zhenya Liu & Yuhao Mu, 2022. "Optimal Stopping Methods for Investment Decisions: A Literature Review," IJFS, MDPI, vol. 10(4), pages 1-23, October.
    4. A. N. ShiryaevM. V. Zhitlukhin & M. V. Zhitlukhin, 2012. "Disorder detection problems with applications in finance," Economics Discussion Paper Series 1229, Economics, The University of Manchester.
    5. repec:hum:wpaper:sfb649dp2006-057 is not listed on IDEAS
    6. Asaf Cohen, 2015. "Parameter Estimation: The Proper Way to Use Bayesian Posterior Processes with Brownian Noise," Mathematics of Operations Research, INFORMS, vol. 40(2), pages 361-389, February.
    7. Shiryaev Albert & Novikov Alexander A., 2009. "On a stochastic version of the trading rule “Buy and Hold”," Statistics & Risk Modeling, De Gruyter, vol. 26(4), pages 289-302, July.
    8. Gapeev, Pavel V. & Jeanblanc, Monique, 2021. "First-to-default and second-to-default options in models with various information flows," LSE Research Online Documents on Economics 110750, London School of Economics and Political Science, LSE Library.
    9. Gapeev, Pavel V., 2022. "Discounted optimal stopping problems in continuous hidden Markov models," LSE Research Online Documents on Economics 110493, London School of Economics and Political Science, LSE Library.
    10. Pavel V. Gapeev & Monique Jeanblanc, 2019. "Defaultable Claims In Switching Models With Partial Information," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-18, June.
    11. Belomestny, Denis & Gapeev, Pavel V., 2006. "An iteration procedure for solving integral equations related to optimal stopping problems," SFB 649 Discussion Papers 2006-043, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    12. Thomas Kruse & Philipp Strack, 2019. "An Inverse Optimal Stopping Problem for Diffusion Processes," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 423-439, May.
    13. Savas Dayanik, 2010. "Wiener Disorder Problem with Observations at Fixed Discrete Time Epochs," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 756-785, November.
    14. Savas Dayanik & Semih O. Sezer, 2016. "Sequential Sensor Installation for Wiener Disorder Detection," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 827-850, August.
    15. Tiziano De Angelis & Jhanvi Garg & Quan Zhou, 2022. "A quickest detection problem with false negatives," Papers 2210.01844, arXiv.org.
    16. Christensen, Sören & Irle, Albrecht, 2020. "The monotone case approach for the solution of certain multidimensional optimal stopping problems," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1972-1993.
    17. Bruno Buonaguidi, 2023. "Finite Horizon Sequential Detection with Exponential Penalty for the Delay," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 224-238, July.
    18. Gapeev, Pavel V., 2006. "Integral options in models with jumps," SFB 649 Discussion Papers 2006-068, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    19. Pavel V. Gapeev, 2020. "On the problems of sequential statistical inference for Wiener processes with delayed observations," Statistical Papers, Springer, vol. 61(4), pages 1529-1544, August.
    20. repec:hum:wpaper:sfb649dp2006-043 is not listed on IDEAS
    21. Gapeev, Pavel V., 2006. "Discounted optimal stopping for maxima in diffusion models with finite horizon," SFB 649 Discussion Papers 2006-057, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    22. Antonio Di Crescenzo & Shelemyahu Zacks, 2015. "Probability Law and Flow Function of Brownian Motion Driven by a Generalized Telegraph Process," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 761-780, September.
    23. repec:hum:wpaper:sfb649dp2006-068 is not listed on IDEAS
    24. Pavel V. Gapeev & Monique Jeanblanc, 2020. "Credit Default Swaps In Two-Dimensional Models With Various Informations Flows," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(02), pages 1-28, March.

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