IDEAS home Printed from
   My bibliography  Save this paper

Optimal stopping under probability distortion


  • Zuo Quan Xu
  • Xun Yu Zhou


We formulate an optimal stopping problem for a geometric Brownian motion where the probability scale is distorted by a general nonlinear function. The problem is inherently time inconsistent due to the Choquet integration involved. We develop a new approach, based on a reformulation of the problem where one optimally chooses the probability distribution or quantile function of the stopped state. An optimal stopping time can then be recovered from the obtained distribution/quantile function, either in a straightforward way for several important cases or in general via the Skorokhod embedding. This approach enables us to solve the problem in a fairly general manner with different shapes of the payoff and probability distortion functions. We also discuss economical interpretations of the results. In particular, we justify several liquidation strategies widely adopted in stock trading, including those of "buy and hold", "cut loss or take profit", "cut loss and let profit run" and "sell on a percentage of historical high".

Suggested Citation

  • Zuo Quan Xu & Xun Yu Zhou, 2011. "Optimal stopping under probability distortion," Papers 1103.1755,, revised Feb 2013.
  • Handle: RePEc:arx:papers:1103.1755

    Download full text from publisher

    File URL:
    File Function: Latest version
    Download Restriction: no


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Zuo Quan Xu, 2018. "Quantile optimization under derivative constraint," Papers 1803.02546,
    2. Henderson, Vicky & Hobson, David & Tse, Alex S.L., 2017. "Randomized strategies and prospect theory in a dynamic context," Journal of Economic Theory, Elsevier, vol. 168(C), pages 287-300.
    3. Zuo Quan Xu, 2014. "A Note on the Quantile Formulation," Papers 1403.7269,, revised Apr 2014.
    4. repec:eee:insuma:v:77:y:2017:i:c:p:111-118 is not listed on IDEAS
    5. Paul Viefers & Philipp Strack, 2014. "Too Proud to Stop: Regret in Dynamic Decisions," Discussion Papers of DIW Berlin 1401, DIW Berlin, German Institute for Economic Research.
    6. Zuo Quan Xu, 2013. "A New Characterization of Comonotonicity and its Application in Behavioral Finance," Papers 1311.6080,, revised Jun 2014.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1103.1755. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (arXiv administrators). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.