Optimal stopping under probability distortion

Zuoquan Xu, Xun Yu Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

36 Citations (Scopus)


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."
Original languageEnglish
Pages (from-to)251-282
Number of pages32
JournalAnnals of Applied Probability
Issue number1
Publication statusPublished - 1 Feb 2013


  • Choquet expectation
  • Optimal stopping
  • Probability distortion
  • Probability distribution/qunatile function
  • S-shaped and reverse S-shaped function
  • Skorokhod embedding

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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