We formulate an optimal stopping problem  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  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.  In particular, we show that the optimality of the exit time of an interval (corresponding to the "cut-loss-or-stop-gain" strategy widely adopted in stock trading ) is endogenous for problems with convex distortion functions, including ones where distortion is absent. We also discuss economical interpretations of the results .
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" .
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for a geometric Brownian motion
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, either in a straightforward way for several important cases or in general
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We also discuss economical interpretations of the results.
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show that the optimality of the exit time of an interval (corresponding to the "cut-loss-or-stop-gain" strategy
justify several liquidation strategies
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) is endogenous for problems with convex distortion functions, including ones where distortion is absent. We also discuss economical interpretations of the results
, 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"