Optimal Stopping Under Probability Distortion

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