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.