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 ∗We are grateful for comments from seminar and conference participants at Oxford, ETH, University of Amsterdam, Chinese Academy of Sciences, University of Hong Kong, Chinese University of Hong Kong, Carnegie Mellon, University of Alberta, the 2009 Workshop on Optimal Stopping and Singular Stochastic Control Problems in Finance in Singapore, the 1st Columbia–Oxford Joint Workshop in Mathematical Finance in New York, the 6th World Congress of the Bachelier Finance Society in Toronto, and the 2011 Conference on Modeling and Managing Financial Risks in Paris. We thank Jan Ob lój for many helpful discussions on the Skorokhod embedding problem. Xu acknowledges financial support from a start-up fund of the Hong Kong Polytechnic University, Zhou acknowledges financial support from a start-up fund of the University of Oxford, and both Xu and Zhou acknowledge research grants from the Nomura Centre for Mathematical Finance and the Oxford–Man Institute of Quantitative Finance. †Mathematical Institute and Nomura Centre for Mathematical Finance, University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK, and Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong. ‡Mathematical Institute and Nomura Centre for Mathematical Finance, and Oxford–Man Institute of Quantitative Finance, The University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK, and Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. Email: .