Optimal Stopping Under Probability Distortion

39 Pages Posted: 12 Mar 2011

See all articles by Zuo Quan Xu

Zuo Quan Xu

Hong Kong Polytechnic University

Xun Yu Zhou

Columbia University - Department of Industrial Engineering and Operations Research (IEOR)

Date Written: March 9, 2011

Abstract

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.

Keywords: optimal stopping, probability distortion, Choquet expectation, probability distribution/qunatile function, Skorokhod embedding problem, $S$-shaped and reverse $S$-shaped function

Suggested Citation

Xu, Zuo Quan and Zhou, Xunyu, Optimal Stopping Under Probability Distortion (March 9, 2011). Available at SSRN: https://ssrn.com/abstract=1781732 or http://dx.doi.org/10.2139/ssrn.1781732

Zuo Quan Xu (Contact Author)

Hong Kong Polytechnic University ( email )

Hung Hom
Kowloon, 0
Hong Kong

Xunyu Zhou

Columbia University - Department of Industrial Engineering and Operations Research (IEOR) ( email )

331 S.W. Mudd Building
500 West 120th Street
New York, NY 10027
United States

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