Stochastic Maximum Principle Under Probability Distortion

  • Qizhu Liang
  • Jie XiongEmail author


Within the framework of the cumulative prospective theory of Kahneman and Tversky, this paper considers a continuous-time behavioral portfolio selection problem whose model includes both running and terminal terms in the objective functional. Despite the existence of S-shaped utility functions and probability distortions, a necessary condition for the optimality is derived. The results are applied to a few examples.


Cumulative prospective theory S-shaped utility function Probability distortion Stochastic maximum principle Behavioral portfolio optimization 

AMS subject classifications

Primary 93E20 Secondary 91G80 



We would like to than an anonymous reviewer for his/her constructive comments and suggestions which improve this paper substantially.


  1. 1.
    Allais, M.: Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’ecole americaine. Econometrica 21, 503–546 (1953). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benartzi, S., Thaler, R.H.: Myopic loss aversion and the equity premium puzzle. Q. J. Econ. 110, 73–92 (1995). CrossRefzbMATHGoogle Scholar
  3. 3.
    Duffie, D., Epstein, L.G.: Stochastic differential utility. Econometrica 60, 353–394 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fishburn, P.C.: Nonlinear Preference and Utility Theory. Johns Hopkins University Press, Baltimore (1988)zbMATHGoogle Scholar
  5. 5.
    He, X.D., Zhou, X.Y.: Portfolio choice under cumulative prospect theory: an analytical treatment. Manag. Sci. 57, 315–331 (2011a). CrossRefzbMATHGoogle Scholar
  6. 6.
    He, X.D., Zhou, X.Y.: Portfolio choice via quantiles. Math. Financ. 21, 203–231 (2011b). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jin, H.Q., Zhou, X.Y.: Behavioral portfolio selection in continuous time. Math. Financ. 18, 385–426 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jin, H.Q., Zhou, X.Y.: Greed, leverage, and potential losses: a prospect theory perspective. Math. Financ. 23, 122–142 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47, 263–291 (1979). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Karatzas, I., Lehoczky, J.P., Shreve, S.E., Xu, G.L.: Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optim. 29, 702–730 (1991). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, New York (1998)CrossRefGoogle Scholar
  12. 12.
    Levy, H., Levy, M.: Prospect theory and mean-variance analysis. Rev. Financ. Stud. 17, 1015–1041 (2003). CrossRefGoogle Scholar
  13. 13.
    Lopes, L.L.: Between hope and fear: the psychology of risk. Adv. Exp. Soc. Psychol. 20, 255–295 (1987). CrossRefGoogle Scholar
  14. 14.
    Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51, 247–257 (1969). CrossRefGoogle Scholar
  15. 15.
    Peng, S.G.: A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28, 966–979 (1990). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pham, H.: Continuous-Time Stochastic Control and Optimization with Financial Applications. Springer, New York (2009)CrossRefGoogle Scholar
  17. 17.
    Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shefrin, H., Statman, M.: Behavioral portfolio theory. J. Financ. Quant. Anal. 35, 127–151 (2000). CrossRefGoogle Scholar
  19. 19.
    Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertain. 5, 297–323 (1992). CrossRefzbMATHGoogle Scholar
  20. 20.
    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (2007)zbMATHGoogle Scholar
  21. 21.
    Yaari, M.E.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yong, J.M., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MacauMacauChina
  2. 2.Department of Mathematics and SUSTech International Center for MathematicsSouthern University of Science and TechnologyShenzhenChina

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