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Time-Consistent Conditional Expectation Under Probability Distortion

Author

Listed:
  • Jin Ma

    (Department of Mathematics, University of Southern California, Los Angeles, California 90089)

  • Ting-Kam Leonard Wong

    (Department of Statistical Sciences, University of Toronto, Toronto, Ontario M5G 1Z5, Canada)

  • Jianfeng Zhang

    (Department of Mathematics, University of Southern California, Los Angeles, California 90089)

Abstract

We introduce a new notion of conditional nonlinear expectation under probability distortion. Such a distorted nonlinear expectation is not subadditive in general, so it is beyond the scope of Peng’s framework of nonlinear expectations. A more fundamental problem when extending the distorted expectation to a dynamic setting is time inconsistency , that is, the usual “tower property” fails. By localizing the probability distortion and restricting to a smaller class of random variables, we introduce a so-called distorted probability and construct a conditional expectation in such a way that it coincides with the original nonlinear expectation at time zero, but has a time-consistent dynamics in the sense that the tower property remains valid. Furthermore, we show that in the continuous time model this conditional expectation corresponds to a parabolic differential equation whose coefficient involves the law of the underlying diffusion. This work is the first step toward a new understanding of nonlinear expectations under probability distortion and will potentially be a helpful tool for solving time-inconsistent stochastic optimization problems.

Suggested Citation

  • Jin Ma & Ting-Kam Leonard Wong & Jianfeng Zhang, 2021. "Time-Consistent Conditional Expectation Under Probability Distortion," Mathematics of Operations Research, INFORMS, vol. 46(3), pages 1149-1180, August.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:3:p:1149-1180
    DOI: 10.1287/moor.2020.1101
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