Time-Consistent Conditional Expectation Under Probability Distortion

Jin Ma (), Ting-Kam Leonard Wong () and Jianfeng Zhang ()
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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

Mathematics of Operations Research, 2021, vol. 46, issue 3, 1149-1180

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.

Keywords: Primary: 60H30; Secondary: 35R60; Primary: Dynamic programming/optimal control: Markov: finite state; infinite state; Secondary: Probability: diffusion; probability distortion; time inconsistency; nonlinear expectation (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:46:y:2021:i:3:p:1149-1180

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