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Time-consistent conditional expectation under probability distortion

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  • Jin Ma
  • Ting-Kam Leonard Wong
  • Jianfeng Zhang

Abstract

We introduce a new notion of conditional nonlinear expectation under probability distortion. Such a distorted nonlinear expectation is not sub-additive 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 towards 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, 2018. "Time-consistent conditional expectation under probability distortion," Papers 1809.08262, arXiv.org, revised Jun 2020.
    • Handle: RePEc:arx:papers:1809.08262
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    References listed on IDEAS

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