We introduce a new notion of conditional nonlinear expectation where the underlying probability scale is distorted by a weight function . Such a distorted nonlinear expectation is not sub-additive in general, so 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  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  this conditional expectation corresponds to a parabolic differential equation , hence even a backward stochastic differential equation, which involves the law of the underlying diffusion. This work is the first step towards a new understanding of nonlinear expectations beyond capacity theory , and will potentially be a helpful tool for solving time-inconsistent stochastic optimization problems.
We introduce a new notion of conditional nonlinear expectation under probability distortion . Such a distorted nonlinear expectation is not sub-additive in general, so 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.
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where the underlying probability scale is distorted by a weight function
under probability distortion
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introduce a so-called distorted probability and
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in the continuous time model
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, hence even a backward stochastic differential equation, which
whose coefficient
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beyond capacity theory
under probability distortion