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.
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