Time-consistent conditional expectation under probability distortion

TL;DR

This paper develops a new time-consistent conditional expectation framework under probability distortion, overcoming the limitations of traditional nonlinear expectations and enabling better handling of time-inconsistency in stochastic models.

Contribution

It introduces a novel construction of a time-consistent nonlinear expectation under probability distortion, extending the theory beyond Peng's framework.

Findings

Constructs a time-consistent conditional expectation with the tower property.

Shows the continuous-time version satisfies a parabolic differential equation.

Provides a foundation for solving time-inconsistent stochastic optimization problems.

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.