Jin Ma, University of Southern California

Time Consistent Conditional Expectation under Probability Distortion

Abstract

We introduce a new notion of conditional nonlinear expectation in the case where the underlying probability scale is distorted by a weight function. Such a distorted nonlinear expectation is non-sub-additive in general, hence beyond the scope of Peng's well-known framework of nonlinear expectations. A more fundamental problem when extending such distorted expectation to a dynamic setting is the time inconsistency, that is, the usual “tower property” fails. We show that, by localizing the probability distortion and restricting to a smaller class of random variables, it is possible to construct a conditional expectation is such a way that it coincides with the original nonlinear expectation at time zero, but it also has a time-consistent dynamics in the sense that the tower property remains valid. Furthermore, we show that this conditional expectation can be associate to a partial 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.

This is a joint work with Ting-Kam Leonard Wong and Jianfeng Zhang.