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 ˇ§tower propertyˇ¨ 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. |