$g$-Expectation of Distributions

Abstract: We define $g$-expectation of a distribution as the infimum of the $g$-expectations of all the terminal random variables sharing that distribution. We present two special cases for nonlinear $g$ where the $g$-expectation of distributions can be explicitly derived. As a related problem, we introduce the notion of law-invariant $g$-expectation and provide its sufficient conditions. Examples of application in financial dynamic portfolio choice are supplied.