$g$-Expectation of Distributions

TL;DR

This paper introduces the concept of $g$-expectation of distributions, providing explicit formulas for special cases, and explores law-invariant $g$-expectation with applications in financial portfolio optimization.

Contribution

It defines $g$-expectation of distributions, derives explicit formulas for certain nonlinear cases, and introduces law-invariant $g$-expectation with practical financial applications.

Findings

Explicit $g$-expectation formulas for special nonlinear cases

Introduction of law-invariant $g$-expectation with sufficient conditions

Application examples in financial dynamic portfolio choice

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.