On $g$-expectations and filtration-consistent nonlinear expectations

TL;DR

This paper advances the theory of $g$-expectations and filtration-consistent nonlinear expectations by establishing new representation theorems under relaxed conditions, enhancing their application in risk measures and asset pricing.

Contribution

It introduces two new conditions enabling the representation of ${ mf{F}}$-expectations as $g$-expectations, including quadratic cases, without assumptions on the second variable in BSDEs.

Findings

Established comparison and invariant representation theorems for BSDEs without assumptions on $z$.

Proved that under new conditions, any ${ mf{F}}$-expectation can be represented as a $g$-expectation.

Provided representation theorems for multidimensional and quadratic ${ mf{F}}$-expectations.

Abstract

In this paper, we obtain a comparison theorem and a invariant representation theorem for backward stochastic differential equations (BSDEs) without any assumption on the second variable $z$. Using the two results, we further develop the theory of $g$-expectations. Filtration-consistent nonlinear expectation (${\cal{F}}$-expectation) provides an ideal characterization for the dynamical risk measures, asset pricing and utilities. We propose two new conditions: an absolutely continuous condition and a (locally Lipschitz) domination condition. Under the two conditions respectively, we prove that any ${\cal{F}}$-expectation can be represented as a $g$-expectation. Our results contain a representation theorem for $n$-dimensional ${\cal{F}}$-expectations in the Lipschitz case, and two representation theorems for $1$-dimensional ${\cal{F}}$-expectations in the locally Lipschitz case, which contain quadratic ${\cal{F}}$-expectations.