Representation Theorems for Quadratic ${\cal F}$-Consistent Nonlinear Expectations

TL;DR

This paper extends the theory of filtration-consistent nonlinear expectations to quadratic growth cases, establishing fundamental properties and a representation theorem that characterizes such expectations as quadratic g-expectations.

Contribution

It introduces a new domination condition for quadratic nonlinear expectations and proves they are representable as quadratic g-expectations with deterministic, continuous generators.

Findings

Fundamental martingale properties hold for quadratic expectations.

Quadratic nonlinear expectations are characterized as quadratic g-expectations.

The generator must be deterministic, continuous, and simple in form.

Abstract

In this paper we extend the notion of ``filtration-consistent nonlinear expectation" (or "${\cal F}$-consistent nonlinear expectation") to the case when it is allowed to be dominated by a $g$-expectation that may have a quadratic growth. We show that for such a nonlinear expectation many fundamental properties of a martingale can still make sense, including the Doob-Meyer type decomposition theorem and the optional sampling theorem. More importantly, we show that any quadratic ${\cal F}$-consistent nonlinear expectation with a certain domination property must be a quadratic $g$-expectation. The main contribution of this paper is the finding of the domination condition to replace the one used in all the previous works, which is no longer valid in the quadratic case. We also show that the representation generator must be deterministic, continuous, and actually must be of the simple form.