On the representation for dynamically consistent nonlinear evaluations: uniformly continuous case

TL;DR

This paper establishes a representation theorem for dynamically consistent nonlinear evaluations using solutions of BSDEs with Lipschitz and uniformly continuous generators, advancing the understanding of risk measures and pricing models.

Contribution

It proves that any uniformly continuous ${\

Findings

Representation of ${\cal{F}}$-evaluations by BSDE solutions.

Existence of Lipschitz in $y$ and uniformly continuous in $z$ generators.

General domination condition ensures the representation.

Abstract

A system of dynamically consistent nonlinear evaluation (${\cal{F}}$-evaluation) provides an ideal characterization for the dynamical behaviors of risk measures and the pricing of contingent claims. The purpose of this paper is to study the representation for the ${\cal{F}}$-evaluation by the solution of a backward stochastic differential equation (BSDE). Under a general domination condition, we prove that any ${\cal{F}}$-evaluation can be represented by the solution of a BSDE with a generator which is Lipschitz in $y$ and uniformly continuous in $z$.