Representation theorems for dynamic convex risk measures

TL;DR

This paper establishes representation theorems for dynamic convex risk measures using g-expectations with quadratic or linear growth generators, under a domination condition, and provides dual representations with penalty terms or probability measure sets.

Contribution

It proves that dynamic convex risk measures satisfying a domination condition can be represented as g-expectations with convex or sublinear generators, extending the theoretical framework.

Findings

Representation as g-expectation with convex or sublinear generator.

Dual representation involving penalty terms or probability measures.

Applicability to dynamic convex and coherent risk measures.

Abstract

In this paper, we prove that under the domination condition: \begin{equation*} {\cal{E}}^{-\mu,-\nu}[-\xi|{\cal{F}}_t]\leq\rho_t(\xi)\leq{\cal{E}}^{\mu,\nu}[-\xi|{\cal{F}}_t],\quad \forall\xi\in \mathcal{L}^{\exp}_T\ (\text{resp.}\ L^2(\mathcal{F}_T)),\ \forall t\in[0,T], \end{equation*} where ${\cal{E}}^{\mu,\nu}$ is the $g$-expectation with generator $\mu|z|+\nu|z|^2, \mu\geq0, \nu\geq0$, the dynamic convex (resp. coherent) risk measure $\rho$ admits a representation as a $g$-expectation, whose generator $g$ is convex (resp. sublinear) in the variable $z$ and has a quadratic (resp. linear) growth. As an application, we show that such dynamic convex (resp. coherent) risk measure $\rho$ admits a dual representation, where the penalty term (resp. the set of probability measures) is characterized by the corresponding generator $g$.