Representation theorems for dynamic convex risk measures

In this paper, we prove that under the domination condition: \begin{equation*} {\cal{E}}^{-μ,-ν}[-ξ|{\cal{F}}_t]\leqρ_t(ξ)\leq{\cal{E}}^{μ,ν}[-ξ|{\cal{F}}_t],\quad \forallξ\in $\mathcal{L}$^{\exp}_T\ ($\text{resp.}$\ L^2($\mathcal{F}$_T)),\ \forall t\in[0,T], \end{equation*} where ${\mathcal{E}}^{\mu ,\nu }$ is the $g$-expectation with generator $\mu \mathrm{\mid }z\mathrm{\mid }+\nu \mathrm{\mid }z{\mathrm{\mid }}^2,\mu \ge 0,\nu \ge 0$, 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 $g$.