Representation theorems for dynamic convex risk measures

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 $g$.