The framework of this paper is that of risk measuring under uncertainty, which is when no reference probability measure is given. To every regular convex risk measure on ${ \cal C}_b(\Omega)$, we associate a unique equivalence class of probability measures on Borel sets, characterizing the riskless non positive elements of ${ \cal C}_b(\Omega)$. We prove that the convex risk measure has a dual representation with a countable set of probability measures absolutely continuous with respect to a certain probability measure in this class. To get these results we study the topological properties of the dual of the Banach space $L^1(c)$ associated to a capacity $c$. As application we obtain that every $G$-expectation $\E$ has a representation with a countable set of probability measures absolutely continuous with respect to a probability measure $P$ such that $P(|f|)=0$ iff $\E(|f|)=0$. We also apply our results to the case of uncertain volatility.
Categories: q-fin.RM math.PR