# Risk measuring under model uncertainty - Quantitative Finance > Risk Management

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Abstract: The framework of this paper is that of risk measuring under uncertainty,which is when no reference probability measure is given. To every regularconvex risk measure on ${\cal C} b\Omega$, we associate a unique equivalenceclass of probability measures on Borel sets, characterizing the riskless nonpositive elements of ${\cal C} b\Omega$. We prove that the convex riskmeasure has a dual representation with a countable set of probability measuresabsolutely continuous with respect to a certain probability measure in thisclass.To get these results we study the topological properties of the dual of theBanach space $L^1c$ associated to a capacity $c$.As application we obtain that every $G$-expectation $\E$ has a representationwith a countable set of probability measures absolutely continuous with respectto a probability measure $P$ such that $P|f|=0$ iff $\E|f|=0$. We alsoapply our results to the case of uncertain volatility.