Correction to: Fatou closedness under model uncertainty

Positivity - There is an error in Proposition 3.10. In fact, the stated proof only shows     

Fatou closedness under model uncertainty

We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of \({\mathcal P}\)-quasisure bounded random variables, where \({\mathcal P}\) is a (possibly non-dominated) class of probability measures. Applications of our results lie within robust versions the Fundamental Theorem of Asset Pricing or dual representation of convex risk measures.     

Disentangling price,risk and model risk: V&R measures

We propose a method to assess the intrinsic risk carried by a financial position X when the agent faces uncertainty about the pricing rule assigning its present value. Our approach is inspired by a new interpretation of the quasiconvex duality in a Knightian setting, where a family of probability measures replaces the single reference probability and is then applied to value financial positions. Diametrically, our construction of Value and Risk measures is based on the selection of a basket of claims to test the reliability of models. We compare a random payoff X with a given class of derivatives written on X, and use these derivatives to “test” the pricing measures. We further introduce and study a general class of Value and Risk measures \( R(p,X,\mathbb {P})\) that describes the additional capital that is required to make X acceptable under a probability \(\mathbb {P}\) and given the initial price p paid to acquire X.     

Conditionally evenly convex sets and evenly quasi-convex maps

Evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces. We introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem. This notion is then applied to obtain the dual representation of conditionally evenly quasi-convex maps, which turns out to be a fundamental ingredient in the study of quasi-convex dynamic risk measures.