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ISIPTA'07 -
FIFTH INTERNATIONAL SYMPOSIUM ON

IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS

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Charles University, Faculty of Mathematicsand Physics

Prague, Czech Republic

16-19 July 2007

## ELECTRONIC PROCEEDINGS

## Volker Krätschmer

# On sigma-additive robust representations of convex risk measures for unbounded financial positions in the presence of uncertainty of the market model

### Abstract

Recently, Frittelli and Scandolo extend the notion of risk measures, originally introduced by Artzner, Delbaen, Eber and Heath, to the risk assessment of abstract financial positions, including pay offs spread over different dates, where liquid derivatives are admitted as financial instruments, and unbounded fincial positions are also allowed. Convex risk measures may be viewed as convex upper previsions for unbounded gambles, a notion originally introduced by Pelessoni and Vicig. The paper deals with sigma-additive robust representations of convex risk measure, that means envelope theorems in terms of $\sigma-$additive probability
measures. We shall focus on the aspect that the investor is faced with uncertainty about the market model. It turns out that
the results may be applied for the case that a market model is available, and that they encompass as well as improve criteria obtained for robust representations of convex risk measures in the genuine sense by Delbaen, Föllmer and Schied, and Krätschmer.

** Keywords. ** Convex risk measures, convex upper previsions, model uncertainty, sigma-additive robust representation, Fatou property, nonsequential Fatou property, strong sigma-additive robust representation, Krein-Smulian theorem, Greco theorem, inner Daniell stone theorem, general Dini theorem, Simons' lemma.

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** Authors addresses: **

Institute of Mathematics

Berlin University of Technology

Strasse des 17. Juni 136

12623 Berlin

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