On s-additive robust representation of convex risk measures for unbounded financial positions in the presence of uncertainty about the market model

On σ-additive robust representation of convex risk measures for
unbounded financial positions in the presence of uncertainty
about the market model

Volker Kratschmer *t

Institute of Mathematics, Berlin University of Technology,

10623 Berlin, Germany

Abstract

Recently, Frittelli and Scandolo ([9]) extend the notion of risk measures, originally introduced by Artzner, Delbaen,
Eber and Heath ([1]), to the risk assessment of abstract financial positions, including pay offs spread over different
dates, where liquid derivatives are admitted to serve as financial instruments. The paper deals with σ-additive
robust representations of convex risk measures in the extended sense, dropping the assumption of an existing
market model, and allowing also unbounded financial positions. The results may be applied for the case that a
market model is available, and they encompass as well as improve criteria obtained for robust representations of
the original convex risk measures for bounded positions ([4], [7], [16]).

KEYWORDS: Convex risk measures, model uncertainty, σ-additive robust representation, Fatou property, non-
sequential Fatou property, strong σ-additive robust representation, Krein-Smulian theorem, Greco theorem, inner
Daniell stone theorem, general Dini theorem, Simons’ lemma.

JEL CLASSIFICATION G10

AMS CLASSIFICATION 91B30, 91B16, 28A12

*This research was supported by Deutsche Forschungsgemeinschaft through the SFB 649 “Economic Risk”.

^tE-mail: [email protected]

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