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



0 Introduction

The notion of risk measures has been introduced by Artzner, Delbaen, Eber and Heath (cf. [1]) as the key
concept to found an axiomatic approach for risk assessment of fincancial positions. Technically, risk measures are
functionals defined on sets of financial positions, satisfying some basic properties to qualify riskiness consistently.
An outcome of such a functional, that means the risk of a position, is usually interpreted as the capital requirement
of the position to become an acceptable one. Genuinely, risk measures has been defined for one-period positions.
Recently Fritelli and Scandolo ([9]) provide a general framework which extends considerations to abstract financial
positions including pay off streams with liquid derivatives as hedging positions. Applied to the risk assessment of
pay off streams such general risk measures are used for an a priori qualification, which means to take the static
perspective. In contrary the dynamic risk assessment take into account adjustments time after time. Readers who
interested in this topic are referred to e.g. [8], [18], [21].

The main goal of this paper is to investigate risk measures ρ which admit a robust representation of the form

ρ(X) = sup(-Λ(X) - β(Λ)),
Λ

where X denotes a financial position, Λ a linear form on the set of financial positions, and β stands for a penalty
function on the set of linear forms. Special attention will be paid to the problem when these representing linear
forms may in turn be represented by (
σ-additive) probability measures. We shall speak of a robust representation
of
ρ by probability measures or a σ-additive robust representation. Necessarily, only so-called convex risk mea-
sures, that means risk measures which are convex mappings, may have such a robust representation. The basic
assumption of this paper is that the investors are uncertain about the market model underlying the outcomes of
the financial positions. Within this setting a robust representation by probability measures offers an additional
economic interpretation of the risk measures. As suggested by Follmer and Schied (cf. [7]) such a representation
means that an investor has a set of possible models for the outcomes of the financial positions in mind, and
evaluates the worst expected losses together with some penalty costs for misspecification w.r.t. these models.

The problem of σ-additive robust representations of convex risk measures in the genuine sense has been completely
solved in the case that the investors have market models at hand. Ruszczynski and Shapiro showed that convex
risk measures always admit robust representations by probability measures if for any real
p > 1 every integrable
mapping of order
p is available (cf. [19]). However the used methods can not be applied to essentially bounded
positions. Drawing on methods from functional analysis, Delbaen as well as Foollmer and Schied succeeded in
giving a full characterization (cf. [4], [7]) by the so-called Fatou property. As pointed out by Delbaen, the Fatou
property fails to be sufficient in general when the investor is faced with model uncertainty. Moreover, the problem
of
σ-additive robust representation is still open when a market model is not available. Restricting considerations
on bounded one-period positions, Foollmer and Schied (in [7]) suggested a strict sufficient criterion, Kroatschmer



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