showed that it is in some sense also necessary, and he adds some more general conditions ([16]).
This paper may be viewed as a continuation of the studies in [7] as well as in [16]. The generalizations will be
proceeded in several directions. First of all multiperiod positions and liquid hedging instruments will be allowed.
Secondly we shall drop the assumptions that only bounded positions are traded. This is in accordance with
empirical evidences that the distributions of risky assets often show heavy tails. Thirdly we want to investigate
the issue of strong robust representations by probability measures in the sense that the optimization involved in the
σ-additive robust representation has a solution. This is a quite important technical issue from the practical point
of view. In many cases the calculation of outcomes of risk measures has to be employed by numerical optimization
algorithms, and the most customary ones assume the existence of solutions. In presence of a market model, Jouini,
Schachermayer and Touzi (cf. [11]) have given a full solution to the problem of strong robust representations.
Finally, the criteria should encompass the results already derived within a fixed market model.
The paper is organized as follows. Section 1 introduces the concept of Frittelli and Scandolo to define risk
measures in general, and some representation results of risk measures will be presented as starting points for the
investigations afterwards. The following section 2 deals with the question when the Fatou property might be used
as a sufficient condition. In general, as a rule a nonsequential counterpart is more suitable unless in some special
cases. However, it also seems that even the nonsequential Fatou property is appropriate in quite exceptional
situations only. Therefore an alternative general criterion is offered in section 3, extending a former result in [16]
to unbounded positions, within a nontopological framework. It will be used for strong robust representations of
risk measures by probability measures in section 4. We shall succeed in giving a complete solution. In particular
the aboved mentioned strict criterion by Follmer and Schied will turn out to be necessary and sufficient. The
investigations of the sections 1 - 4 will then be applied to recover in section 5 the already known representation
results within a given market model. The proofs of the main results will be provided separately in the sections 7,
8 and 9 as well as in appendix B. They rely on some technical arguments gathered in section 6 and a measure
theoretical tool presented in appendix A.
1 Some basic representations of convex risk measures
Let us fix a set Ω. Financial positions will be expressed by mappings X ∈ Rω. As a special case Ω = Ω × T with
Ω denoting a set of scenarios, equipped with a family (Ft)t∈T of σ-algebras, and T being a time set, we may
consider financial positions X ∈ Rω×t with X(∙,t) being Ft-measurable for every t ∈ T. They may be viewed
as discounted pay off streams, liquidated at the dates from the time set. The available financial positions are
gathered by a nonvoid vector subspace X ⊆ Rω containing the constants. Sometimes we shall in addition assume
that X ∧ Y := min{X, Y }, X ∨ Y := max{X, Y } ∈ X for X, Y ∈ X. In this case X is a so-called Stonean vector