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

max (-Λ(X) - βρ (Λ)) for every X ∈ X. Moreover, ρ(X) = sup  (-Λ(X) - βρ (Λ)) holds for every X ∈ X if

^λ∈^x+^π                                                      Λ∈X+^π ∩X'

and only if ρ is lower semicontinuous w.r.t. τ.

The proof may be found in Appendix B.

The aim of the paper is to improve the representation results by allowing only representing linear forms which
are in turn representable by probability measures. For notational purposes let us introduce the counterpart of βρ
w.r.t. the probability measures on the σ-algebra σ(X) on Ω generated by X

αρ : M1 →] - ∞, ∞], P → sup (-EP [X] - ρ(X)).

X∈X

Here M1 is defined to consist of all probability measures P on σ(X) such that all positions from X are P -integrable,
and EP [X] denotes the expected value of X w.r.t. P . We shall speak of a robust representation by probability
measures from M or a σ-additive robust representation of ρ w.r.t. M if M ⊆ M1 nonvoid, and
ρ(X) = sup (-EP [X] - αρ(P)) for every X ∈ X. As an immediate consequence of Proposition 1.1 we obtain a first
P∈M

characterization of such representations.

Proposition 1.2 Let F be a vector space of bounded countably additive set functions on σ(X) which separates
points in X such that each X ∈ X is integrable w.r.t. any μ ∈ F. Then in the case that the set M₁ (F) of all
P ∈ M1 ∩ F with EP|C = π is nonvoid

ρ(X) =  sup (-EP [X] - αρ (P)) for all X ∈ X

P∈M1 (F)

if and only if ρ is lower semicontinuous w.r.t. weak topology σ(X, F) on X induced by F.

Remark 1.3 Retaking assumptions and notations from Proposition 1.2, ρ admits a robust representation in terms
of M1 (F) if F contains the Dirac measures, and if lim inf ρ(Xi) ≥ ρ(X) holds for every net (Xi)i∈I in X which
i
converges pointwise to some X ∈ X.

In general the lower semicontinuity of ρ w.r.t. the topology from Proposition 1.2 is not easy to verify. Therefore we
are looking for more accessible conditions. The considerations will be based on the idea to reduce the investigations
to bounded financial positions. As shown in Lemma 6.5 below, in case of X being a Stonean vector lattice, this
may be achieved if the linear forms from the domain of βρ are representable by finitely additive set functions
in the sense explained there. Fortunately, we might express this condition equivalently by the property that
lim ρ(-λ(X - n)⁺) = ρ(0) ((X - n)⁺ positive part of X - n) is satisfied for every λ > 0 and any nonnegative
n→∞

X ∈ X, which is obviously true if all positions in X are bounded (cf. Proposition 6.6 below).

Before going into the development of criteria for σ-additive representations let us collect some necessary conditions.

In the case that the positions from X are essentially bounded mappings w.r.t. a reference probability measure of

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