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

The sequential condition (2.1) is closely related with the concepts of double limit relations. For a comprehensive
exposition the reader is referred to [15]. In general one may try to apply double limit relations to the sets Ar from
(2.1) and suitable sets of bounded countably additive set functions on σ(X).

The main result of this section relies on the following assumption, denoting by B(Ω) the space of all bounded
real-valued mappings on Ω.

(2.2) The sets Br := {X ∈ Xb ∣ sup |X(ω)∣ ≤ r} (r > 0) are closed w.r.t. the topology of pointwise convergence
ω∈Ω

on B(Ω).

Theorem 2.2 Let either X = Xb or X be a Stonean vector lattice such that lim ρ(-λ(X - n)⁺) = ρ(0) holds for
n→∞

any nonnegative X ∈ X, λ > 0. Furthermore let F denote a vector space of bounded countably additive set functions
on σ(X) which contains all Dirac measures as well as at least one probability measure P with EP |C = π such that
every X ∈ X is integrable w.r.t. any μ ∈ F. Additionally, F is supposed to be complete w.r.t. the seminorm ∣∣ ∙ ∣∣F,
defined by ∣∣μ∣∣F := sup{∣ ʃ X dμ∣ ∣ X ∈ X_b, sup ∣X(ω)∣ ≤ 1}. Consider the following statements:
ω∈Ω

.1 ρ satisfies the nonsequential Fatou property.

.2 ρ has a σ-additive robust representation w.r.t. M1 ∩ F.

.3 ρ fulfills the Fatou property.

If (2.2) is valid, then .1 ⇒ .2 ⇒ .3, and all statements are equivalent provided that condition (2.1) holds in
addition. In the case that the sets Ar from (2.1) are even relatively compact w.r.t. the weak topology σ(X, F) we
have .1 ⇔ .2 ⇒ .3.

The proof will be performed in section 9.

Remark 2.3 The nonsequential Fatou property is not necessary for a σ-additive representation of risk measures.
Take for example X the space of all boundend Borel-measurable mappings on R, and define ρ by ρ(X) = -EP [X],
where P denotes any probability measure which is absolutely convex w.r.t. the Lebesgue-Borel measure on R.
Obviously, on one hand ρ is a convex risk measure w.r.t. the identity on R, having a trivial σ-additive robust
representation. On the other hand, consider the set I of the cofinite subsets of R, directed by set inclusion, and the
net (Xi)i∈I of all its indicator mappings. It converges pointwise to 0, but unfortunately lim inf ρ(Xi) = -1 < 0 =
i

ρ(0).

Remark 2.4 Let F be any vector space of bounded countably additive set functions on σ(X) such that each X ∈ X
is integrable w.r.t. every μ ∈ F, and such that Xb separates points in F. Additionally, F is supposed to be closed
w.r.t. the norm of total variation. Then the sets Ar from (2.1) are relatively σ(X, F)-compact if and only if Xb
may be identified via evaluation mapping with the topological dual of F w.r.t. the norm of total variation (cf. proof
in section 9).

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