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

In the case of an at most countable Ω, we have a simplified situation which admits an application of the full
Theorem 2.2. The reason is that then the topology of pointwise convergence on the space B(Ω) is metrizable.

Corollary 2.5 Let Ω be at most countable, and let X ⊆ B(Ω) be sequentially closed w.r.t. the pointwise topology
on B(Ω). Then ρ has a robust representation by probability measures from M₁ if and only it satisfies the Fatou
property, or equivalently, if and only if ρ is continuous from above.

As another application of Theorem 2.2 we shall retain in the proof of Theorem 5.3 below the above mentioned
result that in face of a market model the Fatou propery describes equivalently robust representations of convex risk
measures for essentially bounded positions by probability measures. Unfortunately, it is unclear whether we may
avoid in Theorem 2.2 condition (2.2) in order to guarantee a σ-additive robust representation of risk measures by
the nonsequential Fatou property. Moreover, the nonsequential Fatou property is unsatisfactory in the way that
it does not work for trivial representations like those indicated in Remark 2.3. However, we may only provide
a sufficient substitution by the Fatou property under the quite restrictive condition (2.1). So it seems that in
presence of model uncertainty the Fatou property and its nonsequential counterpart are appropriate conditions for
σ-additive representations of convex risk measures in quite exceptional situations only. Therefore we shall look
for alternatives in the following section.

3 Robust representation of convex risk measures by inner regu-
lar probability measures

Throughout this section let X be a Stonean vector lattice, and let L ⊆ X denote any Stonean vector lattice
∞

which contains C as well as generates σ(X) and which induces the set system S := { ^T X_n^-1 ([xn, ∞[) | Xn ∈
n=1

L nonnegative, bounded, xn > 0}. Additionally, let E consist of all bounded sup Yn , where (Yn)n is a sequence
n

of nonnegative bounded positions from L. In view of the inner Daniell-Stone theorem (cf. [14], Theorem 5.8,
final remark after Addendum 5.9) every probability measure P ∈ M1 has to be inner regular w.r.t. S, i.e.
P(A) = sup P(B) for every A ∈ σ(X). So within this setting we are dealing with robust representations of ρ by
A⊇B∈S

probability measures from M1 (S) defined to consist of all probability measures belonging to M1 which are inner
regular w.r.t. S and which represent π on C. We are ready to formulate the general representation result w.r.t.
inner regular probability measures.

Theorem 3.1 Let ∆c (c ∈] - ρ(0), ∞[) gather all P ∈ M1 (S) with αρ(P) ≤ c, and let ρ satisfy the following
properties.

(1) ρ(X) = sup inf ρ(Z) for all nonnegative bounded X ∈ X,
X≤Y ∈E Y ≥Z∈X

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