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

Now let condition (2.2) be valid, and let us assume that ρ satisfies the nonsequential Fatou property. Using Dirac
measures δ_ω (ω ∈ Ω), we may define for any J ∈ F' a mapping XJ ∈ R^ω via XJ(ω) := J(R ∙ δ_ω). Each XJ is
bounded because |XJ(ω)| ≤ ∣∣J∣∣ holds for every ω ∈ Ω. Furthermore for any J ∈ F' there exists a uniformly
bounded net (Xi)i∈I in Xb such that (e(X_i))i∈I converges to J w.r.t. σ(F', F). In particular XJ is the pointwise
limit of (Xi)i∈I, which means that it belongs to Xb due to (2.2). Hence the mapping ρ : F' → R, J → ρ(XJ) is
well defined with ρ(e(X )) = ρ(X) for X ∈ Xb.

For every r > 0 and any net (Ji)i∈I in ρ^-1(] — ∞, 0]) ∩ rB^, we may select by Banach-Alaoglu theorem a subnet
(Ji(_k))_k∈K and some J ∈ rB_F/ such that (Ji(_k))_k∈K converges to J w.r.t. σ(F', F). Then (XJ_i(_k) )_k∈K is a uniformly
bounded net in Xb which converges pointwise to XJ. Since ρ fulfills the nonsequential Fatou property, we obtain

P(J) = P(XJ) ≤ liminf ρ(XJ_i(_k) ) = liminf ρ(Ji_w) ≤ 0.
k         ()       k

Thus the sets ρ^-1(] — ∞, 0]) ∩rBp, (r > 0) are σ(F', F)-compact, which means that ρ^-1(] — ∞, 0]) is closed w.r.t.
σ(F',F) by Krein-Smulian theorem. Now it is easy to check that ρ^-1(] — ∞, 0]) ∩ Xb is closed w.r.t. σ(Xb,F),
which implies that all level sets ρ^-1 (] — ∞, c]) ∩ Xb (c ∈ R) are σ(Xb, F)—closed due to the translation invariance
of ρ. This shows statement .2, drawing on Propositions 1.2, 6.6 and Lemma 6.5. As a further consequence we have
equivalence of the statements .1 - .3 under (2.1), (2.2) in view of Lemma 2.1.

If we strengthen condition (2.2) by the assumption that the sets Ar from (2.1) are compact w.r.t. σ(X, F), it
remains to show the implication .2 ⇒ .1. Indeed statement .2 implies that ρ is lower semicontinuous w.r.t. the weak
topology σ(X, F) by Propositions 1.2. Furthermore for any uniformly bounded net (Xi)i∈I in Xb with pointwise
limit X ∈ Xb we may suppose without loss of generality that (Xi)i∈I is a net in some Ar due to translation
invariance of ρ. Then, drawing on relative σ(X, F)—compactness of Ar , the mapping X is the σ(X, F)—limit of
(Xi)i∈I. This implies liminf ρ(Xi) ≥ ρ(X), and completes the proof.                                          H

i

Proof of Remark 2.4:

Let ^ denote the evaluation mapping from Xb into the topological dual F' of F w.r.t. the norm of total variation.

It is isometric w.r.t. the sup norm ∣∣ ∙ ∣∣∞ and the operator norm ∣∣ ∙ ∣∣. Then the if part is obvious in view of
Banach-Alaoglu theorem. Conversely, translation invariance and relative σ(X, F)—compactness of the sets Ar
imply the σ(F',F)—compactness of the sets ^(Xb) ∩ rB_F/ (r > 0), where B_f/ denotes the unit ball w.r.t. ∣∣ ∙ ∣∣.
This means that ^(Xb) is closed w.r.t. σ(F',F) due to the Krein-Smulian theorem, and then e(X) = F' because
Xb separates points in F, and thus ^(Xb) is dense w.r.t. σ(F',F). The proof is finished.                         H

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