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

∞

that lim Q(An) = 1 whenever (An)_n is an isotone sequence in S with (J An = Ω. Fortunately, this follows
n→∞                                                      _n=1

immediately from the second part of statement .4 and

βρ (Λ) + ρ(0) + inf   ρ(Z) - ρ(λ)

1_A ≥Z∈X

| Q(Ω) - Q(An)∣ ≤ ----------------n---------------- for all λ> 0;

λ

Note that βρ(Λ) ≥ Eq[-λ1A_η,] — inf   ρ(λZ) + (λQ(Ω) + ρ(λ) — ρ(0)).                                    H

1A_n ≥Z ∈X

9 Proofs of results from section 2

Proof of Lemma 2.1:

It remains to show the if part. For this purpose let (Xi)i∈I denote a uniformly bounded net in Xb with pointwise

limit X ∈ Xb. Setting c := lim inf ρ(Xi) and fixing ε > 0, we may find a subnet (X_i(_k) )k∈K with ρ(X_i(_k) ) < c + ε
i

for every k ∈ K. Hence (X_i(_k) + c + ε)k∈K is a net in some Ar which converges pointwise to X + c + ε. So in view

of condition (2.1) X + c + ε is also the pointwise limit of some sequence (Yn )n from Ar. If ρ satisfies the Fatou

property we may conclude ρ(X + c + ε) ≤ lim inf ρ(Yn) ≤ 0, and hence ρ(X) ≤ c + ε. This completes the proof.

n→∞

Proof of Theorem 2.2:

Let us retake assumptions and notations from Theorem 2.2

The implication .2 ⇒ .3 is always valid as indicated in Proposition 1.4.

Let us now introduce the space F consisting of all real linear forms on Xb which are representable by some μ ∈ F.
The operator norm ∣∣ ∙ ∣∣_j^ on F w.r.t. the sup norm ∣∣ ∙ ∣∣∞ satisfies ∣∣ R ∙ dμ∣∣p = ∣∣μ∣∣p for every μ ∈ F. Since F is
supposed to be complete w.r.t ∣∣ ∙ ∣∣p, (F, ∣∣ ∙ ∣∣p) is a Banach space. The topological dual F' of F will be endowed
with the respective operator norm ∣∣ ∙ ∣∣, and Bp, denotes the unit ball in F'.

Since F contains all Dirac measures, Xb may be embedded isometrically into F' w.r.t. ∣∣ ∙ ∣∣∞ and ∣∣ ∙ ∣∣ by
the evaluation mapping ^ : Xb → F'. Next let us fix an arbitrary J ∈ F' outside the closure cl(e(X_b) ∩ Bp,) of
e(X_b) ∩Bp, w.r.t. the weak * topology σ(F', F). By Hahn-Banach theorem we may find some σ(F', F)-continuous
real linear form Λ on F' with

sup{Λ(J) | J ∈ cl(e(Xb) ∩ Bp,)} < Λ(J).

In addition there is some μ ∈ F with Λ(J) = J(R ∙ dμ) for any J ∈ F'. Without loss of generality we may assume
∣∣μ∣∣p = 1. Since ^ is isometric, we obtain then

X dμ | X ∈ Xb, sup |X(ω)| ≤ 1} = ∣∣μ∣∣p = 1.
ω∈Ω

ττ          -/-"V^v ∖ '' τ->     ∙      ' τFf rι∖ 1         ∙    τ->

Hence e(Xb) ∩ Bp, is σ(F' ,F)—dense in Bp,.

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