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

Fenchel-Legendre transform βρ and its counterpart αρ for the probability measures.

Lemma 6.1 Let X₁, X₂ ∈ X with X₁ ≤ X₂. Then there exists some c* ∈] — ρ(0), ∞[ such that the representation

Proof:

Let Y ∈ X with X₁ ≤ Y ≤ X. By Proposition 1.1 there is some Λ ∈ β^-1(R) with ρ(Y) = —Λ(Y) — β_ρ(Λ). Then

βρ(Λ) = —Λ(2Y) — ρ(Y) + Λ(Y) ≤ ρ(2Y) + βρ(Λ) — ρ(Y) + Λ(Y) = ρ(2Y) — 2ρ(Y) ≤ ρ(2Xχ) — 2ρ(X2).

Therefore any c > max{ρ(2X₁) — 2ρ(X₂), — ρ(0)} is as required.                                               I

Lemma 6.2 Let X ∈ X with X ≤ inf Z, where E ⊆ X is assumed to be downward directed, i.e. for Z1 , Z2 ∈ E
Z∈E

there is some Z ∈ E with Z ≤ min{Z1 , Z2}. Furthermore let Λ ∈ β_ρ^-1 (R).

Then inf Λ(Z) = Λ(X) if inf p(—λ(Z — X)) = ρ(0) for arbitrary λ > 0.
Z∈E               Z∈E

Proof:

For arbitrary λ > 0 and every Z ∈ E we have β_ρ(Λ) ≥ —A(—λ(Z — X)) — p(—λ(Z — X)), and therefore by

0 ≤ inf Λ(Z — X) ≤ ^βρ(ΔL÷₂^p(⁰) .
Z∈E                  λ

Finally, by taking λ ↑ ∞, we obtain inf Λ(Z — X) = 0 because 0 ≤ βρ (Λ) + ρ(0) < ∞. The proof is now complete.
Z∈E

We may divide the domain of β_ρ into the classes {β_ρ ≤ c} (—p(0) = inf βρ < c < ∞). The following topological
property of these classes is crucial.

Lemma 6.3 {βρ ≤ c} is compact w.r.t. the product topology on R^X for every c ∈] — ρ(0), ∞[.

Proof:

Let c ∈] — ρ(0), ∞[, and let (Λi)i∈I be a net in {βρ ≤ c} which converges to some Λ ∈ R^X w.r.t. the product
topology. Obviously, Λ is a positive linear form on X which extends π. Furthermore

—Λ(X) — ρ(X) = lim(—Λ_i(X) — ρ(X)) ≤ limsupβ_ρ(Λ_i) ≤ c for X ∈ X.

i_i

Therefore {βρ ≤ c} is closed w.r.t. the product topology, and the proof may be completed by the application of
Tychonoff’s theorem because {β_ρ ≤ c} ⊆ X [—c — ρ(X),c + ρ(XX)].                                   И

X ∈ X

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