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

Moreover, due to statement .2 and Greco’s representation theorem again, there exists some m ∈ N such that
∣Eq[X⁺ - (X^- ∧ n)] - Eq[(X ⁺ ∧ m) - (X^- ∧ n)]∣, ∣ρ(X⁺ - (X^- ∧ n)) - ρ((X⁺ ∧ m) - (X^- ∧ n))∣ < ^ε

We may conclude by

-EQ[X] - ρ(X) < -EQ[(X⁺ ∧m) - (X^- ∧n)] - ρ((X⁺ ∧m) - (X^- ∧n))+ε≤ sup (-EQ[Y] - ρ(Y)) +ε

and then sup (-EQ [X] - ρ(X)) = sup (-EQ [X] - ρ(X)). The rest of statement .3 follows easily from statements
X∈X                   X∈X_b

.1, .2.

proof of .4:

By assumption ρ : X → R, X → sup (-E_p[X] - α_ρ(P)), is a well defined convex risk measure w.r.t. π with ρ ≤ ρ
P∈M

and ρ(X) = ρ(X) for bounded X ∈ X. In particular β^-1(R) ⊆ β^-1(R), which implies that every Λ ∈ β^-1 (R) is
representable by a probability content due to the assumptions on βρ. Therefore the statements .1, .2 are also valid
for ρ, following the same line of reasoning used in the proof of them. Then firstly, fixing ε > 0, we may find for
X ∈ X an integer n ∈ N with

∣ρ(X) - ρ(X⁺ - (X^- ∧ n))∣, ∣ρ(X) - ρ(X⁺ - (X^- ∧ n))∣ <4 .

Furthermore there is some m ∈ N such that

∣ρ((X⁺ ∧ m) - (X^- ∧ n)) - ρ(X⁺ - (X^- ∧ n))∣, ∣ρ((X⁺ ∧ m) - (X^- ∧ n)) - ρ(X⁺ - (X^- ∧ n))∣ < ^ε .

Thus ∣∕5(X) - ρ(X)∣ < ε, and hence ∕5(X) = ρ(X), which completes the proof.                                 H

In order to apply Lemma 6.5 we are now interested in conditions on ρ that ensure that linear forms from the
domain of βρ are representable by probability contents. We shall succeed in providing a full characterization.

Proposition 6.6 Let X be a Stonean vector lattice. Then every linear form from β_ρ^-1 (R) is representable by a
probability content if and only if lim ρ(-λ(X - n)⁺) = ρ(0) for every λ > 0 and nonnegative X ∈ X.
n→∞

Proof:

For the if part let Λ ∈ β_ρ^-1 (R). Then by assumption and Lemma 6.2 the sequence Λ((X - n)⁺) _n converges to 0
for nonnegative X ∈ X. Hence, due to Greco’s representation theorem (cf. [14], Theorem 2.10 with Remark 2.3)
Λ is representable by a probability content.

Conversely, let every linear form from the domain of βρ be representable by a probability content, and let λ > 0 as
well as X ∈ X be nonnegative. In view of Lemma 6.1 there is some c* ∈]-ρ(0), ∞[ with ρ(Y) = sup   (-Λ(Y)-

Λ∈{βρ ≤c∙}

βρ(Λ)) for any Y ∈ X with -λX ≤ Y ≤ 0. In particular

∣ρ(-λ(X - n) ⁺ ) - ρ(0)∣ ≤ sup   (Λ(λ(X - n) ⁺ )) for every n.

Λ∈{βρ≤c*}

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