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

representable by a probability content if there is some probability content Q such that every X ∈ X is
integrable w.r.t. Q and Λ(X) = EQ [X].

If X is a Stonean vector lattice, then X ∧ Y = min{X, Y }, X ∨ Y = max{X, Y } ∈ X for X, Y ∈ X in particular
X⁺ := X ∨ 0, X^- := (-X) ∨ 0 ∈ X for any X ∈ X. In this case, if all linear forms from the domain of βρ are
representable by probability contents, then ρ is concentrated on the bounded positions, and as a consequence ρ
admits a robust representation by probability measures if its restriction to the bounded positions does so.

Lemma 6.5 Let X be a Stonean vector lattice, and let every linear form Λ ∈ β_ρ^-1 (R) be representable by some
probability content Q on σ(X). Then we can state:

.1 The sequence ρ(X⁺ - (X^- ∧ n)) _n converges to ρ(X) for every X ∈ X.

.2 The sequence ρ((X ∧ m) - Y ) _m converges to ρ(X - Y ) for nonnegative X, Y ∈ X, Y being bounded.

.3 inf{∣ρ((X⁺ ∧ m) — (X^- ∧ n)) — ρ(X)∣ ∣ m, n ∈ N} = 0 for X ∈ X, and in addition

sup (-EQ [X] - ρ(X)) = sup (-EQ [X] - ρ(X))
X∈X                   X∈X_b

for every probability content Q on σ(X) such that each X ∈ X is integrable w.r.t. Q .

.4 If M ⊆ α_ρ^-1 (R) with ρ(X) = sup (—EQ [X] — αρ(Q)) for all bounded X ∈ X, then

Q∈M

ρ(X) = sup (—EQ [X] — αρ (Q)) for all X ∈ X.

Q∈M

Proof:

The most important tool of the proof is Greco’s representation theorem. The reader is kindly referred to [14]
(Theorem 2.10 with Remark 2.3).

Since any Λ ∈ β_ρ^-1 (R) is representable by a probability content, statement .2 follows immediately from Greco’s
representation theorem and Lemma 6.4.

proof of .1:

Let X ∈ X, and let ε > 0. Then there exists some probability content Q with βρ(EQ |X) < ∞ such that the
inequality ρ(X) — ε < —EQ [X⁺] — βρ(EQ |X) + EQ [X^-] holds. Application of Greco’s representation theorem leads
then to

ρ(X) — ε< -Eq[X⁺] — βρ(EQ∣X)+limEq[X^- ∧ n] ≤ limρ(X⁺ — (X^- ∧ n)) ≤ ρ(X)
nn

proof of .3:

Let Q be a probability content on σ(X) such that every X ∈ X is integrable w.r.t. Q, and let X ∈ X. Then for
ε > 0 we may choose by statement .1 and Greco’s representation theorem some n ∈ N with

∣ — Eq [X] — ρ(X) — '—Eq [X⁺ — (X^- ∧ n)] — ρ(X⁺ — (X^- ∧ n))´ ∣ < ɪ

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