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

A Appendix

Proposition A.1 Let (Ω, F ) be a measurable space, and let {0, Ω} ⊆S⊆F be stable under finite union and
countable intersection, generating F. Furthermore, for every A ∈ S there exists a sequence (An)n in S such that

^∞

Ω \ A = U A_n. Then every probability content Q on F is a probability measure if and only if lim Q(A_n) = 1
n=1                                                                       ^n→∞

∞

holds for any isotone sequence (A_n)_n in S with (J A_n = Ω.

n=1

Proof:

Let Q be a probability content on F, and let us denote φ := Q |S. The only if part of the statement is obvious.

For the if part we want to show

∞

(*) lim φ(A_n) = φ(A) for every isotone sequence (A_n)_n in S with (J A_n = A ∈S
n→∞                                               _n=1

∞

Since for A, B ∈ S with A ⊆ B there is by assumption an isotone sequence (An )n in S with    An = B \ A,

n=1

we may apply a version of the general extension theorem by Konig (cf. [13], Theorem 7.12 with Proposition 4.5).

Hence condition (*) together with the assumptions on S guarantee a probability measure P on the σ-algebra

F^e with P |S = φ, and P(A) = sup P(B) for every A ∈ F. In particular P ≤ Q which implies P = Q due to
A⊇B∈S

additivity of Q and P . Therefore it remains to prove the condition (*).

proof of (*):

∞

Let (An)n be an isotone sequence in S with    An = A ∈ S. By assumption there exists some isotone sequence

n=1

∞∞

(B_n)_n in S with U B_n = Ω \ A. Then (B_n ∪ A)_n and (B_n ∪ A_n)_n are isotone sequences in S with (J (B_n ∪ A_n) =
n=1                                                                            n=1

^∞

Ω = U (B_n ∪ A), and therefore lim (φ(B_n) + φ(A_n)) = 1 = lim φ(B_n) + φ(A). Hence lim φ(A_n) = φ(A),
_n=1                   n→∞                   n→∞                 n→∞

which shows (*), and completes the proof.

B Appendix

Proof of Proposition 1.1:

Let X ∈ X, and let ρ₊(X, ∙) : X → R denote the respective rightsided derivative of ρ at X defined by ρ+(X,Y) : =

^ρ(X^+hY)—^ρ(X). It it well known from convex analysis that P+(X, ∙) is well defined and sublinear satisfying
ρ₊ (X, Y - X) ≤ ρ(Y) - ρ(X) for all Y ∈ X. Then we may choose by Hahn-Banach theorem some linear form Λ

ΛΛ.           Z        Z

on X with Λ ≤ P+(X, ∙). Moreover, we obtain for Z ≥ 0

~             Z

Λ(Z) ≤ P+(X, (X + Z) - X) ≤ ρ(X + Z) - ρ(X) ≤ 0.

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