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 An. Then every probability content Q on F is a probability measure if and only if lim Q(An) = 1
n=1                                                                       n→∞

holds for any isotone sequence (An)n in S with (J An = 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(An) = φ(A) for every isotone sequence (An)n in S with (J An = 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

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

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

∞∞

(Bn)n in S with U Bn = Ω \ A. Then (Bn A)n and (Bn An)n are isotone sequences in S with (J (Bn An) =
n=1                                                                            n=1

Ω = U (Bn A), and therefore lim (φ(Bn) + φ(An)) = 1 = lim φ(Bn) + φ(A). Hence lim φ(An) = 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) : =

lim
h→0+


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.

25



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