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

4 Strong robust representation of convex risk measures by prob-

ability measures

We want to look for conditions which induce a strong robust representation of ρ by probability measures in the
sense that

ρ(X) = _Pm_∈Max (-EP [X] - αρ(P))

holds for any X ∈ X. The considerations are reduced to a Stonean vector lattice X being stable w.r.t. countable
convex combinations of antitone sequences of financial positions. In this case the following result gives a complete
answer to the problem of strong robust representations.

Theorem 4.1 Let X be a Stonean vector lattice and let us assume that for every antitone sequence (Xn)n in
∞∞

X with Xn ∖ 0 and each sequence (λ_n)_n in [0,1] with ∑ λ_n = 1 there is some pointwise limit ɪɪ λ_nX_n of
n=1                                       n=1

m

(   λnXn)m belonging to X. Then the following statements are equivalent:

n=1

.1 ρ(X) = max (-EP [X] - αρ (P)) holds for every X ∈ X.

.2 ρ(Xn) ∖ ρ(x) for Xn 7 X.

.3 lim ρ(-λ(X — n) ⁺ ) = ρ(0) hold for arbitrary nonnegative X ∈ X,λ> 0, and ρ(X_n) ∖ ρ(X) for any isotone
n→∞

sequence (X_n)_n of bounded positions from X with X_n / X, X being bounded.

In any of these cases every linear form from the domain of βρ is representable by a probability measure. Moreover,
for any Stonean vector lattice L ⊆ X which contains C as well as generates the same σ-algebra as X and induces
∞

the set system S consisting of all sets ^T X_n^-1 ([xn , ∞[) (Xn ∈ L ∩ Xb nonnegative, xn > 0) any of the statements
n=1

.1 - .3 is implied by

.4 lim ρ(-λ(X — n) ⁺ ) = ρ(0) hold for arbitrary nonnegative X ∈ X,λ> 0, and inf   p(λZ) ∖ ρ(λ) for

^n→∞                                                                  1A_n ≥Z∈X

∞

λ> 0 whenever (1A_n)_n is an isotone sequence of indicator mappings of subsets An ∈ S with (J An = Ω.

We have even equivalence of the statements .1 - .4 if the indicator mappings 1A (A ∈ S) belong to X.

The proof may be found in section 8.

For bounded one-period positions, Theorem 4.1 enables us to give an equivalent characterization of convex risk
measures that admit strong robust representations by probability measures.

Corollary 4.2 Let F denote some σ-algebra on Ω, and let X consist of all bounded F—measurable real-valued
mappings. Then the following statements are equivalent:

.1 ρ(X) = max (—EP [X] — αρ (P)) holds for every X ∈ X.

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