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



(*) sup (-EP [X] - ρ(X)) = sup (-EP [X] - ρ(X)) = sup (-EP [X] - inf ρ(Z)) = sup (-EP [X] -

XXb                    XX+b                    XE           X≥ZX         XL+b

ρ(X)) =: c < .

Let X X be nonnegative. It is an upper envelope of an isotone sequence (Xn)n of nonnegative simple
σ(
X)-measurable mappings. Hence, in view of the monotone convergence theorem it remains to show that the
sequence (E
P [Xn])n is bounded from above. Indeed, bearing (*) in mind, EP [Xn] - ρ(-Xn) c for each n, which
implies sup E
P[Xn] c + ρ(-X). The proof is now complete.                                               H

n

Proof of Proposition 1.4:

In view of Proposition 1.2 ρ is lower semicontinuous w.r.t. the weak topology σ(X, F) where F is the space of
all bounded countably additive set functions on σ(
X) such that every X X is integrable w.r.t. any μ F.
This implies that ρ satisfies the Fatou property, and thus ρXb is continuous from above. The remaining part of
Proposition 1.4 follows immediately from Lemma 7.1.                                                      
H

Proof of Theorem 3.1:

Let X+b , L+b consist of all nonnegative bounded positions from X and L respectively. For any Λ from the domain
of β
ρ assumption (4) ensures it may be represented by a probability content Q in the sense explained just before
Lemma 6.5 (cf. Proposition 6.6). Then condition (3), Lemma 6.2 and the inner Daniell-Stone theorem (cf. [14],
Theorem 5.8, final remark after Addendum 5.9) provide a probability measure P on σ(
X) with P(A) = sup P(B)
ABS

for every A σ(X) such that EP [Z] = Λ(Z) holds for every Z L Xb (note that L Xb generates σ(X)
by assumption on
L since L is a Stonean vector lattice). Then any X X is P -integrable by condition (2)
with Lemma 7.2. In particular we may define for every Z
L with positive and negative part Z+ and Z-
respectively, via Yn := Z+ n - Z- n a sequence (Yn)n in L Xb which converges pointwise to Z and satisfies by
dominated convergence as well as the Greco theorem (cf. [14], Theorem 2.10) the idenities Λ(Z) = lim Λ(Y
n) =
n→∞

lim EP[Yn] = EP [Z]. This also means P M1 (S) because C L. Applying Lemma 7.2 again, and bearing
n→∞

Lemma 6.5 with Proposition 6.6 in mind, we obtain αρ(P) = sup (-EP [X] - ρ(X)) βρ(Λ). Summarizing the
X L+b
discussion we have shown

(*) For every Λ from the domain of βρ there is some P M1(S) such that EP|L = ΛL and αρ(P) βρ(Λ).

After these preliminary considerations we are ready to prove Theorem 3.1.

Statement .1 is borrowed from [17] (p.12 there).

proof of statement .2:

In order to verify statement .2 we may use (*), and it remains to show that the sets ∆c (c ]ρ(0), [) are compact
w.r.t. the topology τ
L introduced in statement .1. For this purpose let (Pi)iI be a net in ∆c with arbitrary

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