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] -

X∈Xb                    X∈X+b                    X∈E           ^X≥Z∈X         X∈L+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 (EP [Xn])n is bounded from above. Indeed, bearing (*) in mind, EP [Xn] - ρ(-Xn) ≤ c for each n, which
implies sup EP[X_n] ≤ 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 ρ∣X_b 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 βρ 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)
A⊇B∈S

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(Yn) =
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 β_ρ there is some P ∈ M₁(S) such that E_P|L = Λ∣L and α_ρ(P) ≤ β_ρ(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)i∈I be a net in ∆c with arbitrary

20

<<   18   19   20   21   22   23   24   >>

More intriguing information