a given market model the so-called Fatou property plays a prominent role. Adapting this concept, we shall say
that a risk measure ρ fulfill the Fatou property if lim inf ρ(Xn) ≥ ρ(X) whenever (Xn)n denotes a uniformly
n→∞
bounded sequence in X which converges pointwise to some bounded X ∈ X. The Fatou property implies obviously
that ρ∖Xb is continuous from above, defined to mean ρ(Xn) ∕' ρ(X) for Xn ∖ X. Both conditions coincide if
sup Xn ∈ X for any uniformly bounded sequence (Xn)n in X.
n
Proposition 1.4 Let ρ admit a σ-additive robust representation w.r.t. some nonvoid M ⊆ M1 , then ρ satisfies
the Fatou property and ρ∖Xb is continuous from above. Moreover, if X is a Stonean vector lattice, and if L ⊆ X
denotes any Stonean vector lattice which contains C as well as generates σ(X), then ρ(X) = sup inf ρ(Z)
X≤Y ∈E Y ≥Z∈X
for every bounded nonegative X ∈ X, where E := {sup Yn ∖ Yn ∈ L, Yn ≥ 0, sup Yn bounded}.
The proof may be found in section 7.
As mentioned in the introduction, a robust representation of ρ by probability measures is not guaranteed in general
by the Fatou property or continuity from above, even if X contains bounded positions only. In the next section
we shall investigate additional conditions to guarantee the sufficiency by the Fatou property and a nonsequential
counterpart.
2 Representation of convex risk measures by probability mea-
sures and the Fatou properties
It will turn out by the investigations within this section that in the case of uncertainty about the market model
the nonsequential counterpart of the Fatou property takes over partly the role that the Fatou property plays when
a reference probability measure is given. We shall say that ρ satisfies the nonsequential Fatou property if
lim inf ρ(Xi) ≥ ρ(X) holds whenever (Xi)i∈I is a uniformly bounded net in X which converges pointwise to some
i
bounded X ∈ X. The following condition provides an important special situation when the Fatou property and its
nonsequential counterpart are equivalent.
(2.1) For any r > 0, every Z ∈ Xb from the closure of Ar := {X ∈ Xb ∖ ρ(X) ≤ 0, sup ∖X (ω)∖ ≤ r} w.r.t. the
ω∈Ω
topology of pointwise convergence on Xb is the pointwise limit of a sequence in Ar .
Lemma 2.1 Under (2.1) ρ satisfies the nonsequential Fatou property if and only if it fulfills the Fatou property.
The proof is delegated to section 9.
Remark: