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

c ∈]ρ(0), ∞[. So (Ep∣X)_i∈ι is a net in {β_ρ ≤ c}, which in turn is compact w.r.t. to product topology on RX by

Lemma 6.3. Therefore there exist a subnet (P_i(_k))k∈K and some Λ ∈ {βρ ≤ c} with lim EP_i(k) [X] = Λ(X) for
k     ()

every X ∈ X. Then (Pi(k))k∈K converges to some P ∈ ∆c due to (*). This finishes the prove of statement .2.

proof of .3:

Drawing on Lemma 6.5 with Proposition 6.6 and the translation invariance of ρ it remains to show

ρ(X) =   sup (-EP [X] - αρ (P)) for all X ∈ X+b.

P∈M1(S)

For this purpose let X ∈ X+b , and let ε > 0. Then by (1), (2) and definition of E we may find an isotone sequence

(Yn)n in L+b with X ≤ sup Yn ∈ E and ρ(X) < inf ρ(Yn) + ε. In view of Lemma 6.1 and (*) there is some
nⁿ

c* ∈] — ρ(0), ∞[ with ρ(Y_n) = sup (—EP[Y_n] — α_ρ(P)) for any n. Furthermore F_n(P) := — EP[Y_n] — α_ρ(P) defines
P∈∆_c.

an antitone sequence of mappings Fn := ∆_c* → R which are upper semicontinuous w.r.t. the relative topology of

TL (see Lemmata 7.2, 6.5 again) such that F_n ∖ F, defined by F (P) := —E_p[Y ] — α_ρ(P). Due to .2 we may apply
the generalized Dini lemma (cf. [12], Theorem 3.7) and obtain

ρ(X) — ε < infρ(Yn) = sup (—EP [Y] — αρ(P)) ≤ sup (—EP[X] — αρ(P)) ≤ ρ(X),
ⁿ           P∈∆_c,                        P∈∆_c,

which completes the proof.                                                                               H

Proof of Remark 3.4:

Obviously, σ(X) = B(Ω) ® B(T), and any closed subset A of Ω × T w.r.t. the metrizable topology tω × TT may be
∞

described by A = ^T X_n^-1 ([xn , ∞[) for some sequence (Xn)n of nonnegative continuous mappings and a sequence
n=1

(xn)n of positive real numbers.

Now let X ∈ L. Then X(∙,t) is B(Ω)—measurable for t ∈ T and X(ω, ∙) is continuous w.r.t. t_t for ω ∈ Ω, which
implies X ∈ X because TT is separably metrizable (cf. e.g. [3], Lemma III-14). Therefore L ⊆ X, and σ(X) is
generated by S. Then the statement of Remark 3.4 follows immediately from Theorem 3.1.                   H

8 Proof of Theorem 4.1

Obviously, .2 ⇒ .3, and statement .2 implies statement .4 if the indicator mappings 1A (A ∈ S) belong to X. So
in view of Corollary 3.3 and Proposition 1.1 it remains to prove the implications .1 ⇒ .2 and .4 ⇒ .3.

proof of .1 ⇒ .2:

∞

Let Q consist of all sequences (λn)n in [0, 1] with λn = 1. Aditionally, let (Xn)n be any isotone sequence in
n=1
∞

X which converges pointwise to some X ∈ X. Then by assumption    λnXn is a well defined member of X for

n=1

21

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