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

continuous real-valued mappings on SL₁+,. The inverse mapping Φ ¹ : Φ(L∞(Ω,F,P)) → L∞(Ω,F,P) may be
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used to define the convex risk measure ρ := ρ◦ Φ ¹ on X := Φ(L∞(Ω,F,P)) w.r.t. to the identity π on R. Notice
that Φ(R) consists of all constant real-valued mappings on SL₁₊ .

Furthermore let F be the linear span of the Dirac measures δ<_g> (< g >∈ SL₁+ ). For any ν ∈ F there is
some < g >∈ L₁(Ω, F, P) such that R Φ(< X >) dν = R Xg dP holds for every < X >∈ L∞(Ω,F, P), and
∣∣ν∣∣f := sup{∣ R Φ(< X >) dν∣ ∣ sup ∣Φ(< X >)(< g >)∣ ≤ 1} = Il < g > H1. Here ∣∣∙∣∣1 denotes the Li—norm.

<g>∈SL₁₊

r

Conversely, for each < g > from L1(Ω, F, P) with arbitrary representation < g >=    λi < gi > (r ∈ N, λi ∈ R,

i=1
r

< gi >∈ SL₁+ ; i = 1,...,r), we may define ν := P λ_iδ<_gi> ∈ F which satisfies ʃ Φ(< X >) dν = ʃ Xg dP for
i=1

every < X >∈ L∞(Ω,F,P). Therefore F is complete w.r.t. the seminorm ∣∣ ∙ ∣f, and in order to apply Theorem
2.2 we have to show that the conditions (2.1), (2.2) are fulfilled for the sets Ar := ρ^-1(] — ∞, 0]) ∩ Br and
~ . . . . . . . . .

Br := {Φ(< X >) ∈ X ∣ sup   ∣Φ(< X >)(< g >)∣ ≤ r} (r > 0).

<g>∈SL₁₊

For this purpose fix r > 0. Since L∞ (Ω, F, P) represents the norm dual of L1(Ω, F, P), the application of the
Banach-Alaoglu theorem yields that Φ^-1(Br) is σ(L∞(Ω, F, P), L1(Ω, F, P))—compact. This in turn implies by
construction that Br is compact w.r.t. the topology σ (X, F) of pointwise convergence.

Moreover, by definition of Φ, the mapping φ : X → L₁(Ω, F, P), Φ(< X >) →< X >, is injective, and continuous
w.r.t. σ(X, F) and the weak topology on L1(Ω, F, P). Since the closure cl(Ar) w.r.t. σ(X, F) is even compact,
the restriction ^∣cl(A_r) : cl(A_r) → φ(d(Ar)) is a homeomorphism w.r.t. the associated relative topologies. In
particular φ(d(Ar)) is the weak closure of φ(Ar), and hence, by Eberlein-Smulian theorem, every element is the
limit point of a sequence in φ(Ar) w.r.t. the weak topology. Therefore each point from Cl(Ar) is the pointwise
limit of a sequence in Ar.

Now in view of Proposition 1.4 the relationships .1 ⇒ .2 ⇔ .3 are clear. The implication .3 ⇒ .1 follows from
Theorem 2.2 by the following argument. Let (Xn)n be a sequence in L∞ (Ω, F, P) and let X ∈ L∞ (Ω, F, P) such
that (Φ(< Xn >)n is uniformly bounded and converges pointwise to Φ(< X >). We may find a subsequence
(X_i(_n))n with lim inf ρ(Xn) = lim ρ(X_i(_n)). Since the P —essential sup norm on L∞ (Ω, F, P) coincides with
n→∞        n→∞

the operator norm w.r.t. ∣∣ ∙ ∣∣₁, the sequence (X_n)_n is P —essentially bounded. Therefore Komlos’ subsequence
theorem (cf. [7], Lemma 1.69) guarantees a sequence (Yn)n of convex combinations Yn from {X_i(_m) | m ≥ n}
which converges P —a.s. pointwise to X and satisfies liminf ρ(X_n) ≥ liminf ρ(Y_n).                             H

n→∞       n→∞

6 Some auxiliary results

Throughout this section we want to gather some technical arguments which will be often used when proving
the main results of the paper. In the following ρ denotes a convex risk measure w.r.t. π associated with the

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