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



In view of Greco’s representation theorem we have inf Λ(λ(X - n)+) = 0 for any Λ βρ-1 (R). So by Lemma 6.3
n

we may apply the general Dini lemma as in the proof of Lemma 6.4 to conclude lim ρ(-λ(X n) + ) ρ(0)| = 0.
n→∞

This completes the proof.                                                                                H

Remark: If X is a Stonean vector lattice which consists of bounded positions only, then Proposition 6.6 is trivial.

7 Proof of Proposition 1.4 and Theorem 3.1

Throughout this section let L, E and S as in the context of Proposition 1.4 and Theorem 3.1. Furthermore X is
assumed to be a Stonean vector lattice. The next two results are for preparation.

Lemma 7.1 Let X+b consist of all nonnegative bounded X X, and let P be a probability measure on σ(X). Then
EP [X] = inf  EP [Y ] for every X X+b, and sup (EP [X] ρ(X)) = sup (EP [X] inf ρ(Z)).

XY E                             XX+b                   XE           XZX

Proof:

Let us use the abbreviations c := sup (EP [X] ρ(X)) and d := sup (EP [X] inf ρ(Z)). Setting T :=
XX+b                          XE           XZX

r

{Ω\ A A S}, we have { P λi1Gi r N, λ1,..., λr ]0, [, G1,..., Gr T} E, where 1A denotes the indicator
i=1

mapping of the subset A (cf. [14], Proposition 3.2). Since L generates σ(X), the inner Daniell-Stone theorem tells
us that P satisfies

P(A) = sup{P(B) A B S} = inf{P(B) A B T}

for every A σ(X) (cf. [14], Theorem 5.8, final remark after Addendum 5.9).

Every nonnegative bounded function from X may be described as a lower(!) envelope of a sequence of simple
σ(
X)-measurable mappings. This implies EP[X] = inf{Ep[Y] X Y E} for all bounded nonnegative
X
X. In particular c d. Moreover for any X E and ε > 0 there is some Y X+b with Y X and
inf ρ(Z) + ε > ρ(Y). This implies the inequalities
EP[X] inf ρ(Z) ≤ —EP[Y] ρ(Y) + ε c + ε. Hence
xzX                                                  xzX

d c, which completes the proof.                                                                         H

Lemma 7.2 Let P be a probability measure on σ(X) with sup (EP [X] ρ(X)) < , where L+b = L X+b
X
L+b

with X+b consisting of all nonnegative positions from Xb. If condition (2) of Theorem 3.1 is satisfied, then every

X X is P —integrable, and sup (EP [X] ρ(X)) = sup (EP [X] ρ(X)).

X Xb                    X L+b


Proof:

We have sup (EP[X] inf ρ(Z)) sup (EP [X] ρ(X)) by definition of E and condition (2) of Theorem
XE          XZX        XL+b

3.1. Moreover, L+b E, and therefore the application of Lemma 7.1 with translation invariance of ρ leads to

19



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