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 ∈ β_ρ^-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)).

X≤Y ∈E                             _X∈X_+b                   X∈E           X≥Z∈X

Proof:

Let us use the abbreviations c := sup (—EP [X] — ρ(X)) and d := sup (—EP [X] — inf ρ(Z)). Setting T :=
X∈X_+b                          X∈E           X≥Z∈X

r

{Ω\ A ∣ A ∈ S}, we have { P λ_i1G_i ∣ r ∈ N, λ₁,..., λ_r ∈]0, ∞[, G₁,..., G_r ∈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{E_p[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
x≥z∈X                                                  x≥z∈X

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)).

Proof:

We have sup (—EP[X] — inf ρ(Z)) ≤ sup (—EP [X] — ρ(X)) by definition of E and condition (2) of Theorem
X∈E          X≥Z∈X        X∈L_+b

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

19

<<   17   18   19   20   21   22   23   >>

More intriguing information