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

(2) inf ρ(Z) = inf ρ(Z) for Y ∈ E,

Y ≥Z∈X       Y ≥Z∈L

(3) ρ(X_n) ∖ ρ(X) for any isotone sequence (X_n)_n of bounded positions X_n ∈ L with X_n ∕' X ∈ L, X bounded,

(4) lim ρ(-λ(X - n)⁺) = ρ(0) for every nonnegative X ∈ X and λ > 0.
n→∞

Then we may state:

.1 The initial topology τ_L on M1 (S) induced by the mappings ψX : M1 (S) → R, P → EP [X], (X ∈ L) is
completely regular and Hausdorff.

.2 Each ∆c (c ∈] - ρ(0), ∞[) is compact w.r.t. τ_L, and furthermore for every Λ from the domain of βρ there is
some P ∈ M₁(S) with Λ∣L = E_P|L and α_ρ(P) ≤ β_ρ(Λ).

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

P∈M1(S)

The proof of Theorem 3.1 is delegated to section 7.

Remarks 3.2 In view of Proposition 1.4, assumption (1) in Theorem 3.1 is necessary for a robust representation
of ρ by probability measures. Let us now point out some special situations where the assumptions on ρ, imposed in
Theorem 3.1, may be simplified:

.1 If X is restricted to bounded positions, then assumption (4) is redundant. Also (1), (2) hold in general in
the case X = L.

.2 By Lemma 6.4 below assumption (3) is fulfilled in general whenever L+b, consisting of all nonnegative
bounded X ∈ L, is a so-called Dini cone, i.e. inf sup Xn (ω) = sup inf Xn (ω) for any antitone sequence
ⁿ ω∈Ω          ω∈Ω ⁿ

(Xn)n in L+b with pointwise limit in L+b. The most prominent Dini cones are the cones of nonnegative
upper semicontinuous and nonnegative continuous real-valued mappings on compact Hausdorff spaces due to
the general Dini lemma (cf. [12], Theorem 3.7).

.3 If E ⊆ X, then assumptions (1), (2) read as follows:

(1) ρ(X) = sup ρ(Y ) for all nonnegative bounded X ∈ X,
X≤Y ∈E

(2) ρ(Y ) = inf ρ(Z) for Y ∈ E.
Y ≥Z∈L

We may specialize to X = L, and a direct application of Theorem 3.1 in combination with Lemma 6.4 below
leads to the following condition to ensure that every linear form Λ from the domain of βρ is representable by a
probability measure. Note that here M1(S) = M1.

Corollary 3.3 Let X be a Stonean vector lattice, and let lim ρ(-λ(X -n)⁺) = ρ(0) be valid for every nonnegative
n→∞

X ∈ X, λ > 0. Then every linear form from the domain of βρ is representable by some probability measure from M1
if and only if ρ(X_n) ∖ ρ(X) whenever (X_n)_n is an isotone sequence of bounded positions from X which converges
pointwise to some bounded X ∈ X.

<<   7   8   9   10   11   12   13   >>

More intriguing information