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 ZX       Y ZL

(3) ρ(Xn) ρ(X) for any isotone sequence (Xn)n of bounded positions Xn L with Xn ∕' 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 τL on M1 (S) induced by the mappings ψX : M1 (S) R, P EP [X], (X L) is
completely regular and Hausdorff.

.2 Eachc (c ] - ρ(0),[) is compact w.r.t. τL, and furthermore for every Λ from the domain of βρ there is
some
P M1(S) with ΛL = EP|L and αρ(P) ≤ βρ(Λ).

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

PM1(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 ωΩ          ωΩ 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,
XY E

(2) ρ(Y ) = inf ρ(Z) for Y E.
YZL

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(Xn) ρ(X) whenever (Xn)n is an isotone sequence of bounded positions from X which converges
pointwise to some bounded
X X.



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