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

Proof:

Let Lp(Ω, F, P) be equipped with the order >, defined by < X >^< Y > if X ≥ Y P —a.s., which induces the
operations of minimum and maximum. It is known that (Lp(Ω, F, P), ∣∣ ∙ ∣∣_p, >) (∣∣ ∙ ∣∣_p Lp — norm) is a Banach
lattice, and therefore all positive linear forms w.r.t. > are continuous w.r.t. ∣∣ ∙ ∣∣_p (cf. [10], p. 151/152, Corollary
3). Next notice that β_ρ(Λ) < ∞ implies that Λ(Z) = 0 holds for Z = 0 P —a.s., so that Λ(< X >) := Λ(X)
describes a well defined positive linear form on L_p(Ω, F, P) w.r.t. > . Then the claimed representation of ρ follows
immediately from Proposition 1.1 and the representation result for norm-continuous linear forms on Lp(Ω, F, P).

■

In the case of X = L∞(Ω, F, P) we may generalize the equivalent characterization of strong robust representations
for ρ shown in [11].

Theorem 5.2 Let X = L∞(Ω,F,P), and ρ(X) = ρ(Y) for X = Y P a.s.. Then ρ(X) = max (—Eq[X] —
Q∈M1

αρ(Q)) for all X ∈ L∞(Ω, F, P) if and only if ρ(Xn) ∖ ρ(X) for Xn / X P — a.s..

Proof:

The statement follows immediately from Theorem 4.1 since the condition ρ(X_n) ∖ ρ(X) for X_n / X P —a.s. is
equivalent with the property ρ(X_n) ∖ ρ(X) for X_n / X.                                                  H

The next result retains an equivalent characterization of the robust representations for ρ which may be found in
[7] (Theorem 4.31).

Theorem 5.3 Let X := L∞(Ω, F, P) and π be the identity on C = R. Furthermore ρ is supposed to satisfy
ρ(X) = ρ(Y) for X = Y P —a.s.. If M1 (P) denotes the set of probability measures on F which are absolutely
continuous w.r.t. P, then the following statements are equivalent.

.1 ρ(X)=  sup (—Eq[X] — αρ(Q)) for all X ∈ L∞(Ω,F,P).

Q∈M1(P)

.2 ρ(Xn) / ρ(X) for Xn ∖ X P —a.s..

.3 liminf ρ(X_n) ≥ ρ(X) whenever (X_n)_n is a uniformly P —essentially bounded sequence in L∞(Ω,F,P) with
n→∞

Xn → X P —a.s..

Proof:

First of all, ρ' : L∞(Ω, F, P) → R, ρ(< X >) := ρ(X) is well defined.

Next let SL₁+ := {< g >∈ L₁(Ω,F,P) | g ≥ 0 P —a.s., EP[g] = 1} be endowed with the relative topology of
the L1—norm topology on L₁(Ω,F,P). We may introduce, via Φ(< X >)(< g >) = EP[Xg], an injective vector
space homomorphism Φ from L∞(Ω, F, P) onto a vector subspace of C_b(SL₁+ ), defined to consist of the bounded,

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