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

.2 ρ(Xn) ∖ p(X) for Xn / X.

.3 ρ(λ1A_n ) ∖ ρ(λ) for λ > 0 whenever (1A_n )_n is an isotone sequence of indicator mappings of subsets A_n ∈ F
∞

with U A_n = Ω.

n=1

Let us adapt Theorem 4.1 to the special situations of Remarks 3.4, 3.5.

Corollary 4.3 In the special context of Remark 3.4 with the notations introduced there all the statements .1 - .4
of Theorem 4.1 are equivalent.

Remark:

Corollary 4.3 generalizes a result for one-period positions (cf. [16], Theorem 1).

Remark 4.4 Let T = [0,T], C = R, let (F_t)_t∈T be a filtration of σ-algebras on a set of scenarios Ω, and let X
be the set of cadlag positions. Then all statements .1 - .4 from Theorem 4.1 are equivalent, choosing L to be the
vector space spanned by the stochastic intervals [S, ∞[ (cf. Remark 3.5).

5 Robust representations of convex risk measures in presence of
given market models

Througout this section we assume that we have a market model with a reference probability measure P on a
σ-algebra F on the set of scenarios Ω. In the following we shall retain, and partly generalize, already known
results concerning the σ-additive robust representations of the convex risk measure ρ within the setting of a
market model. The point is that they may be derived from the results presented in the sections 1, 2 and 4. We
shall use the following notations. The spaces of P -integrable mappings of order p ∈ [1, ∞[ and P -essentially
bounded mappings will be denoted by L_p(Ω, F, P) and L∞(Ω,F, P) respectively. For p ∈ [1, ∞] the symbol
L_p(Ω,F,P) will be used for the space formed by identifying functions in L_p(Ω,F,P) that agree P —a.s.. The
equivalence class of any X ∈ L_p(Ω, F, P) will be indicated by < X > .

The first result may be found in [19] for π being the identity on R. Using Propostion 1.1, we obtain a slight
generalization.

Proposition 5.1 Let X = L_p(Ω, F, P) (p ∈ [1, ∞[) with conjugate space L_q(Ω, F, P). Furthermore let ρ(X) = ρ(Y)
for X = Y P a.s.. If M₁(q) denotes the set of all Q ∈ M₁ having some P —density from L_q(Ω, F, P), then

ρ(X) = max (—Eq[X] — α_p(Q)) for all X ∈ Lp(Ω, F, P).

Q∈M1(q)

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