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

Remark:

Corollary 3.3 extends a respective result for bounded one-period positions ([16], Proposition 3).

Let us now consider some special situations where Theorem 3.1 might be used.

Remark 3.4 Let Ω = Ω × T with Ω denoting a set of scenarios, equipped with a metrizable topology tω as well as
the induced σ-algebra B(Ω), and T being a time set, endowed with a separably metrizable topology τ_τ as well as
the generated σ-algebra B(T). Furthermore let X consist of all bounded real-valued mappings on Ω × T which are
measurable w.r.t. the product σ-algebra B(Ω) ® B(T), and let L be the set of all bounded real-valued mappings on
Ω × T which are continuous w.r.t. the product topology tω × τ_τ. Finally S is defined to gather the closed subsets of
Ω × T w.r.t. the metrizable topology tω × τ_τ. Using the introduced notations, σ(X) = B(Ω) ®B(T) is generated by S,
L ⊆ X, and we may restate Theorem 3.1 with E being the space of all bounded nonnegative lower semicontinuous
mappings on Ω × T. This version generalizes an analogous result for the one-period positions (cf. [16], Theorem
2), and will be proved in section 7.

We may also utilize Theorem 3.1 for cadlag positions.

Remark 3.5 Let T = [0,T], C = R, let (F_t)_t∈T be a filtration of σ-algebras on some nonvoid set Ω, and let X
be the set of cadlag positions. Then σ(X) is the so-called optional σ-algebra. We may associate for stopping
times S1 , S2 , S1 ≤ S2, the stochastic interval [S1 , S2 [, defined by [S1 , S2 [(ω, t) := 1 if S1 (ω) ≤ t < S2 (ω), and
[S1 , S2 [(ω, t) := 0 otherwise. I stands for the set of all such stochastic intervals. It can be shown that σ(X) is
generated by the stochastic intervals [S, ∞[ (cf. [5], IV, 64).

For L let us choose the vector space spanned by the stochastic intervals [S, ∞[, which is also spanned by the positions
max [Si , S^ei [ (r ∈ N), where [Si , S^ei [ ∈ I for i ∈ {1, ..., r}. Moreover, L is indeed a Stonean vector lattice, and
i∈{1,...,r}
r            _-1

{X^-1 ([x, ∞[) | X ∈ L nonnegative, x > 0} is an algebra consisting of all the subsets ^S [Si , S^ei[   ({1}) with r ∈ N

i=1

and [Si, S^ei[ ∈ I for i ∈ {1, ..., r}.

Using the introduced notations, we may restate Theorem 3.1.

Theorem 3.1 may be used as a basis to derive conditions for a strong robust representation of ρ, i.e. a σ-additive
robust representation where solutions of the associated optimization problems exist. We shall succeed in finding a
full characterization in the next section.

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