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

(λ_n)_n ∈ Q. Then for arbitrary (λ_n)_n ∈ Q, m ∈ N and every ω ∈ Ω

m∞         m                m

X1(ω)(1 - ^Xλn) ≤ ^X λnXn(ω) - ^XλnXn(ω) ≤ X(ω)(1 - ^Xλn).
n=1      n=1            n=1                      n=1

This implies

∞m

(*) lim Λ(^P λnXn) -Λ(^P λnXn) =0 for anyΛ∈ βρ^-1(R).
^m→∞   n=1          n=1

Let us define ∆_c := {α_ρ ≤ c}. Then in view of Lemma 6.1 and statement .1 there exists some real c* > —ρ(0) with

ρ(Y) =   max  (-Λ(Y) - βρ(Λ)) = max (-EP [Y] - αρ(P))

Λ∈{βρ≤c*}                        ^p∈^δc*

∞

for Y ∈ {X, ^P λnXn | (λn )n ∈ Q}.
n=1

The sequence (f_n)_n in R{^βρ≤^{c }}, defined by f_n(Λ) = —Λ(X_n) — β_ρ(Λ), is uniformly bounded because

^{-2c -} p(^-X) ≤ ^— ∕^βρ(Λ.) ^- p(^-X) ^- ∕^βρ(Λ.) ≤ ^fn (Λ.) ≤ ρ(Xn) ≤ ρ(Xι).

∞

Furthermore for (λn)n ∈ Q the mapping    λnfn is well defined with

n=1
∞                m           m∞

X λnfn(Λ) = lim (—Λ(X λnXn) — βρ(Λ) X λn) = —Λ(X λnXn) — βρ(Λ)
m→∞
n=1                      n=1               n=1           n=1

for β_ρ (Λ) ≤ c* due to (*). Hence
∞∞    ∞     ∞

sup X λ_nf_n (Λ) = ρ(X λnXn) = max (—Ep^X λ_nX_n] — α_p (P)) = max X λnfn(Ep∣X).

^Л^вр^*} n = 1              n = 1          ^p∈^δc*       n=ι                    ^p∈^δc* n = ι

The application of Simons’ lemma (cf. [20], Lemma 2) and the monotone convergence theorem leads then to

ρ(X) = sup (—EP[X] — αρ(P)) = sup limsupfn(EP∣X) ≥ inf{   sup f(Λ) ∣ f ∈ co({fn ∣ n ∈ N})},

P∈∆_c.                        P∈∆_c.    n                      Λ∈{β_ρ≤c*}

where co({f_n ∣ n ∈ N}) denotes the convex hull of {f_n ∣ n ∈ N} in R{^βρ≤^{c }}. Thus for ε > 0 there is some convex
rr

combination f =    λifn_i with n1 < ... < nr and ρ(X) + ε > sup   f(Λ) = ρ(   λifn_i ). In particular the

i=1                                                    Λ∈{βρ≤c*}             i=1

inequalities ρ(X) + ε > ρ(Xn ) ≥ ρ(X) hold for n ≥ nr , which implies lim ρ(Xn ) = ρ(X).
n→∞

proof of .4 ⇒ .3:

Let X+b gather all nonnegative X ∈ Xb . Drawing on Corollary 3.3 it suffices to prove that every linear form
Λ from the domain of βρ is representable by a probability measure. So let Λ belong to β_ρ^-1 (R). Then by part
of statement .4 as well as Proposition 6.6, the linear form Λ is representable by some probability content Q on
σ(X) in the sense as introduced just before Lemma 6.5. We want to apply Proposition A.1 (cf. appendix A) to
∞               ∞ ∞               _-1

verify that Q is a probability measure. Since Ω\ T Xn¹([xn, ∞[) = S S (xn — (Xn ∧ Xn))  ([m¹, ∞[) holds

n=1                  n=1 m=1

for every pair (Xn )n , (xn)n of sequences in L ∩ X+b and ]0, ∞[ respectively, it remains to show by assumption

22

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