A quantitative approach to carbon price risk modeling - CiteSeerX

which with (40) implies that the following inclusions hold almost surely: Calculating
left limit λ ↑ ξ i t(ω), we have
{ξ ∗i
t ∈]0, λ i ]} ⊆ {A ∗ t − E i t ≥ 0} ⇔ {A ∗ t − E i t < 0} ⊆ {ξ ∗i
t = 0} (41)
For the right limit λ ↓ ξ i t(ω), we obtain
{ξ ∗i
t ∈ [0, λ i [} ⊆ {A ∗ t − E i t ≤ 0} ⇔ {A ∗ t − E i t > 0} ⊆ {ξ ∗i
t = λ i }. (42)
The assertions (41) and (42) give (35) and (36) respectively.
Finally, it remains to show the existence of the solution to the global optimization
problem (14) formulated in the Proposition 1.
Proof. Let us utilize techniques from functional analysis. First, note that L N 1 ,
equipped with the norm
T∑
−1 N∑
‖Ξ‖ = E(|Ξ i t|)
t=0 i=1
is a Banach space with dual L N ∞ . The canonical bilinear form is
〈Ξ, ξ〉 :=
T∑
−1
t=0 i=1
N∑
E(Ξ i tξt) i Ξ ∈ L N 1 , ξ ∈ L N ∞.
Next, we consider the weak topology σ(L N ∞, L N 1 ) on LN ∞ (see [12]). Note that
in this topology, the neighborhood basis of a point ξ ∈ L N ∞ is given by all finite
intersections of sets
B ξ (Ξ, δ) := {ξ ′ ∈ L N ∞ : |〈Ξ, ξ ′ − ξ〉| < δ}, Ξ ∈ L N 1 , δ > 0. (43)
In other words, σ(L N ∞, L N 1 ) is the weakest topology for which any linear functional
L N ∞ → R, ξ ↦→ 〈Ξ, ξ〉, Ξ ∈ L N 1 (44)
is continuous. A function f : L N ∞ → R is lower semicontinuous at ξ if for each
ε > 0 there exist a neighborhood B ξ of ξ such that f(ξ ′ ) > f(ξ) − ε for all
ξ ′ ∈ B ξ , the function is called lower semicontinuous, if it is lower semicontinuous
at each point. This generalization of continuity is useful since lower semicontinuous
functions attain their minima on compact sets. We use this property to show the
existence of ξ ∗ , being a minimizer of ξ ↦→ E(−G(ξ)) on U . To proceed so, we need
to prove that ξ ↦→ E(−G(ξ)) is lower semicontinuous with respect to σ(L N ∞, L N 1 ).
Given ξ ∈ L N ∞ , write
E(−G(ξ)) =
T∑
−1
t=0 i=1
N∑
E(Etξ i t) i + πE((Γ − Π(ξ)) + ).
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