A quantitative approach to carbon price risk modeling - CiteSeerX

Note that since ξ and ξ ′ coincide at times 0, . . . , t−1, this definition indeed yields
an adapted process ξ ′ ∈ U . With (38), we have the decomposition
G(ξ ′ ) = 1 M G(ξ) + 1 Ω\M G(ξ ∗ ),
which gives a contradiction to the optimality of ξ ∗ :
E(G(ξ ′ )) = E(E(1 M G(ξ) + 1 Ω\M G(ξ ∗ )|F t ))
= E(1 M E(G(ξ)|F t ) + 1 Ω\M E(G(ξ ∗ )|F t ))
> E(1 M E(G(ξ ∗ )|F t ) + 1 Ω\M E(G(ξ ∗ )|F t )) = E(G(ξ ∗ )).
To prove (35) and (36) we consider for each λ in the countable set Q := [0, λ i ] ∩ Q
a strategy ξ(λ, i) ∈ U defined by
{
ξs k λ if s = t and k = i
(q, i) =
,
else
ξ ∗k
s
That is, ξ(λ, i) coincides with ξ ∗ with the exception of time t, where only for
the agent i the fuel switch intensity is changed from ξt
∗i to a deterministic value
λ ∈ Q. The policy ξ(λ, i) satisfies
Π(ξ(λ, i)) = Π(ξ ∗ ) − (ξt ∗i − λ)
F (ξ(λ, i)) = F (ξ ∗ ) − (ξt
i∗ − λ)Et
i
for all λ ∈ Q. (39)
Since the set Q is countable due to (37), there exists a set ˜Ω with P (˜Ω) = 1 such
that
E(G(ξ ∗ |F t ))(ω) − E(G(ξ(λ, i)|F t ))(ω)
≥
(ω) − λ|
|ξ ∗i
t
Using (39) and (13), we conclude from this inequality that
0 ≤ − ξ∗i t (ω) − λ
|ξt ∗i(ω)
− t(ω)
λ|Ei
0 for all ω ∈ ˜Ω with λ ≠ ξ ∗i
t (ω).
−E(π (Γ T − Π(ξ ∗ )) + − (Γ T − Π(ξ ∗ ) + (ξt ∗i − λ)) +
|ξt ∗i
| F t )(ω) (40)
− λ|
holds for all ω ∈ ˜Ω with λ ≠ ξt ∗i (ω). Let us denote the term in in (40) by
D(ξ ∗ , λ)(ω) Approaching ξt ∗i (ω) by λ ∈ Q \ {ξt ∗i (ω)}, we apply dominated convergence
theorem to obtain
lim
λ↑ξt ∗(ω) D(ξ∗ , λ)(ω) = −E ( )
π1 {Γ−Π(ξ ∗ )≥0} | F t (ω) for ξ
∗i
t (ω) ∈]0, λ i ],
lim
λ↓ξt ∗(ω) D(ξ∗ , λ)(ω) = E ( )
π1 {Γ−Π(ξ ∗ )>0} | F t (ω) for ξ
∗i
t (ω) ∈ [0, λ i [.
Now (15) gives
E ( π1 {Γ−Π(ξ ∗ )≥0} | F t
)
= E
(
π1{Γ−Π(ξ ∗ )>0} | F t
)
= A
∗
t
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