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For a multidimensional Lévy process satisfying the Assumptions
14.3.4
,weagain
have a sector condition as shown in [134].
Lemma 14.5.1
Let X be a Lévy process with characteristic triplet (
,ν,
0
) where
the Lévy measure satisfies
(
14.16
), (
14.17
).
Then
,
there exist constants C
i
>
0,
i
=
1
,
2
,
3,
Q
i
=
1
|
d
i
=
1
|
d
2
ρ
i
,
2
ρ
i
ψ(ξ)
≥
C
1
ξ
|
|
ψ(ξ)
| ≤
C
2
ξ
|
+
C
3
.
Using this, we can show as in Theorem 10.4.3
Theorem 14.5.2
Let X be a Lévy process with characteristic triplet (
Q
,ν,
0
) where
the Lévy measure satisfies
(
14.16
), (
14.17
).
Then
,
there exist constants C
i
>
0,
i
=
1
,
2
,
3,
such that for all ϕ,φ
H
ρ
(
d
) the following holds
:
∈
R
a
J
(ϕ, φ)
J
(ϕ, ϕ)
2
H
ρ
(
2
L
2
(
|
|≤
C
1
ϕ
H
ρ
(
R
d
)
φ
H
ρ
(
R
d
)
,
≥
C
2
ϕ
d
)
−
C
3
ϕ
d
)
.
R
R
In particular
,
for every u
0
∈
L
2
(
R
d
) there exists a unique solution to the problem
(
14.20
).
For multidimensional payoffs
g
which satisfy the growth condition (8.10), we
can localized to a bounded domain
G
=
(
−
R,R)
d
d
,
R>
0 which again cor-
responds to approximating the option price by a knock-out barrier option. Simi-
lar to Theorem 8.3.1 and Theorem 10.5.1, we obtain that the barrier option price
converges to the option price exponentially fast in
R
if the Lévy measure satisfies
(
14.15
). Therefore, we have the localized problem
⊂ R
L
2
(J
;
H
ρ
(G))
H
1
(J
L
2
(G))
such that
Find
u
R
∈
∩
;
a
J
(u
R
,v)
∈
H
ρ
(G) ,
a.e. in
J,
(∂
t
u
R
,v)
+
=
0
,
∀
v
(14.21)
u
R
(
0
)
=
u
0
|
G
,
which as a unique solution for every payoff
g
satisfying the growth condition (8.10).
14.6 Wavelet Discretization
Since the diffusion part has been already discussed in Sect. 8.4,weset
0 and
only consider pure jump models. The main problem is as in the one-dimensional
case the singularity of the Lévy measure at the origin
z
Q
=
=
0 and additionally on
the axis
z
i
=
1
,...,d
. Therefore, we integrate by parts twice the integro-
differential expression for the jump generator as in Lemma
14.2.7
to obtain for
φ,ϕ
∈
H
1
(G)
0,
i
=
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