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S
2
m
(x)
ρ,δ
(m
1
(x
1
),...,m
d
(x
d
)) and symbol ψ(x,ξ)
∈
for some
0
<δ<ρ
≤
1
for
which there exists C>
0
with
2
m
(x)
d
.
ψ(x,ξ)
+
1
≥
C
ξ
∀
x,ξ
∈ R
(16.30)
Ψ
2
m
(x)
Then
,
ρ,δ
satisfies a Gårding inequality in the variable order space
H
m
(x)
(G)
:
There are constants C
1
>
0
and C
2
≥
A
(x, D)
∈
0
such that
∈
H
m
(x)
(G)
2
2
∀
:
≥
C
1
H
m
(x)
(G)
−
C
2
u
a(u, u)
u
u
L
2
(G)
,
(16.31)
and
:
H
m
(x)
(G)
H
−
m
(x)
(G)
∃
λ>
0
such that
A
(x, D)
+
λI
→
(16.32)
∈
H
m
(x)
(G)
.
is boundedly invertible
,
for a(u,v)
=
A
u, v
H
−
m
(x)
(G), H
m
(x)
(G)
,
u, v
Proof
The proof follows along the lines of the proof of [137, Theorem 5], where
the case
d
=
1 was treated.
Note that in the case of an admissible time-homogeneous market model,
a
G
(u, u)
=
a
G
(u, u)
holds for
u
∈
V
G
and
a
G
(
·
,
·
)
as in (
16.28
).
Theorem 16.5.6
The problem
(
16.28
)
-
(
16.29
)
for an admissible time-homogeneous
market model X with symbol ψ(x,ξ) with initial condition g
∈
H
=
L
2
(
R
d
) and
Y
≥
1,
Q
=
0
or Q
≥
Q
0
>
0
has a unique solution
.
Proof
We obtain from Lemma
16.4.3
S
Y
(x)
1
,δ
ψ(x,ξ)
∈
for
Y
≥
1
,Q
=
0
,
ψ(x,ξ)
∈
S
1
,δ
for
Q
≥
Q
0
>
0
.
Theorem
16.5.1
implies
Y
(x)
ψ(x,ξ)
+
1
≥
C
ξ
for
Y
≥
1
,Q
=
0
,
(16.33)
2
ψ(x,ξ)
+
1
≥
C
ξ
for
Q
≥
Q
0
>
0
,
(16.34)
d
. An application of Theorem
16.5.5
implies the claimed result.
for all
x,ξ
∈ R
16.6 Numerical Examples
In this section the implementation of numerical solution methods for the Kol-
mogorov equations for admissible market models with inhomogeneous jump mea-
sures using the techniques described above is discussed. We assume the risk-neutral
dynamics of the underlying asset to be given by
S(t)
=
S(
0
)e
rt
+
X(t)
,
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