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In the following lemma, we characterize the symbol classes of admissible market
model. This is crucial for the well-posedness of the pricing equation as discussed in
Sect.
16.5
.
Lemma 16.4.3
The symbol ψ(x,ξ) of a time-homogeneous admissible market
model given as
1
2
ξ,Q(x)ξ
ψ(x,ξ)
=−
i
b(x), ξ
+
1
ν(x,
d
z)
e
i
z,ξ
+
+
−
i
z, ξ
d
0
=
z
∈R
is contained in the following symbol classes
:
⎧
⎨
ψ(x,ξ)
∈
S
1
,δ
for Q(x)
≥
Q
0
>
0
,
S
Y
(x)
1
,δ
ψ(x,ξ)
∈
for Q
=
0
,γ
=
0
,
⎩
ψ(x,ξ)
∈
S
2
m
(x)
1
,δ
for Q
=
0
,γ
=
0
,
max
(Y
i
(x
i
),
1
)
2
where δ
∈
(
0
,
1
) and
m
i
(x
i
)
=
,
i
=
1
,...,d
.
Proof
We have, analogous to [134, Proposition 3.5],
1
ν(x,
d
z)
≤
C
1
d
d
−
e
i
z,ξ
+
i
z, ξ
Y
i
(x
i
)
,
∀
ξ
∈ R
:
1
|
ξ
i
|
R
d
i
=
for some positive constant
C
1
,C
2
,C
3
>
0. The following estimate holds for the
diffusion and the drift component:
≤
d
d
1
2
d
2
∀
ξ
∈ R
:
ξ,Q(x)ξ
C
2
1
|
ξ
i
|
and
|
i
b(x), ξ
| ≤
C
3
1
|
ξ
i
|
.
i
=
i
=
Remark 16.4.4
The partially degenerate case
Q
=
0, but
Q
≯
0 can be analyzed as
in [134, Remark 4.9]. Note that in the case
γ
0 additional assumptions
on the behavior of
Y
i
(x
i
)
at 1 are necessary in order to ensure the smoothness of
=
0 and
Q
=
=
m
i
(x
i
)
,
i
1
,...,d
.
The infinitesimal generator
A
of a time-homogeneous admissible market model
X
reads
A
ϕ(x)
=
A
Tr
ϕ(x)
+
A
BS
ϕ(x)
+
A
J
ϕ(x),
A
Tr
ϕ(x)
b(x)
∇
=
ϕ(x),
2
tr
Q
(x)D
2
ϕ(x)
,
1
(16.20)
A
BS
ϕ(x)
=
ϕ(x
ϕ(x)
ν(x,
d
z),
z
∇
A
J
ϕ(x)
=
+
z)
−
ϕ(x)
−
d
R
for
ϕ
∈
C
∞
(
R
d
)
.
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