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≤
≤
where
a
i,j
(x)
,1
i, j
d
are continuously differentiable functions such that
a
i,j
(x)
=
a
j,i
(x)
and
d
2
2
κ
1
|
ξ
|
≤
a
i,j
(x)ξ
i
ξ
j
≤
κ
2
|
ξ
|
i,j
=
1
d
d
holds for all
x
∈ R
and
ξ
∈ R
for some constants 0
<κ
1
≤
κ
2
. Besides, we assume
d
∂a
i,j
∂x
i
=
0
,
i
=
1
for any
j
=
1
,.
.
.,d
and
c(x)
is a continuous and bounded function satisfying
0
<
c
≤
c(x)
≤
c<
∞
. In this situation, we obtain the following representation
f
of
A
=
f(
A
)
for
u
∈
D(
A
)
,
A
=
L(x, D)
.
∞
f(
A
)u
=
f (L(x, ξ ))u
+
λR
λ
K
λ
(x, D)uμ(
d
λ),
(16.18)
0
where
K
λ
(x, D)
=
(L(x, D)
+
λ
id
)
◦
q
λ
(x, D)
−
id
,
1
L(x, ξ )
+
λξ
,
q
λ
(x, ξ )
=
d
L(x, ξ )
=
a
i,j
ξ
i
ξ
j
+
c(x)
i,j
=
1
and
R
λ
denotes the resolvent of
0 for constant sym-
bols, which proves Theorem
16.3.8
. In general, both terms in (
16.18
)havetobe
considered. We refer to Carr [34] for a generalization of the VG model.
A
at
λ
. We remark that
K
λ
≡
Remark 16.3.10
The consideration of symbols of the type
a(x,ξ)
, where
a(x,ξ)
is
a Lévy symbol for all
x
∈ R
is therefore in general not equivalent to a construction
f
is
not given as
f(ψ(x,ξ))
. It has a more involved structure as described above. This
observation was made by [7], where some asymptotic expansion of the difference in
terms of the symbol under certain assumptions on the structure of the process was
provided.
via subordination, i.e. for
A
=
L(x, D)
with symbol
ψ(x,ξ)
the symbol of
A
16.4 Admissible Market Models
We now formulate the requirements for market models which are admissible for our
pricing schemes in terms of the marginals and the copula function. These require-
ments not only ensure existence and uniqueness of a solution of the corresponding
pricing problem, but also ensure that the presented FEM based algorithms are fea-
sible.
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