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Assumption 16.1.5
The characteristic triplet
(b(x), Q(x), ν(x,
d
z))
of a Feller pro-
cess in
d
R
satisfies the following conditions:
d
.
(I)
(b(x), Q(x), ν(x,
d
z))
is a Lévy triplet for all fixed
x
∈ R
→
B
∩R
2
)ν(x,
d
z)
is continuous for all
B
d
)
.
∧|
|
∈
B
R
(II) The mapping
x
(
1
z
(
d
\{
0
}
(III) There exists a Lévy measure
ν(z)
s.t.
2
)ν(x,
d
z)
≤
2
)ν(
d
z) <
∞
,
0
≤
(
1
∧ |
z
|
(
1
∧ |
z
|
B
∩R
d
\{
0
}
B
∩R
d
\{
0
}
d
)
.
(IV) The functions
x
→
b(x)
and
x
→
Q(x)
are continuous and bounded.
d
,B
∈ R
∈
B
R
for all
x
(
Our aim is to conclude that there exists a Feller process whose generator is a
PDO for a symbol that satisfies Assumption
16.1.5
. Therefore, it suffices to verify
the assumptions of Theorem
16.1.3
.
Lemma 16.1.6
Let (b(x), Q(x), ν(x,
d
z)) be the characteristic triplet of a process
X taking values in
d
ψ(x,D),C
0
d
))
R
−
R
that satisfies Assumption
16.1.5
.
Then
,
(
(
extends to a Feller generator
,
where ψ(x,ξ) is given by
1
2
ξ,Q(x)ξ
ψ(x,ξ)
=−
i
b(x), ξ
+
1
ν(x,
d
z).
i
z, ξ
e
i
z,ξ
+
+
−
(16.7)
2
1
+|
z
|
d
R
\{
0
}
Proof
Condition (I) of Assumption
16.1.5
implies that the corresponding Feller
symbol is negative definite. Conditions (III) and (IV) imply (a) of
16.1.3
, Condi-
tions (II) and (III) imply (b), and (c) follows from (II) and (IV).
Remark 16.1.7
Note that real price market models, as well as Ornstein-Uhlenbeck
models do not fit into our modelling framework due to Assumption (a) in Theo-
rem
16.1.3
, as they do not admit a uniform estimate in the state space variable.
The numerical methods presented in the following can in many cases be straightfor-
wardly extended to this kind of models.
In order to apply available tools from pseudodifferential calculus, we need to im-
pose stronger assumptions on the characteristic triplets of the considered processes.
We state the assumptions needed at the end of Sect.
16.4
. In particular, smoothness
of the characteristic triplet in the state variable
x
. Numerical experiments indicate
strongly that these assumptions can be weakened.
16.2 Variable Order Sobolev Spaces
For later use, we shall introduce anisotropic and variable order Sobolov spaces. We
start with the definition of fractional order isotropic spaces and define for a positive
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