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is the solution of
d
, T,x)
d
,
∂
t
v
+
A
v
=
0 n
J
× R
=
g(x)
in
R
(11.1)
where
A
denotes the infinitesimal generator of
X
. We consider processes
X
where
A
is given by
1
2
tr
[
Q
(x)D
2
f(x)
]+
b(x)
∇
f(x)
+
c(x)f(x)
(
A
f )(x)
=
f(x
f(x)
ν(
d
z),
z
∇
+
+
z)
−
f(x)
−
(11.2)
d
R
d
d
×
d
,
b
: R
d
d
,
c
: R
d
d
where
Q
: R
→ R
→ R
→ R
and
ν
a Lévy measure in
R
satisfying
R
and
2
d
min
{
1
,
|
z
|
}
ν(
d
z) <
∞
>
1
|
z
i
|
ν(
d
z) <
∞
,
i
=
1
,...,d
.
|
z
|
Definition 11.1.1
We call a process
X
a parametric Markovian market model with
admissible parameter set
S
η
,if
(i) For all
η
∈
S
η
X
is a strong Markov process with respect to
(Ω,
F
,
F
,
P
)
,
(ii) The infinitesimal generator
A
of the semigroup generated by
X
has the form
S
η
→{
Q
}
(
11.2
), and the mapping
η
,b,c,ν
is infinitely differentiable.
Examples of Markov processes
X
and their infinitesimal generators are given
by the one-dimensional diffusion (4.2), the multidimensional diffusion (8.4), the
general stochastic volatility model (9.27) and the one-dimensional Lévy pro-
cess (10.14).
We calculate the sensitivities of the solution
v
of (
11.1
) with respect to parame-
ters in the infinitesimal generator
A
and with respect to solution arguments
x
and
t
.
We write
A
(η
0
)
for a fixed parameter
η
0
∈
S
η
to emphasize the dependence of
A
on
η
0
and change the time to time-to-maturity
t
t
. For sensitivity computa-
tion, it will be crucial below to admit a non-trivial right hand side. Accordingly, we
consider the parabolic problem
→
T
−
d
,
0
,x)
d
,
∂
t
u
−
A
(η
0
)u
=
f
in
J
× R
=
u
0
in
R
(11.3)
with
u
0
=
g
. For the numerical implementation, we truncate (
11.3
) to a bounded
d
and impose boundary conditions on
∂G
. Typically,
G
is
d
-
dimensional hypercube, i.e.
G
domain
G
⊂ R
=
k
=
1
(a
k
,b
k
)
for some
a
k
,b
k
∈ R
,
b
k
>a
k
,
k
1
,...,d
, as shown in the localization Theorems 4.3.1, 8.3.1, 9.4.1 and 10.5.1.
With a parametric Markovian market model
X
in the sense of Definition
11.1.1
with parameter set
=
S
η
and infinitesimal generator
A
(η
0
)
as in (
11.2
),
η
0
∈
S
η
,we
associate to
A
(η
0
)
the Dirichlet form
a(η
0
;·
,
·
)
:
V
×
V
→ R
via
a(η
0
;
u, v)
:= −
A
(η
0
)u, v
V
∗
,
V
, ,v
∈
V
,
where we consider the abstract setting as given in Sect. 3.2 with Hilbert spaces
V
⊂
H
≡
H
∗
⊂
V
∗
.
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