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a
C
(t
a
J
(t
|
;
|≤|
;
|+|
;
|
a(t
ϕ,φ)
ϕ,φ)
ϕ,φ)
≤
C
1
ϕ
W
φ
W
+
C
4
ϕ
H
α/
2
(G
R
)
φ
H
α/
2
(G
R
)
≤
2max
{
C
4
,C
5
}
ϕ
V
φ
V
.
We deduce that the weak formulation to (
15.18
)
L
2
(J
H
1
(J
L
2
(G
R
))
such that
Find
w
∈
;
V)
∩
;
(15.19)
(∂
t
w,v)
+
a(t
;
w,v)
=
0
,
∀
v
∈
V,
a.e. in
J,
g(e
x
)
=
w(
0
)
L
2
(G
R
)
.
∈
admits a unique solution for every
g
15.4 Wavelet Discretization
As in the previous chapters, the discretization of the weak formulation (9.29)ofthe
general pure diffusion SV model, (
15.17
) of the jump-diffusion SV model of Bates
or (
15.19
) of the BNS model is based on the sparse tensor product space
V
L
and the
hp
-dG time stepping scheme.
In order to describe the stiffness matrix
A
for these SV models, we introduce
as in Chap. 9, weighted matrices
M
w(x
i
)
,
B
w(x
i
)
and
S
w(x
i
)
. In contrast to Chap. 9,
however, we need to define them with respect to the wavelet basis
{
ψ
,k
}
, compare
with (13.12)-(13.14),
b
i
ψ
i
,k
i
(x
i
)ψ
i
,k
i
(x
i
)w(x
i
)
d
x
i
0
≤
i
,
i
≤
L
k
i
∈∇
i
,k
i
∈∇
i
M
w(x
i
)
:=
,
(15.20)
a
i
b
i
ψ
i
,k
i
(x
i
)ψ
i
,k
i
(x
i
)w(x
i
)
d
x
i
0
≤
i
,
i
≤
L
k
i
∈∇
i
,k
i
∈∇
i
S
w(x
i
)
:=
,
(15.21)
a
i
b
i
ψ
i
,k
i
(x
i
)ψ
i
,k
i
(x
i
)w(x
i
)
d
x
i
0
≤
i
,
i
≤
L
k
i
∈∇
i
,k
i
∈∇
i
B
w(x
i
)
:=
.
(15.22)
a
i
B
κ
As an example, consider the (transformed) Bates model with operator
A
as in
(
15.15
). Then, the corresponding stiffness matrix
A
κ
is given by
A
κ
=
A
κ
B
1
M
1
−
λ
0
A
J
M
1
,
⊗
⊗
+
λ
0
κ
where
A
κ
) and
A
J
has to be replaced by
⊗
is as in (9.32) (where
⊗
is the stiff-
ness matrix to the jump operator
f (x))ν
0
(
d
ζ)
. Note that
A
J
(f (x
+
ζ)
−
can be
R
implemented and (wavelet-) compressed as described in Chap. 12.
Applying the
hp
-dG time stepping to the semi-discrete problem leads then after
decoupling, as explained in the previous chapters, to linear systems of the form
λ
M
w
k/
2
A
=:
+
=
s
,
(15.23)
B
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