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In-Depth Information
d
a
J
(ϕ, φ)
=
∂
i
ϕ(x
+
z
i
)∂
i
φ(x)k
−
2
(z
i
)
d
x
d
z
i
i
R
G
i
=
1
d
∂
I
ϕ(x
z
I
)φ(x)U
I
(z
I
)
d
x
d
z
I
.
−
+
(14.22)
i
R
G
i
=
2
|
I
|=
i
I
1
<
···
<
I
i
Using the spline-wavelet basis
ψ
,
k
=
L
,
k
i
∈
∇
i
of
V
L
(see Sect. 13.1) of the Finite Element space, we need to compute the
stiffness matrix
ψ
1
,k
1
···
ψ
d
,k
d
,0
≤
1
+···+
d
≤
d
A
J
∂
i
ψ
,
k
(x
+
z
i
)∂
i
ψ
,
k
(x)k
−
2
(
,
k
),(
,
k
)
=
(z
i
)
d
x
d
z
i
i
R
G
i
=
1
d
∂
I
ψ
,
k
(x
+
z
I
)ψ
,
k
(x)U
I
(z
I
)
d
x
d
z
I
.
−
R
i
G
i
=
2
|
I
|=
i
I
1
<
···
<
I
i
We define the one-dimensional mass matrix
M
i
as in (13.12) and additionally
R
R
A
(
,k
),(,k)
:=
ψ
,k
(y)ψ
,k
(x)k
−
2
(y
−
x)
d
y
d
x,
i
−
R
−
R
A
(
I
,
k
I
),(
I
,
k
I
)
:= −
∂
I
ψ
I
,
k
I
(y)ψ
I
,
k
I
(x)U
I
(y
−
x)
d
y
d
x,
[−
R,R
]
|
I
|
[−
R,R
]
|
I
|
|
I
|
where
I
=
(
i
)
i
∈
I
,0
≤
i
≤
L
,
k
I
=
(k
i
)
i
∈
I
,
k
i
∈∇
i
,
I
⊂{
1
,...,d
}
,
>
1.
Then, we can write the jump stiffness matrix as
d
A
(
I
,
k
I
),(
I
,
k
I
)
j
M
j
A
(
,
k
),(
,
k
)
=
(
j
,k
j
),(
j
,k
j
)
.
i
=
|
I
|=
i
I
1
<
···
<
I
i
∈
I
c
1
As in the diffusion case, we can then compute the jump stiffness matrix
A
J
as a
sparse tensor product using the matrices
A
I
and
M
j
.
Since the Black-Scholes operator is a local operator, there are only
(
2
L
L
d
−
1
)
non-zero entries in
A
BS
. But as already discussed in the one-dimensional case, the
stiffness matrix for the jump part is densely populated. Again using wavelet com-
pression, we can reduce the number of non-zero entries in
A
J
to
O
O
(
2
L
L
2
(d
−
1
)
)
.We
also need a smoothness assumption on the Lévy density
k
d
∂
n
k(z)
C
0
C
|
n
|
|
−
α
∞
z
i
|
−
n
i
−
1
,
|
|≤
n
|!
z
1
|
∀
z
i
=
0
,
(14.23)
i
=
d
for
C
0
,C >
0,
α
=
α
∞
and a multi-index
n
=
(n
1
,...,n
d
)
∈ N
0
.
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