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14.6.2 Fully Discrete Scheme
We can again use the norm equivalences (13.8) to precondition our linear systems.
Denote by
D
the diagonal matrix with entries 2
α
1
1
2
α
d
d
+···+
for an index corre-
sponding to level
=
(
1
,...,
d
)
. Then, (13.8)for
s
i
=
α
i
/
2,
i
=
1
,...,d
implies
that
u
,
A
J
u
2
∼
u
H
α
/
2
(G)
∼
u
,
D
u
,
A
J
D
−
1
/
2
)
is bounded, independent
and we obtain that the condition number
κ(
D
−
1
/
2
of the level
L
.
Using the
hp
-dG timestepping method (see Sect. 12.3) for the time discretization,
we have the (perturbed) fully discrete scheme
1
)N
L
Find
u
m
(r
m
+
∈ R
such that for
m
=
1
,...,M,
C
m
⊗
A
J
u
m
k
2
I
m
(
ϕ
m
M
)
u
m
−
1
,
⊗
M
+
=
⊗
(14.27)
u
0
=
u
0
.
The linear systems can again be decoupled and preconditioned using
D
.Wehave
the following extension to the one-dimensional result Theorem 12.3.4.
Theorem 14.6.3
Let X be a Lévy process with characteristic triplet (
Q
,ν,γ)where
Q
>
0
and Lévy density k satisfies Assumption
14.3.4
and
(
14.23
).
Moreover
,
let
max
i
=
1
{
a
−
2
(p
+
α/
2
)
i
a
−
(p
+
α)
i
+
}
be sufficiently small and assume that the payoff
|
G
∈
H
s
(G) for some
0
<s
g
≤
1.
Choose M
=
r
=
O
(L) and use in each time step
(L
5
) GMRES iterations
.
Then
,
the fully discrete Galerkin scheme with incomplete
GMRES gives
O
p
−
1
U
dG
(T )
C N
−
s
L
(
log
2
N
L
)
(d
−
1
)s
+
ε
,s
1
,
where C>
0
is a constant independent of h and U
dG
denotes the
(
perturbed
)
hp -dG
approximation
.
u(T )
−
L
2
(G)
≤
:=
p
−
1
+
dp
−
We give a numerical example.
Example 14.6.4
Let
d
=
2 and consider two independent variance gamma pro-
cesses [118] with parameter
σ
=
0
.
3,
ϑ
=
0
.
25 and
θ
=−
0
.
3. We set the com-
pression parameter
a
i
=
8, the absolute
value of the entries in the stiffness matrix
A
J
and the compressed matrix
A
J
are
shown in Fig.
14.6
. As in Example 12.3.5, large entries are colored red. One again
clearly sees that the compression scheme neglects small entries.
For a geometric basket option with maturity
T
a
i
=
1,
i
=
1
,...,
2,
p
=
2,
p
=
2. For
L
=
=
1 and strike
K
=
1, we compute
in Fig.
14.7
the
L
∞
-error at maturity
t
(K/
2
,
3
/
2
K)
2
.In
the discretization, we use the sparse tensor wavelet basis and the (perturbed)
hp
-dG
time stepping with
M
=
O
(L)
graded time steps. As in Sect. 13.5.1,wealsosolved
the problem on the full grid to illustrate the “curse of dimension”.
=
T
on the subset
G
0
=
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