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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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