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≤
s
i
≤
−
=
for 0
p
1
/
2,
i
1
,...,d
. Similarly, due to the intersection structure (
13.4
),
we obtain
∞
2
(
2
2
s
1
1
2
2
s
d
d
)
2
2
u
H
s
(G)
+···+
|
u
,
k
|
u
H
s
(G)
,
(13.8)
i
=
0
k
i
∈∇
i
for 0
1
,...,d
.
We study the approximation property of the sparse tensor product space
V
L
for
functions
u
≤
s
i
≤
p
−
1
/
2,
i
=
s
(G)
. To this end, we define the sparse projection operator
P
L
:
∈
H
→
V
L
by truncating the wavelet expansion (
13.6
)
P
L
u
:=
L
2
(G)
u
,
k
ψ
,
k
.
|
|
1
≤
L
k
i
∈∇
i
d
For a multi-index
α
∈ R
>
0
, denote by
α
∗
:=
min
{
α
i
}
. The next result is taken from
[133].
Theorem 13.1.2
Assume
0
≤
s
i
≤
p
−
1
/
2
and s
i
<t
i
≤
p
,
i
=
1
,...,d
.
Then
,
for
∈
H
s
(G) there holds
u
C
2
−
(
t
−
s
)
∗
L
H
=
0
or t
i
=
∀
u
if
s
p,
i,
t
(G)
−
P
L
u
u
H
s
(G)
≤
C
2
−
(
t
−
s
)
∗
L
L
(d
−
1
)/
2
u
H
otherwise.
t
(G)
We can also state the approximation rate in terms of the dimension of the sparse
tensor product space
N
L
=
O
(
2
L
L
d
−
1
)
. For example, if
u
p
(G)
, we obtain
∈
H
−
P
L
u
CN
−
p
L
(
log
2
N
L
)
(p
+
1
/
2
)(d
−
1
)
+
ε
ε>
0
.
Hence, for the wavelets of polynomial degree 1 with approximation order
p
u
L
2
(G)
≤
u
H
p
(G)
,
∀
=
2as
in Example 12.1.1, the approximation rate is, up to logarithmic terms, the same as in
one dimension, compare with the finite element interpolation estimate (3.36), where
with
h
(N
−
1
)
we have
CN
−
2
=
O
u
−
I
N
u
L
2
(G)
≤
u
H
2
(G)
. Thus, the curse of
CN
−
p/d
L
dimension
u
−
P
L
u
L
2
(G)
≤
u
H
p
(G)
(see also (8.24)) of the full tensor
product space
V
L
can be avoided by the sparse tensor product space.
In the next section, we use the sparse tensor product space
V
L
to discretize the
weak formulation of the pricing equation for multi-asset options in a BS market.
13.2 Sparse Wavelet Discretization
Recall the weak formulation of the localized pricing problem of multi-asset options
(8.13). Its space semi-discretization using the sparse tensor product space
V
L
⊂
H
0
(G)
reads
;
V
L
)
;
V
L
)
such that
L
2
(J
H
1
(J
Find
u
L
∈
∩
∈
V
L
,
a.e. in
J,
a
BS
(
(∂
t
u
L
,v)
+
u
L
,v)
=
0
,
∀
v
(13.9)
=
u
L
(
0
)
u
L,
0
.
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