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the compact Sobolev embedding
H
m
(G)
C
0
(G)
only holds if
m>d/
2. We as-
⊂
sume that the operator
P
N
satisfies the following approximation property (compare
with (3.36))
Ch
s
−
t
u
−
P
N
u
H
t
(G)
≤
u
H
s
(G)
,t
=
0
,
1
,
0
≤
t
≤
s
≤
2
.
(8.23)
The construction of
P
N
having the property (
8.23
) for a large class of finite element
spaces including the tensor product space
V
N
in (
8.19
) can be found, e.g. in [27,
Sect. 4.8] or [64, Sect. 1.6]. Thus, replacing in the proof of Theorem 3.6.5 the nodal
interpolant
I
N
by
P
N
and using the approximation property (
8.23
), we find
C
1
(J
H
2
(G))
C
3
(J
H
−
1
(G))
.
Let u
m
(x)
Theorem 8.4.3
Assume u
R
∈
;
∩
;
:=
u
R
(t
m
,x) and u
N
:=
u
N
(t
m
,x)
∈
V
N
,
defined via
(
8.21
),
with V
N
as in
(
8.19
),
θ<
2
where N
1
= ··· =
N
d
.
Assume for
0
≤
also
(3.30).
Then
,
the following er-
ror bound holds
:
M
−
1
u
M
u
N
2
u
m
+
θ
u
m
+
θ
N
2
H
1
(G)
−
L
2
(G)
+
k
0
−
m
=
Ch
2
T
Ch
2
2
≤
T
H
2
(G)
+
max
0
u(t)
∂
t
u(s)
H
1
(G)
d
s
≤
t
≤
0
C
k
2
0
2
∗
∂
tt
u(s)
d
s
if
0
≤
θ
≤
1
,
+
k
4
0
2
∗
1
=
∂
ttt
u(s)
d
s
if θ
2
.
u
M
As in the univariate case, using
k
=
O
(h)
and
θ
=
1
/
2, one can show that
−
u
N
2
(h
2
)
. Since
n
(h
−
1
)
, the dimension of the
L
2
(G)
=
O
:=
N
1
= ··· =
N
d
=
O
n
d
, and the rate of convergence with
respect to the mesh width
h
can be expressed in terms of the number of degrees of
freedom
N
finite element space
V
N
is
N
:=
dim
V
N
=
u
M
−
u
N
2
L
2
(G)
=
O
(N
−
2
/d
).
(8.24)
The exponential decay of the rate of convergence with respect the dimension
d
of
the problem, or equivalently, the exponential growth of
N
with respect to
d
is called
the
curse of dimension
. Therefore, full tensor product spaces
V
N
lead to inefficient
schemes for large
d
.
Example 8.4.4
For
d
=
2 we consider two different options: A basket option with
payoff
g(s)
=
(α
1
s
1
+
α
2
s
2
−
K)
+
, and a better-of-option with payoff
g(s)
=
{
s
1
,s
2
}
=
=
Q
=
max
.Wesetstrike
K
100, and maturity
T
1, covariance matrix
(σ
i
σ
j
ρ
ij
)
1
≤
i,j
≤
2
with volatilities
σ
=
(
0
.
4
,
0
.
1
)
and correlation
ρ
12
=
0
.
2. Further-
more, we let the interest rate
r
=
0
.
3 in the payoff of the
basket option. The option values are shown in Fig.
8.1
where we used finite elements
for the discretization.
Using analytic formulas (see, e.g. [165]), we can compute the
L
∞
-convergence
rate on a subset
G
0
=
0
.
01 and
α
1
=
0
.
7,
α
2
=
(K/
2
,
3
/
2
K)
2
T
. The convergence rates
are shown in Fig.
8.2
. It can be seen that we obtain the optimal convergence rate
⊂
G
at maturity
t
=
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