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C
0
(G)
·
H
m
(G)
.
We are now able to give the weak formulation of Eq. (
8.6
). It reads
H
0
(G)
:=
L
2
(J
H
1
(
d
))
H
1
(J
L
2
(
d
))
such that
Find
u
∈
;
R
∩
;
R
a
BS
(u, v)
H
1
(
d
),
a.e. in
J,
(∂
t
u, v)
+
=
0
,
∀
v
∈
R
(8.8)
u(
0
)
=
u
0
,
g(e
x
)
and the bilinear form
a
BS
(
H
1
(
d
)
H
1
(
d
)
where
u
0
(x)
:=
·
,
·
)
:
R
×
R
→ R
is
given by
r
1
2
a
BS
(ϕ, φ)
ϕ)
Q
∇
μ
∇
=
(
∇
φ
d
x
+
ϕφ
d
x
+
ϕφ
d
x.
(8.9)
d
d
d
R
R
R
Proposition 8.2.1
Assume the symmetric matrix
Q
is positive definite
,
i
.
e
.
there ex-
ists a constant γ>
0
such that x
Q
γx
x
,
d
.
Then there exist constants
x
≥
∀
x
∈ R
1
,
2
,
3,
such that for all ϕ,φ
∈
H
1
(
R
d
) the following holds
:
C
i
>
0,
i
=
a
BS
(ϕ, φ)
|
|≤
C
1
ϕ
H
1
(
R
d
)
φ
H
1
(
R
d
)
,
a
BS
(ϕ, ϕ)
2
H
1
(
2
L
2
(
≥
C
2
ϕ
d
)
−
C
3
ϕ
d
)
.
R
R
Proof
By Hölder's inequality,
d
d
1
2
a
BS
(ϕ, φ)
|
|≤
1
|
Q
ij
|
d
|∇
||∇
|
+
1
|
μ
i
|
d
|∇
||
|
ϕ
φ
d
x
ϕ
φ
d
x
R
R
i,j
=
i
=
r
+
d
|
ϕ
||
φ
|
d
x
≤
C
1
ϕ
H
1
(
R
d
)
φ
H
1
(
R
d
)
.
R
and
R
2
1
d
∂
x
i
(ϕ
2
)
d
x
Furthermore, by the positive definiteness of
Q
d
∂
x
i
ϕϕ
d
x
=
R
=
0,
2
γ
r
1
a
BS
(ϕ, ϕ)
2
d
x
2
d
x
≥
d
|∇
ϕ
|
+
d
|
ϕ
|
R
R
1
2
γ
2
H
1
(
2
L
2
(
≥
ϕ
d
)
−|
r
−
γ/
2
|
ϕ
d
)
.
R
R
H
1
(
d
)
,
We apply the abstract existence result Theorem 3.2.2 in the spaces
V
=
R
d
)
, a unique solution to the prob-
lem (
8.8
). As in the one-dimensional case described in Sect. 4.3, we need to re-
formulate the problem on a bounded domain where the condition
u
0
(x)
L
2
(
d
)
and obtain, for every
u
0
∈
L
2
(
H
=
R
R
g(e
x
)
=
∈
L
2
(
d
)
can be weakened. We require similar to (4.10) the following polynomial
growth condition on the multidimensional payoff function: There exist
C>
0,
q
≥
R
1
such that
C
1
q
d
d
g(s
1
,...,s
d
)
≤
s
i
+
for all
s
∈ R
≥
0
.
(8.10)
i
=
1
This condition is satisfied by all standard multi-asset options like, e.g. basket, rain-
bow, spread or power options.
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