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Using the boundedness of the derivatives of
f
, the linear growth of the coefficient
Σ
(
8.3
) and proceeding as in the proof of Proposition 4.1.1, we find that the process
0
i
=
1
j
=
1
∂
x
i
f(X
τ
)Σ
i,j
(X
τ
)
d
W
τ
, is a martingale with respect to the filtration
of
W
.
Repeating the arguments which lead to Theorem 4.1.4 yields
Let V
∈
C
1
,
2
(J
× R
d
)
∩
C
0
(J
× R
d
) with bounded derivatives in
Theorem 8.1.3
x be a solution of
d
,V T,x)
d
,
∂
t
V
+
A
V
−
rV
=
0
in J
× R
=
g(x)
in
R
(8.5)
with
A
as in
(
8.4
).
Then
,
V(t,x)can also be represented as
= E
e
−
t
r(X
s
)
d
s
g(X
T
)
x
.
V(t,x)
|
X
t
=
We now consider the pricing of a multi-asset option in a Black-Scholes market
model, i.e. we assume in (
8.1
) that
b
i
(s)
=
rs
i
,
Σ
ij
(s)
=
Σ
ij
s
i
,1
≤
i, j
≤
d
, with
d
r
≥
0 and
Σ
ij
≥
0 constants. Furthermore, we denote by
μ
∈ R
the column vector
r)
. Switching to log-price
x
i
=
given by
μ
:=
(
Q
11
/
2
−
r,...,
Q
dd
/
2
−
log
(s
i
)
,
i
=
1
...,d
, and to time-to-maturity
t
→
T
−
t
, we obtain that
v(t,x
1
,...,x
d
)
:=
t,e
x
1
,...,e
x
d
)
solves
V(T
−
BS
v
d
,
0
,x)
g(e
x
)
d
.
∂
t
v
−
A
+
rv
=
0in
J
× R
=
in
R
(8.6)
Herewith, by a slight abuse of notation, we let
g(e
x
)
g(e
x
1
,...,e
x
d
)
, and
BS
:=
A
denotes the differential operator with constant coefficients
1
2
tr
BS
f )(x)
D
2
f(x)
μ
∇
(
A
:=
[
Q
]−
f(x).
8.2 Variational Formulation
The variational formulation of (
8.6
) will require Sobolev spaces for functions of
several variables. For
m
∈ N
and a bounded Lipschitz and simply connected domain
d
, it is sufficient to introduce
H
m
(G)
as
H
m
(G)
:=
u
∈
L
2
(G)
:
D
n
u
∈
L
2
(G)
for
⊆ R
G
|≤
m
.
(8.7)
Here,
D
n
u
has to be understood in the weak sense, i.e.
D
n
u
is the weak derivative of
|
n
u
satisfying
G
D
n
uϕ
d
x
1
)
|
n
|
G
uD
n
ϕ
d
x
,
C
0
(G)
. The space
H
m
(G)
is
=
(
−
∀
ϕ
∈
equipped with the norm
2
m
D
n
u
2
u
H
m
(G)
:=
L
2
(G)
,
|
n
|≤
H
1
(G)
=
G
|
+
G
|∇
2
2
d
x
2
d
x
.For
G
d
in particular,
u
u
|
u
|
⊂ R
as above, we intro-
duce spaces
H
0
(G)
consisting of trace-zero functions,
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