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n
-valued Brownian
motion. We assume that the
i
th component of the process
X
evolves according to
R
We describe the process
X
in more detail. Let
W
be an
n
Σ
ij
(X
t
)
d
W
t
,X
0
=
d
X
t
=
Z
i
,i
b
i
(X
t
)
d
t
+
=
1
,...,d.
(8.1)
j
=
1
n
satisfy the
usual Lipschitz continuity and linear growth condition, i.e. there exists a constant
C>
0 such that for all
x,y
d
d
,
Σ
d
d
×
: R
→ R
: R
→ R
Herewith, we assume the coefficients
b
d
∈ R
|
b(x)
−
b(y)
|+|
Σ(x)
−
Σ(y)
|≤
C
|
x
−
y
|
,
(8.2)
|
b(x)
|+|
Σ(x)
|≤
C(
1
+|
x
|
),
(8.3)
1
≤
i
≤
m,
1
≤
j
≤
n
|
m
×
n
2
where by
|·|:R
→ R
≥
0
,
A
→ |
A
|:=|
A
|
F
=
A
ij
|
we de-
note the Frobenius norm (Euclidean norm) of an
m
×
n
-matrix
A
. We further assume
(Z
1
,...,Z
d
)
∈ R
d
the existence of a random variable
Z
=
which is independent
2
of the
σ
-algebra generated by
W
and satisfies
. Under these assump-
tions, the existence and uniqueness result of the scalar case (Theorem 1.2.6) carries
over to the SDE (
8.1
). To prove a multidimensional analog to Theorem 4.1.4,we
need a generalization of Propositions 4.1.1, 6.2.1 to arbitrary dimensions. To this
end, denote by
E[|
Z
|
]
<
∞
ΣΣ
the covariance matrix of the process
X
.
Remark 8.1.1
We note that
0
Q
s
d
s
Q
:=
=
X, X
t
, where
X, X
t
denotes the quadratic
variation process.
A
Proposition 8.1.2
Denote by
the infinitesimal generator of X which is
,
for func-
tions f
∈
C
2
(
R
d
) with bounded derivatives
,
given by
1
2
tr
(x)D
2
f(x)
b(x)
∇
(
A
f )(x)
=
[
Q
]+
f(x),
(8.4)
−
0
(
using the notation as in
(2.2).
Then
,
the process M
t
:=
f(X
t
)
A
f )(X
s
)
d
s is
a martingale with respect to the filtration of W
.
X
i
,X
j
Proof
By the multidimensional Itô formula, we have with d
t
=
Q
ij
(X
t
)
d
t
d
d
1
2
∂
x
i
f(X
t
)
d
X
t
+
X
i
,X
j
d
f(X
t
)
=
∂
x
i
x
j
f(X
t
)
d
t
i
=
1
i,j
=
1
d
n
Σ
ij
(X
t
)
d
W
t
b(X
t
)
∇
=
f(X
t
)
d
t
+
∂
x
i
f(X
t
)
i
=
1
j
=
1
d
1
2
(D
2
f)
ij
(X
t
)
+
Q
ij
(X
t
)
d
t
i,j
=
1
b(X
t
)
∇
f(X
t
)
+
d
t
1
2
tr
[
Q
(X
t
)D
2
f(X
t
)
]
=
d
n
Σ
ij
(X
t
)
d
W
t
.
+
∂
x
i
f(X
t
)
i
=
1
j
=
1
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