Information Technology Reference
In-Depth Information
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
 
Search WWH ::




Custom Search