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where
(x,
∞
)
for
x
≥
0
,
I(x)
=
−∞
]
(
,x
for
x<
0
.
-marginal tail integral
U
I
of
X
is
Furthermore, for
I
⊂{
1
,...,d
}
nonempty the
I
the tail integral of the process
X
I
:=
(X
i
)
i
∈
I
.
The next result, [100, Theorem 3.6], shows that essentially any Lévy process
X
=
(X
1
,...,X
d
)
can be built from univariate marginal processes
X
j
,
j
1
,...,d
and Lévy copulas. It can be viewed as a version of Sklar's theorem for Lévy copulas.
=
Theorem 14.2.6
(Sklar's theorem for Lévy copulas)
For any Lévy process X with
state space
d
,
there exists a Lévy copula F such that the tail integrals of X satisfy
U
I
(x
I
)
R
F
I
((U
i
(x
i
))
i
∈
I
=
),
(14.2)
and any x
I
∈ R
|
I
|
\{
for any nonempty
I
⊂{
1
,...,d
}
0
}
.
The Lévy copula F is
unique on
i
=
1
Ran
U
i
.
Conversely
,
let F be a d-dimensional Lévy copula and U
i
,
i
1
,...,d
,
tail inte-
grals of univariate Lévy processes
.
Then
,
there exits a d-dimensional Lévy process X
such that its components have tail integrals U
i
and its marginal tail integrals satisfy
(
14.2
).
The Lévy measure ν of X is uniquely determined by F and U
i
,
i
=
=
1
,...,d
.
Using partial integration, we can write the multidimensional Lévy measure in
terms of the Lévy copula.
Lemma 14.2.7
Let f(z)
∈
C
∞
(
R
d
) be bounded and vanishing on a neighborhood
of the origin
.
Furthermore
,
let X be a d-dimensional Lévy process with Lévy mea-
sure ν
,
Lévy copula F and marginal Lévy measures ν
j
,
j
=
1
,...,d
.
Then
,
d
f(z)ν(
d
z)
=
f(
0
+
z
j
)ν
j
(
d
z
j
)
d
R
R
j
=
1
d
∂
I
f(
0
z
I
)F
I
((U
k
(z
k
))
k
∈
I
)
d
z
I
.
(14.3)
+
+
j
R
j
=
2
|
I
|
=
j
I
1
<
···
<
I
j
Proof
We proceed by induction with respect to the dimension
d
:
For
d
=
1, integration by parts yields
∞
f(z)ν(
d
z)
=−
lim
b
f(b)ν(I(b))
+
lim
f(a)ν(I(a))
→∞
a
→
0
+
0
∞
+
∂
1
f(z)ν(I(z))
d
z,
0
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