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{
Σ
ε
}
ε
∈
(
0
,
1
]
there holds
Assume further that for some family of non-singular matrices
Σ
−
1
ε
Q
ε
Σ
−
→
I
d
,
as ε
→
0
,
ε
d
.
Then
,
for all ε
where
I
d
denotes the identity matrix in
R
∈
(
0
,
1
]
there exists a
càdlàg process R
ε
such that
(d)
=
γ
ε
t
+
Σ
ε
W
t
+
N
t
+
R
t
,
X
t
(14.30)
in the sense of equality of finite dimensional distributions
.
Furthermore
,
we have for
all T>
0, sup
t
∈[
0
,T
]
|
)
−→
(
P
Σ
−
1
ε
0
where γ
ε
,
N
ε
are given in
(
14.29
)
and W is a d-dimensional standard Brownian motion independent of N
ε
.
R
t
|
0,
as ε
→
We give an example of the decomposition (
14.28
) into small and large jumps.
(X
1
,...,X
d
)
be a
d
-dimensional Lévy process with
Lévy measure
ν
and marginal Lévy measures
ν
i
,
i
=
Example 14.7.2
Let
X
1
,...,d
. To obtain
ν
ε
in
d
=
=
1,
we simply cut off the small jumps, i.e.
ν
ε
=
ν
1
.
(14.31)
{|
z
|
>ε
}
For
d>
1 the Lévy measure
ν
ε
could be obtained by
ν
ε
where jumps
are neglected if the jump size in all directions is small. But the corresponding one-
dimensional Lévy measures
ν
i
,
i
=
=
ν
1
{
z
∞
>ε
}
1
,...,d
are then not of the form (
14.31
). If we
choose
ν
ε
=
ν
1
,
(14.32)
{
min
{
z
1
,...,z
d
}
>ε
}
the corresponding one-dimensional Lévy measures
ν
i
,
i
1
,...,d
again satisfy
(
14.31
). We consider the Clayton Lévy copula model as explained in Sect.
14.3.2
with the density
k
given by (
14.14
)for
d
=
(
0
.
5
,
1
.
2
)
.
The corresponding regularized density
k
ε
,
ν
ε
(
d
z)
=
k
ε
(z)
d
z
as in (
14.32
)for
ε
=
0
.
01 is plotted in Fig.
14.8
.
=
2,
ϑ
=
0
.
5,
η
=
0
.
5 and
α
=
We now consider a
d
-dimensional pure jump process
X
with characteristic triplet
(
0
,ν,γ)
where the Lévy measure
ν
satisfies (10.11). Let
γ
be chosen according to
Lemma 10.1.5 such that
e
X
j
,
j
=
1
,...,d
are martingales. The covariance matrix
Q
=
R
d
zz
ν(
d
z)
. For any
ε>
0 the process
X
can be approximated by
a compound Poisson process
Y
1
is given by
as in (
14.29
) where the small jumps are neglected
as in (
14.32
),
Y
1
,t
=
γ
1
t
N
t
.
+
(14.33)
is again such that
e
Y
ε,j
The characteristic triplet of
Y
1
is
(
0
,ν
ε
,γ
1
)
and
γ
1
=
1
,...,d
are martingales. A better approximation can be obtained by replacing the
small jumps with a Brownian motion which yields a jump-diffusion process
Y
2
,
Y
2
,t
=
,
j
1
γ
2
t
N
t
,
Σ
ε
W
t
+
+
(14.34)
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