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d
=
∈ R
distribution
μ
of
X
at
t
1is
Q
-stable, i.e. for any
r>
0 there exists a
c
such that
1
α
1
z
1
,...,r
1
α
d
z
d
)e
i
c,z
.
μ(z)
r
=
μ(r
(14.8)
μ(z)
=
e
−
ψ(z)
, it follows from
(
14.8
) that the characteristic exponent of an
α
-stable process satisfies for any
r>
0
Since we have for the characteristic function
1
α
1
ξ
1
,...,r
1
α
d
ξ
d
)
d
.
ψ(r
=
r
ψ(ξ),
∀
ξ
∈ R
(14.9)
We assume that the Lévy measure
ν
has a Lévy density
k
,i.e.
ν(
d
z)
=
k(z)
d
z
and
obtain for
Q
=
0 that
1
α
1
ξ
1
,...,r
1
α
d
ξ
d
)
ψ(r
1
cos
d
k
sym
(z)
d
z
1
α
i
ξ
i
z
i
=
−
r
d
R
i
=
1
1
α
1
z
1
,...,r
−
1
α
d
z
d
)r
−
1
α
1
−···−
1
α
d
d
z.
)
k
sym
(r
−
=
(
1
−
cos
ξ,z
d
R
Now using (
14.9
), the Lévy density has to satisfy
1
α
1
z
1
,...,r
−
1
α
d
z
d
)
1
α
1
+···+
1
α
d
k
sym
(z
1
,...,z
d
).
k
sym
(r
−
r
1
+
=
(14.10)
d
A simple example of an
α
-stable Lévy process on
R
is given by the Lévy measure
d
α
i
−
1
−
1
α
1
−···−
1
α
d
2
d
=
1
|
z
i
|
ν(
d
z)
c
j
1
Q
j
d
z,
(14.11)
j
=
1
i
=
0,
2
d
j
1
c
j
>
0. The corresponding marginal processes
X
i
,
i
where
c
j
≥
=
1
,...,d
=
|
−
1
−
α
i
d
z
of
X
are again
α
-stable processes in
R
with Lévy measure
ν
i
(
d
z)
=
c
i
|
z
1
,...,
2
d
. We plot the density (
14.11
)for
where
c
i
depend on
d
,
α
and
c
j
,
j
=
d
=
2,
α
=
(
0
.
5
,
1
.
2
)
and
c
j
=
1,
j
=
1
,...,
4inFig.
14.2
.
14.3.1 Subordinated Brownian Motion
As in the one-dimensional case, we can obtain Lévy processes by
subordination
.
For
d>
1 there are two possibilities. Using a one-dimensional increasing process or
subordinator
G
={
G
t
:
t
≥
0
}
, the resulting process is given by
X
t
=
W
i
G
t
+
θ
i
G
t
,
i
∈ R
,t
∈[
0
,T
]
,
(W
1
,...,W
d
)
is a vector of
d
Brownian motions with
for
i
=
1
,...,d
where
W
=
(σ
i
σ
j
ρ
ij
)
1
≤
i,j
≤
d
. Here,
σ
i
covariance matrix
Q
=
,
i
=
1
,...,d
, is the variance of
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