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14.3 Lévy Models
d
If
{
X
t
:
t
≥
0
}
is a Brownian motion on
R
then, for any
r>
0, the process
{
X
rt
:
r
1
/
2
X
t
:
t
. This property is called
self-similarity
of a stochastic process with index 2. There are many self-similar Lévy
processes other than the Brownian motion, the so-called stable processes.
≥
0
}
is identical in law with the process
{
t
≥
0
}
Definition 14.3.1
Let 0
<α<
2. A Lévy process
X
={
X
t
:
t
≥
0
}
with state space
d
R
is called
α
-
stable
if the distribution
μ
of
X
at
t
=
1is
α
-stable, i.e. for any
r>
0
d
there exists
c
∈ R
such that
μ(r
α
z)e
i
c,z
.
μ(z)
r
=
It is shown in [143, Theorem 14.3] that any Lévy process with the characteristic
triplet
(
Q
,ν,γ)
has an
α
-stable probability measure if and only if
Q
=
0 and if there
d
={
∈ R
: |
x
| =
}
is a finite measure
λ
on the unit sphere
S
x
1
such that
λ(
d
ξ)
∞
0
1
r
1
+
α
d
r,
d
).
ν(B)
=
1
B
(rξ)
B
∈
B
(
R
S
d
A simple example of an
α
-stable Lévy process on
R
is given by the Lévy measure
2
d
|
−
d
−
α
1
Q
j
d
z,
ν(
d
z)
=
c
j
|
z
(14.7)
j
=
1
0,
2
d
j
where
c
j
≥
1
this is the only possible
α
-stable process. The corresponding marginal processes
X
i
,
i
1
c
j
>
0 and
Q
j
denoting the
j
th quadrant. Note that for
d
=
=
=
1
,...,d
of
X
are again
α
-stable processes in
R
with Lévy measure
ν
i
(
d
z)
=
|
−
1
−
α
d
z
where
1
,...,
2
d
.
For
d>
1 the notation of stable processes can be extended by using non-singular
matrices for scaling.
c
i
|
z
c
i
depend on
α
,
d
and
c
j
,
j
=
d
×
d
Definition 14.3.2
Let
Q
∈ R
be a matrix with positive eigenvalues. A Lévy
d
process
X
={
X
t
:
t
≥
0
}
with state space
R
is called
Q
-
stable
if for any
r>
0
d
there exist a
c
∈ R
such that the distribution
μ
of
X
at
t
=
1 satisfies
μ(r
Q
z)e
i
c,z
,
μ(z)
r
=
=
n
=
0
(n
where
r
Q
)
−
1
(
log
r)
n
Q
n
.
!
diag
((
1
/α,...,
1
/α))
,0
<α<
2, we again obtain
α
-stable processes.
An extension of (isotropic)
α
-stable processes are anisotropic
α
-stable processes for
an
α
=
(α
1
,...,α
d
)
with 0
<α
i
<
2,
i
=
For
Q
=
1
,...,d
.
α
−
1
i
Definition 14.3.3
Let 0
<α
i
<
2,
i
=
1
,...,d
and
Q
=
diag
{
:
i
=
1
,...,d
}
.
d
A Lévy process
X
={
X
t
:
t
≥
0
}
with state space
R
is called
α
-stable if the
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