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10.2.2 Pure Jump Models
A popular class of processes is obtained by
subordination
of a Brownian motion
with drift.
Definition 10.2.1
A Lévy process
X
is called a
subordinator
if the sample paths of
X
are a.s. nondecreasing, i.e.
t
1
≥
t
2
⇒
X
t
1
≥
X
t
2
a.s.
0.
The characteristic triplet of a subordinator has the following properties, see, e.g. [40,
Proposition 3.10].
Since
X
0
=
0, it immediately follows by the definition that
X
t
≥
0a.s.
∀
t
≥
Lemma 10.2.2
Let G be a subordinator
.
Then the characteristic triplet (σ
2
,ν,γ)
of G satisfies σ
2
0,
=
0,
γ
≥
0
and ν((
−∞
,
0
]
)
=
1
∧
zν(
d
z) <
∞
.
R
+
By Lemma
10.2.2
and (
10.3
), the characteristic exponent
ψ
of a subordinator is
given by
ψ(ξ)
+
R
+
e
iξz
)ν(
d
z)
.
Now let
G
be a subordinator and
W
a standard Brownian motion. Then, we can
construct a Lévy process
X
by time changing
W
=−
iγξ
(
1
−
X
t
=
σW
G
t
+
θG
t
,σ>
0
,θ
∈ R
,t
∈[
0
,T
]
.
As an example we use as the subordinator a gamma process to obtain a
variance
ga
m
ma process
[118]. We consider a gamma process
G
with Lévy density
k
G
(s)
=
s
ϑ
(ϑs)
−
1
1
e
−
. Then, using [143, Theorem 30.1], the Lévy measure of
X
is given
{
s>
0
}
for
B
∈
B
(
R
)
by
∞
(z
−
θs)
2
2
sσ
2
1
1
ϑs
e
−
√
2
πsσ
2
e
−
s
ϑ
d
s
d
z
ν(B)
=
B
0
e
θs/σ
2
∞
0
1
ϑ
√
2
πσ
2
z
2
2
σ
2
θ
2
2
σ
2
+
1
s
−
(
1
ϑ
)s
d
s
d
z.
1
2
−
1
e
−
s
−
=
B
Using [74, Formula 3.471 (9)] to integrate the second integral, we obtain the Lévy
measure
c
e
−
β
+
|
z
|
|
z
|
d
z,
}
+
c
e
−
β
−
|
z
|
|
z
|
ν(
d
z)
=
1
1
(10.9)
{
z>
0
{
z<
0
}
1
/ϑ
,
β
+
=
c
2
/(
2
σ
2
/ϑ
+
θ
2
+
θ)
,
β
−
=
c
2
/(
2
σ
2
/ϑ
+
θ
2
with
c
=
−
θ)
.
The variance gamma process is a special case of the
tempered stable process
(for
c
also called
CGMY
process in [36]or
KoBoL
in [23]) which has a Lévy
density of the form
=
c
+
=
c
−
c
d
z,
e
−
β
+
|
z
|
|
e
−
β
−
|
z
|
|
ν(
d
z)
=
1
}
+
c
1
(10.10)
+
{
z>
0
−
{
z<
0
}
1
+
α
1
+
α
z
|
z
|
for
c
,c
,β
,β
>
0 and 0
≤
α<
2.
+
−
+
−
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