1.5 Organisation of the paper
We now present more precisely the organisation of our paper:
- in Section 2, we recall some basic results about various representations of self-decomposable
variables, and we complete the discussion of Subsection 1.3 above;
- in Section 3, we consider the simple situation, as in Subsection 1.3, where Y
t
= R
2 N
t, for R
N
a Bessel process of dimension N starting from 0;
- the contents of Sections 4, 5, 6 have already been discussed in the above Subsection 1.4.
We end this introduction with the following negative remark concerning further self- decomposability properties for squared Bessel processes: indeed, it is well-known, and goes
back to Shiga-Watanabe [25], that R
2 N
•, considered as a random variable taking values in CR
+
, R
+
is infinitely divisible. Furthermore, in the present paper, we show and exploit the self- decomposability of
R
0,∞
R
2 N
s dµs for any positive measure µ. It then seems natural to wonder about the self-decomposability of R
2 N
•, but this property is ruled out: the 2-dimensional vectors R
2 N
t
1
, R
2 N
t
1
+ t
2
are not self-decomposable, as an easy Laplace transform computation implies.
2 Sato processes and PCOC’s
2.1 Self-decomposability and Sato processes
We recall, in this subsection, some general facts concerning the notion of self-decomposability. We refer the reader, for background, complements and references, to Sato [24, Chapter 3].
A random variable X is said to be self-decomposable if, for each u with 0 u 1, there is the equality
in law: X
d
= u X + b X
u
for some variable b X
u
independent of X . On the other hand, an additive process U
t
, t ≥ 0 is a stochastically continuous process with càdlàg
paths, independent increments, and satisfying U = 0.
An additive process U
t
which is H-self-similar for some H 0, meaning that, for each c 0, U
c t d
= c
H
U
t
, will be called a Sato process or, more precisely, a H-Sato process. The following theorem, for which we refer to Sato’s book [24, Chapter 3, Sections 16-17], gives
characterizations of the self-decomposability property.
Theorem 2.1. Let X be a real valued random variable. Then, X is self-decomposable if and only if one of the following equivalent properties is satisfied:
1 X is infinitely divisible and its Lévy measure is hx
|x| dx
with h increasing on −∞, 0 and
decreasing on 0, + ∞.
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2 There exists a Lévy process C
s
, s ≥ 0 such that
X
d
= Z
∞
e
−s
dC
s
.
3 For any or some H 0, there exists a H-Sato process U
t
, t ≥ 0 such that X
d
= U
1
. In 2 resp. 3 the Lévy process C
s
resp. the H-Sato process U
t
is uniquely determined in law by X , and will be said to be associated with X . We note that, if X
≥ 0, then the function h vanishes on
−∞, 0, C
s
is a subordinator and U
t
is an increasing process. The relation between C
s
and U
t
was made precise by Jeanblanc-Pitman-Yor [11, Theorem 1]:
Theorem 2.2. If U
t
is a H-Sato process, then the formulae: C
− s
= Z
1 e
−s
r
−H
dU
r
and C
+ s
= Z
e
s
1
r
−H
dU
r
, s
≥ 0 define two independent and identically distributed Lévy processes from which U
t
, t ≥ 0 can be recov-
ered by: U
t
= Z
∞ − log t
e
−sH
dC
− s
if ≤ t ≤ 1
and U
t
= U
1
+ Z
log t
e
sH
dC
+ s
if t
≥ 1. In particular, the Lévy process associated with the self-decomposable random variable U
1
is C
s
= C
− s
H
, s
≥ 0.
2.2 Sato processes and PCOC’s
Kategori