We show that if $(X_s, s\geq 0)$ is a right-continuous process,
$Y_t=\int_0^t\d s X_s$ its integral process and $\tau = (\tau_{\ell}, \ell \geq
0)$ a subordinator, then the time-changed process $(Y_{\tau_{\ell}}, \ell\geq
0)$ allows to retrieve the information about $(X_{\tau_{\ell}}, \ell\geq 0)$
when $\tau$ is stable, but not when $\tau$ is a gamma subordinator. This
question has been motivated by a striking identity in law involving the Bessel
clock taken at an independent inverse Gaussian variable

Bertoin, Jean

Yor, Marc

English

arXiv

We show that if $(X_s, s\geq 0)$ is a right-continuous process, $Y_t=\int_0^t\d s X_s$ its integral process and $\tau = (\tau_{\ell}, \ell \geq 0)$ a subordinator, then the time-changed process $(Y_{\tau_{\ell}}, \ell\geq 0)$ allows to retrieve the information about $(X_{\tau_{\ell}}, \ell\geq 0)$ when $\tau$ is stable, but not when $\tau$ is a gamma subordinator. This question has been motivated by a striking identity in law involving the Bessel clock taken at an independent inverse Gaussian variable

Hal-Diderot

Retrieving information from subordination
Jean Bertoin, Marc Yor
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Retrieving information from subordination
Jean Bertoin∗ Marc Yor †‡
Abstract
We show that if (Xs, s ≥ 0) is a right-continuous process, Yt =
∫ t
0 dsXs its integral
process and τ = (τℓ, ℓ ≥ 0) a subordinator, then the time-changed process (Yτℓ , ℓ ≥ 0)
allows to retrieve the information about (Xτℓ , ℓ ≥ 0) when τ is stable, but not when τ is
a gamma subordinator. This question has been motivated by a striking identity in law
involving the Bessel clock taken at an independent inverse Gaussian variable.
Key words: Time-change, subordinator, information retrieval.
1 Introduction and main statements
1.1 Motivation
In Dufresne and Yor [4], it was remarked that by combining Bougerol’s identity in law (see,
e.g. Bougerol [3] and Alili et al. [1]) and the symmetry principle of De´sire´ Andre´, there is the
identity in distribution for every fixed ℓ ≥ 0
Hτℓ
(law)
= τa(ℓ), (1)
where a(ℓ) = Argsinh(ℓ) = log
(
ℓ+
√
1 + ℓ2
)
,
Ht =
∫ t
0
dsR−2s , t ≥ 0 ,
is the Bessel clock constructed from a two-dimensional Bessel process (Rs, s ≥ 0) issued from
1, and (τℓ, ℓ ≥ 0) is a stable (1/2) subordinator independent from (Rs, s ≥ 0).
∗Laboratoire de Probabilite´s et Mode`les Ale´atoires, UPMC, 175 rue du Chevaleret, 75013 Paris, France.
Email: jean.bertoin@upmc.fr
†Laboratoire de Probabilite´s et Mode`les Ale´atoires, UPMC, 175 rue du Chevaleret, 75013 Paris, France.
Email: deaproba@proba.jussieu.fr
‡Institut Universitaire de France.
1
In [4], the authors wondered whether (1) extends at the level of processes indexed by ℓ ≥ 0, or
equivalently whether (Hτℓ , ℓ ≥ 0) has independent increments. Our main result entails that this
is not the case. Indeed, Theorem 1 below implies that for every ℓ ≥ 0, the filtration (Hˆℓ, ℓ ≥ 0)
generated by (Hτℓ , ℓ ≥ 0) contains the filtration generated by (Rτℓ , ℓ ≥ 0). On the other hand,
((Rs, Hs), s ≥ 0) is a Markov (additive) process, and since subordination by an independent
stable subordinator preserves the Markov property, ((Rτℓ , Hτℓ), ℓ ≥ 0) is Markovian in its own
filtration, which coincides with
(
Hˆℓ, ℓ ≥ 0
)
by Theorem 1. It is immediately seen that for any
ℓ′ > 0, the conditional distribution of Hτ
ℓ+ℓ′
given (Rτℓ , Hτℓ) does not only depend on Hτℓ ,
but on Rτℓ as well. Consequently the process (Hτℓ , ℓ ≥ 0) is not even an autonomous Markov
process. We point out that the process (Rτℓ , ℓ ≥ 0) is Markov (by subordination), and refer to
a forthcoming paper [2] for details on the semigroup of (Rτℓ , Hτℓ).
1.2 Main results
More generally, we consider in this work an Rd-valued process (Xs, s ≥ 0) with right-continuous
sample paths, and (τℓ, ℓ ≥ 0) a stable subordinator with index α ∈ (0, 1). We stress that we do
not require X and τ to be independent. Introduce
Yu =
∫ u
0
dsXs , u ≥ 0 ,
and the right-continuous time-changed processes
Xˆℓ = Xτℓ and Yˆℓ = Yτℓ , ℓ ≥ 0 .
We are interested in comparing the information embedded in the processes Xˆ and Yˆ , respec-
tively. We state our main result.
Theorem 1 The right-continuous filtration (Yˆℓ, ℓ ≥ 0) generated by the process
(
Yˆℓ, ℓ ≥ 0
)
contains the right-continuous filtration (Xˆℓ, ℓ ≥ 0) generated by
(
Xˆℓ, ℓ ≥ 0
)
.
A perusal of the proof (given below in Section 2) shows that Theorem 1 can be extended to
the case when it is only assumed that τ is a subordinator such that the tail of its Le´vy measure
is regularly varying at 0 with index −α, which suggests that this result might hold for more
general subordinators. On the other hand, if (Nℓ, ℓ ≥ 0) is any increasing step-process issued
from 0, such as for instance a Poisson process, then the time-changed process (YNℓ , ℓ ≥ 0) stays
at 0 until the first jump time of N which is strictly positive a.s. This readily implies that the
germ-σ-field ⋂
ℓ>0
σ(YNλ , λ ≤ ℓ)
2
is trivial, in the sense that every event of this field has probability either 0 or 1. Focussing
on subordinators with infinite activity, it is interesting to point out that Theorem 1 fails when
one replaces the stable subordinator τ by a gamma subordinator, as it can be seen from the
following observation.
Proposition 1 Let γ = (γt, t ≥ 0) be a gamma-subordinator and ξ a random variable with
values in (0,∞) which is independent of γ. Then the germ-σ-field
⋂
t>0
σ(ξγs, s ≤ t)
is trivial.
We point out that Proposition 1 holds more generally when γ is replaced by a subordinator
with logarithmic singularity, also called of class (L), in the sense that the drift coefficient is zero
and the Le´vy measure is absolutely continuous with density g such that g(x) = g0x
−1 + G(x)
where g0 is some strictly positive constant and G : (0,∞)→ R a measurable function such that
∫ 1
0
|G(x)|dx <∞ , g(x) ≥ 0 , and
∫ ∞
1
g(x)dx <∞ .
Indeed, it has been shown by von Renesse et al. [5] that such subordinators enjoy a quasi-
invariance property analogous to that of the gamma subordinator, and this is the key to
Proposition 1.
It is natural to investigate a similar question in the framework of stochastic integration.
For the sake of simplicity, we shall focus on the one-dimensional case. We thus consider a
real valued Brownian motion (Bt, t ≥ 0) in some filtration(Ft, t ≥ 0) and an (Ft)-adapted
continuous process (Xt, t ≥ 0), and consider the stochastic integral
It =
∫ t
0
XsdBs , t ≥ 0 .
We claim the following.
Proposition 2 Fix η > 0 and assume that the sample paths of (Xt, t ≥ 0) are Ho¨lder-
continuous with exponent η a.s. Suppose also that (τℓ, ℓ ≥ 0) is a stable subordinator of in-
dex α ∈ (0, 1), which is independent of F∞. Then the right-continuous filtration (Iˆℓ, ℓ ≥ 0)
generated by the subordinate stochastic integral
(
Iˆℓ = Iτℓ , ℓ ≥ 0
)
contains the right-continuous
filtration generated by (|Xτℓ|, ℓ ≥ 0).
The proofs of these statements are given in the next section.
3
2 Proofs
2.1 Proof of Theorem 1.
We first observe that the proof can be reduced to showing that the germ-σ-field Yˆ0 contains
the σ-field generated by Xˆ0 = X0. Indeed, let us take this for granted, fix ℓ > 0 and define
τ ′u = τℓ+u − τℓ and X ′v = Xv+τℓ . Then τ ′ is again a stable(α) subordinator and X ′ a right-
continuous process, and
Yˆℓ+u − Yˆℓ =
∫ τ ′u
0
dvX ′v .
Hence X ′0 = Xˆℓ is measurable with respect to Yˆℓ, and our claim follows.
Thus we only need to verify that X0 is Yˆ0-measurable. In this direction, we shall use the
following version of the Law of Large Numbers for the jumps ∆τs = τs − τs− of a stable
subordinator. Fix any m > 2/α and introduce for any given b ∈ R and ε > 0
Nε,b = Card{s ≤ ε : b∆τs > εm} .
Note that Nε,b ≡ 0 for b ≤ 0. For the sake of simplicity, we henceforth suppose that the tail of
the Le´vy measure of τ is x 7→ x−α, which induces no loss of generality. So for b > 0, Nε,b is a
Poisson variable with parameter
ε(εm/b)−α = bαε1−mα .
Combining a standard argument based on the Borel-Cantelli lemma and Chebychev’s inequality
with monotonicity, we get that for ε = 1/n
lim
n→∞
n1−αmN1/n,b = b
α for all b > 0, almost-surely. (2)
Let us assume that the process X is real-valued as the case of higher dimensions will then
follow by considering coordinates. Set
Jε = Card{s ≤ ε : ∆Yˆs > εm} ,
where as usual ∆Yˆs = Yˆs − Yˆs−. We note that
∆Yˆs −X0∆τs =
∫ τs
τs−
du(Xu −X0) .
4
Hence if we set aε = sup0≤u≤τε |Xu −X0|, then
(X0 − aε)∆τs ≤ ∆Yˆs ≤ (X0 + aε)∆τs ,
from which we deduce
Nε,X0−aε ≤ Jε ≤ Nε,X0+aε .
Since X has right-continuous sample paths a.s., we have limε→0 aε = 0 a.s., and taking ε = 1/n,
we now deduce from (2) that
lim
n→∞
n1−mαJε = (X
+
0 )
α .
Hence X+0 is Yˆ0-measurable, and the same argument also shows that X−0 is Yˆ0-measurable.
This completes the proof of our claim. 
2.2 Proof of Proposition 1.
Let Ω denote the space of ca`dla`g paths ω : [0,∞) → R+ endowed with the right-continuous
filtration (At, t ≥ 0) generated by the canonical process ωt = ω(t), and write Q for the law on
Ω of the process (ξγt, t ≥ 0).
It is well known that for every x > 0 and t > 0, the distribution of the process (xγs, 0 ≤ s ≤ t)
is absolutely continuous with respect to that of the gamma process (γs, 0 ≤ s ≤ t) with density
x−t exp ((1− 1/x)γt) .
Because ξ and γ are independent, this implies that for any event Λ ∈ Ar with r < t
Q (Λ) = E
(
ξ−t exp ((1− 1/ξ)γt) 1{γ∈Λ}
)
.
Observe that
lim
t→0+
ξ−t exp ((1− 1/ξ)γt) = 1 a.s.
and the convergence also holds in L1(P) by an application of Scheffe´’s lemma (alternatively, one
may also invoke the convergence of backwards martingales). We deduce that for every Λ ∈ F0,
we have
Q(Λ) = P(γ ∈ Λ)
and the right-hand-side must be 0 or 1 because the gamma process fulfills the Blumenthal’s 0-1
law. 
5
2.3 Proof of Proposition 2.
The guiding line is similar to that of the proof of Theorem 1. In particular it suffices to verify
that |X0| is measurable with respect to the germ-σ-field Iˆ0.
Because B and τ are independent, the subordinate Brownian motion (Bˆℓ = Bτℓ , ℓ ≥ 0) is a
symmetric stable Le´vy process with index 2α. With no loss of generality, we may suppose that
the tail of its Le´vy measure Π is given by Π(R\[−x, x]) = x−2α. As a consequence, for every
m > 2/α and ε > 0 and b ∈ R, if define
Nε,b = Card{s ≤ ε : |b∆Bˆs|2 > εm} ,
then Nε,b is a Poisson variable with parameter |b|2αε1−mα, and this readily yields
lim
n→∞
n1−αmN1/n,b = |b|2α for all b ∈ R, almost-surely. (3)
Next set
Jε = Card{s ≤ ε : |∆Iˆs|2 > εm} ,
where as usual Iˆs = Iτs , and observe that
∆Iˆs = X0∆Bˆs + (Xτs− −X0)∆Bˆs +
∫ τs
τs−
(Xu −Xτs−)dBu . (4)
Recall the assumption that the paths of X are Ho¨lder-continuous with exponent η > 0, so the
(Ft)-stopping time
T = inf
{
u > 0 : sup
0≤v<u
(u− v)−η|Xu −Xv|2 > 1
}
is strictly positive a.s. In particular, if we write Λε = {τε < T}, then P(Λε) tends to 1 as
ε→ 0+.
We fix a > 0, consider
Kε,a = Card
{
s ≤ ε :
∣∣∣∣
∫ τs
τs−
(Xu −Xτs−)dBu
∣∣∣∣
2
> aεm
}
,
and claim that
lim
ε→0
εαm−1E(Kε,a,Λε) = 0 . (5)
If we take (5) for granted, then we can complete the proof by an easy adaptation of the argument
in Theorem 1. Indeed, we can then find a strictly increasing sequence of integers (n(k), k ∈ N)
6
such that with probability one, for all rational numbers a > 0
lim
k→∞
n(k)1−αmK1/n(k),a = 0 . (6)
We observe from (4) that for any a ∈ (0, 1/2), if |∆Iˆs|2 > εm, then necessarily either
|X0∆Bˆs|2 > (1− 2a)2εm ,
or
|(Xτs− −X0)∆Bˆs|2 > a2εm ,
or ∣∣∣∣
∫ τs
τs−
(Xu −Xτs−)dBu
∣∣∣∣
2
> a2εm .
As
lim
ε→0+
sup
0≤s≤ε
|Xτs− −X0| = 0 ,
this easily entails, using (3) and (6), that
lim sup
k→∞
n(k)1−αmJ1/n(k) ≤ lim
k→∞
n(k)1−αmN1/n(k),(1−2a)−1 |X0|
= (1− 2a)−2α|X0|2α , a.s.
where the identity in the second line stems from (3). A similar argument also gives
lim inf
k→∞
n(k)1−αmJ1/n(k) ≥ (1 + 2a)−2α|X0|2α , a.s.,
and as a can be chosen arbitrarily close to 0, we conclude that
lim
k→∞
n(k)1−αmJ1/n(k) = |X0|2α , a.s.
Hence |X0| is Iˆ0-measurable.
Thus we need to establish (5). As τ is independent of F∞, we have by an application of
Markov’s inequality that for every s ≤ ε
P
(∣∣∣∣
∫ τs
τs−
(Xu −Xτs−)dBu
∣∣∣∣
2
> aεm,Λε | τ
)
≤ 1
aεm
∫ ∆τs
0
dvvη ≤ (∆τs)
1+η
aεm
.
7
It follows that
E(Kε,a,Λε) ≤ E
(∑
s≤ε
(
(∆τs)
1+η
aεm
∧ 1
))
= εc
∫
(0,∞)
dxx−1−α
(
x1+η
aεm
∧ 1
)
= O(ε1−αm/(1+η)) ,
where for the second line we used the fact that the Le´vy measure of τ is cx−1−αdx for some
unimportant constant c > 0. This establishes (5) and hence completes the proof of our claim.

References
[1] L. Alili, D. Dufresne and M. Yor: Sur l’identite´ de Bougerol pour les fonctionnelles ex-
ponentielles du mouvement brownien avec drift. In: M. Yor (ed.): Exponential functional
and principal values related to Brownian motion, pp. 3-14. Bibl. Rev. Mat. Iberoamericana,
Madrid, 1997.
[2] J. Bertoin, D. Dufresne and M. Yor: A remarkable two-dimensional Markov process derived
from Bougerol’s identity. In preparation (May 2010).
[3] Ph. Bougerol: Exemples de the´ore`mes locaux sur les groupes re´solubles. Ann. Inst. H.
Poincare´ Sect. B 19 (1983), 369-391.
[4] D. Dufresne and M. Yor: A two-dimensional extension of Bougerol’s identity in law for
the exponential functional of Brownian motion: the story so far. In preparation
[5] M.-K. von Renesse, M. Yor, L. Zambotti: Quasi-invariance properties of a class of subor-
dinators. Stochastic Process. Appl.188 , (2008), 2038-2057.
8


Retrieving information from subordination

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