# Article6 Sghir Aissa .pdf

Nom original: Article6_Sghir_Aissa.pdf

Ce document au format PDF 1.4 a été généré par TeX output 2014.02.12:1103 / dvipdfmx (20090708); modified using iText 2.1.7 by 1T3XT, et a été envoyé sur fichier-pdf.fr le 17/02/2017 à 20:31, depuis l'adresse IP 105.145.x.x. La présente page de téléchargement du fichier a été vue 388 fois.
Taille du document: 123 Ko (10 pages).
Confidentialité: fichier public

### Aperçu du document

Serials Publications

Communications on Stochastic Analysis
Vol. 7, No. 3 (2013) 373-382

www.serialspublications.com

THE GENERALIZED SUB-FRACTIONAL BROWNIAN MOTION
AISSA SGHIR*

Abstract. In this paper we introduce a self-similar Gaussian process called
the generalized sub-fractional Brownian motion. This process generalizes the
well-known sub-fractional Brownian motion introduced by Bojdecki et al.
. We prove the existence and the joint continuity of the local time of our
process. We use the concept of local nondeterminism for Gaussian process
introduced by Berman  and the analytic method used by Berman  for
the calculation of the moments of local time.

1. Introduction
An extension of standard Brownian motion, (Bm for short), which preserves
many properties of fractional Brownian motion, (fBm for short), but not the
stationarity of the increments, is the so called sub-fractional Brownian motion
S H := {StH ; t ≥ 0}, (sfBm for short). It was introduced by Bojdecki et al. .
It is a continuous centered Gaussian process, starting from zero, with covariance
function
1
S(t, s) = tH + sH − [(t + s)H + |t − s|H ],
2
where H ∈ (0, 2).
Notice that the fBm B H := {BtH ; t ≥ 0} was introduced by Mandelbrot and
Van Ness . It is the unique continuous, centered, Gaussian process, starting
from zero, with covariance function
F (t, s) =

1 H
[t + sH − |t − s|H ],
2

where H ∈ (0, 2). H
2 is called the Hurst parameter of fBm.
The self similarity and stationarity of the increments are two main properties
for which fBm enjoyed success as modeling tool in telecommunications and finance.
Notice that for H = 1 both processes fBm and sfBm are Bm. The sfBm is
H
-self
similar process and its increments satisfy for all s ≤ t,
2
(t − s)H ≤ E[StH − SsH ]2 ≤ (2 − 2H−1 )(t − s)H ,

∀ H ∈ (0, 1]

(2 − 2H−1 )(t − s)H ≤ E[StH − SsH ]2 ≤ (t − s)H ,

∀ H ∈ [1, 2).

Received 2012-7-6; Communicated by the editors.
2010 Mathematics Subject Classification. 60G18.
Key words and phrases. Sub-fractional Brownian motion, bifractional Brownian motion, fractional Brownian motion, local time, local nondeterminism.
* This research is supported by the Doe Foundation.
373

374

AISSA SGHIR

Bardina and Bascompte  and Ruiz de Chavez and Tudor  have obtained for
H ∈ (0, 1), the following decomposition in law of sfBm,
StH d BtH + C1 (H)XtH ,
(1.1)
q
R +∞
H+1
H
, XtH = 0 (1 − e−θt )θ− 2 dWθ and Bm W and fBm
where C1 (H) = 2Γ(1−H)
B H are independent.
For H ∈ (1, 2), Bardina and Bascompte  have obtained the following decomposition in law,
BtH d StH + C2 (H)XtH ,
(1.2)
q
H(H−1)
where C2 (H) = 2Γ(2−H) and Bm W and sfBm S H are independent.
On the other hand, Bardina and Bascompte  have proved that X H is Gaussian, centered, and its covariance function is,
(
Γ(1−H) H
[t + sH − (t + s)H ], ∀H ∈ (0, 1),
H
T (t, s) =
Γ(2−H)
H
H
H
H(H−1) [(t + s) − t − s ], ∀H ∈ (1, 2).
Moreover, X H has a version with trajectories which are infinitely differentiable on
(0, +∞) and absolutely continuous on [0, +∞).
2. The Generalized Sub-fractional Brownian Motion
Now we are ready to introduce and justify the definition and the existence of
our process.
Definition 2.1. The generalized sub-fractional Brownian motion (gsfBm) S H,K
:= {StH,K ; t ≥ 0}, with parameters H ∈ (0, 2) and K ∈ [1, 2) such that HK ∈
(0, 2), is a centered Gaussian process, starting from zero, with covariance function
1
G(t, s) = (tH + sH )K − [(t + s)HK + |t − s|HK ].
2
The case K = 1 corresponds to sfBm with parameter H ∈ (0, 2).
Existence of gsfBm can be shown in the following way:
Consider the process
Yt :=

H,K
BtH,K + B−t
,
22−K

t ≥ 0,

where {BtH,K ; t ∈ R} is the bifractional Brownian motion on the whole real line
with parameters H ∈ (0, 2) and K ∈ (1, 2) such that HK ∈ (0, 2) introduced by
Bardina and Es-Sebaiy . It is easy to see that the covariance function of the
process Yt is precisely G(t, s). Therefore the gsfBm exists.
Theorem 2.2. The covariance function G is symmetric and positive-definite.
Moreover, the gsfBm has the following decomposition in law,
q
where C3 (K) =
S

HK

StH,K = StHK + C3 (K)XtKH ,
K(K−1)
Γ(2−K) ,

(2.1)

H ∈ (0, 2), K ∈ (1, 2) such that HK ∈ (0, 2) and sfBm

with parameter HK ∈ (0, 2) and Bm W are independent.

THE GENERALIZED SUB-FRACTIONAL BROWNIAN MOTION

375

Proof. Using the fact that the gsfBm is Gaussian process, it suffices to see that
1
G(t, s) = ((tH + sH )K − tHK − sHK ) + (tHK + sHK − [(t + s)HK + |t − s|HK ]).
2

We end this section by the following lemma on the regularity of the increments
of gsfBm.
Lemma 2.3. Let T &gt; 0 fixed and assume H ∈ (0, 2) and K ∈ (1, 2) such that
HK ∈ (0, 2). For all 0 ≤ t, s ≤ T and any integer p ≥ 2, there exists a constant
0 &lt; Cp &lt; ∞, such that
p

E[StH,K − SsH,K ]p ≤ Cp |t − s| 2 HK .

(2.2)

Proof. By virtue of (2.1) and the elementary inequality (a + b)2 ≤ 2a2 + 2b2 , we
obtain
E[StH,K − SsH,K ]2 ≤ 2E[StHK − SsHK ]2 + 2C32 (K)E[XtKH − XsKH ]2 .
Therefore,
E[StH,K − SsH,K ]2 ≤ CH,K |t − s|HK + 2C32 (K)CK |tH − sH |2 ,
where we have used in the last inequality the result of Mendy : There exists a
constant 0 &lt; CK &lt; +∞ such that
E[XtK − XsK ]2 ≤ CK |t − s|2 .

(2.3)

Now, we distinguish two cases:
1) Case H ∈ (0, 1). We have |tH − sH | ≤ |t − s|H . Then
h
i
E[StH,K − SsH,K ]2 ≤ |t − s|HK CH,K + 2C32 (K)CK |t − s|H(2−K) .
Since 1 &lt; K &lt; 2 and 0 ≤ t, s ≤ T , there exists a constant 0 &lt; C &lt; ∞, such that
E[StH,K − SsH,K ]2 ≤ C|t − s|HK .
Finally, the fact that the gsfBm is a centered Gaussian process give the desired
estimation.
2) Case H ∈ (1, 2). Making use of the theorem on finite increments for the
function x 7→ xH , there exists ξ ∈ (s, t) such that
|tH − sH | = H|ξ|H−1 |t − s|
≤ C|t − s|.
Consequently,

E[StH,K − SsH,K ]2 ≤ |t − s|HK CH,K + CCK |t − s|2−HK .

Since 0 &lt; HK &lt; 2 and 0 ≤ t, s ≤ T , the proof of Lemma 2.3 is thus concluded.
In the sequel C and Cp denote constants which be different even when they
vary from one line to the next.

376

AISSA SGHIR

3. Local Nondeterminism and Local Time
We begin this section by the definition of the concept of local nondeterminism,
(LND for short). Let J be an open interval on the t axis. Assume that {Xt ; t ≥ 0}
is a zero mean Gaussian process without singularities in any interval of the length
δ, for some δ &gt; 0, and without fixed zeros, i.e., there exists δ &gt; 0, such that

E[Xt − Xs ]2 &gt; 0, whenever 0 &lt; |t − s| &lt; δ,
E(Xt )2 &gt; 0, f or t ∈ J.
To introduce the concept of LND, Berman  defined the relative conditioning
error,
V ar{Xtp − Xtp−1 /Xt1 , ...Xtp−1 }
Vp =
,
V ar{Xtp − Xtp−1 }
where for p ≥ 2, t1 &lt; ... &lt; tp are arbitrary ordered points in J.
We say that the process X is LND on J if for every p ≥ 2,
lim

inf

c→0+ 0&lt;tp −t1 ≤c

Vp &gt; 0.

This condition means that a small increment of the process is not almost relatively
predictable on the basis of a finite number of observations from the immediate
past. Berman  has proved, for Gaussian process, that the LND is characterized
as follows.
Proposition 3.1. A Gaussian process X is LND if and only if for every integer
p ≥ 2, there exists two positive constants δ and Cp such that
!
p
m
X
X
V ar
uj (Xtj − Xtj−1 ) ≥ Cp
u2j V ar(Xtj − Xtj−1 ),
i=1

i=1

for all orderer points t1 &lt; ... &lt; tp are arbitrary points in J with t0 = 0, tp − t1 ≤ δ
and (u1 , ..., uj ) ∈ R.
Remark 3.2. Let T &gt; 0 fixed. Mendy  proved by using (1.1) that the sfBm is
LND on [0, T ] for H ∈ (0, 1).
We end this section by the definition of local time. For a complete survey on
local time, we refer to Geman and Horowitz  and the references therein.
Let X := {Xt ; t ≥ 0} be a real-valued separable random process with Borel
sample functions. For any Borel set B ⊂ R+ , the occupation measure of X on B
is defined as
µB (A) = λ{s ∈ B ; Xs ∈ A},
∀A ∈ B(R),
where λ is the one-dimensional Lebesgue measure on R+ . If µB is absolutely
continuous with respect to Lebesgue measure on R, we say that X has a local
time on B and define its local time, L(B, .), to be the Radon-Nikodym derivative
of µB . Here, x is the so-called space variable and B is the time variable. By
standard monotone class arguments, one can deduce that the local time have a
measurable modification that satisfies the occupation density formula: for every
Borel set B ⊂ R+ and every measurable function f : R → R+ ,
Z
Z
f (Xt )dt =
f (x)L(B, x)dx.
B

R

THE GENERALIZED SUB-FRACTIONAL BROWNIAN MOTION

377

Sometimes, we write L(t, x) instead of L([0, t], x).
Here is the outline of the analytic method used by Berman  for the calculation
of the moments of local time.
For fixed sample function at fixed t, the Fourier transform on x of L(t, x) is the
function
Z
F (u) =
eiux L(t, x)dx.
R

Using the density of occupation formula, we get
Z t
F (u) =
eiuXs ds.
0

Therefore, we may represent the local time as the inverse Fourier transform of this
function, i.e.,

Z +∞ Z t
1
L(t, x) =
eiu(Xs −x) ds du.
(3.1)
2π −∞
0
4. The Existence and the Joint Continuity of Local Time
The purpose of this section is to present sufficient conditions for the existence
of the local time of gsfBm. Furthermore, using the LND approach, we show that
the local time of gsfBm have a jointly continuous version.
Theorem 4.1. Assume H ∈ (0, 2) and K ∈ (1, 2) such that HK ∈ (0, 2). On
each (time-)interval [a, b] ⊂ [0, ∞), the gsfBm admits a local time which satisfies
Z
L2 ([a, b], x)dx &lt; ∞
a.s.
R

For the proof of Theorem 4.1, we need the following lemma. This result on the
regularity of the increments of gsfBm will be the key for the existence and the
regularity of local times.
Lemma 4.2. Assume H ∈ (0, 2) and K ∈ (1, 2) such that HK ∈ (0, 2). There
exists δ &gt; 0 and, for any integer p ≥ 2, there exists a constant 0 &lt; Cp &lt; +∞,
such that
p
E[StH,K − SsH,K ]p ≥ Cp |t − s| 2 HK ,
(4.1)
for all s, t ≥ 0 such that |t − s| &lt; δ.
Proof. By virtue of (2.1) and the elementary inequality (a + b)2 ≥ 12 a2 − b2 , we
obtain
1
E[StH,K − SsH,K ]2 ≥ E[StHK − SsHK ]2 − C32 (K)E[XtKH − XsKH ]2 .
2
Then (2.3) implies that
E[StH,K − SsH,K ]2 ≥ CH,K |t − s|HK − C32 (K)CK |tH − sH |2 .
Now, we distinguish two cases:
1) Case H ∈ (0, 1). We have |tH − sH | ≤ |t − s|H . Then
i
h
E[StH,K − SsH,K ]2 ≥ |t − s|HK CH,K − C32 (K)CK |t − s|H(2−K) .

378

AISSA SGHIR

Since 1 &lt; K &lt; 2, we can choose δ &gt; 0 small enough such that for all t, s ≥ 0 with
|t − s| &lt; δ, we have
h
i
CH,K − C32 (K)CK |t − s|H(2−K) &gt; 0.
Indeed, it suffices to choose

δ&lt;

1/H(2−K)
CH,K
∧1
.
C32 (K)CK

Finally,
E[StH,K − SsH,K ]2 ≥ C|t − s|HK ,
with |t − s| &lt; δ and

h
i
C = CH,K − C32 (K)CK δ H(2−K) .

Since S H,K is a centered Gaussian process, then the proof of this case is done.
2) Case H ∈ (1, 2). Making use of the theorem on finite increments for the
function x 7→ xH , there exists ξ ∈ (s, t) such that
|tH − sH | = H|ξ|H−1 |t − s|
≤ C|t − s|.
Then

E[StH,K − SsH,K ]2 ≥ |t − s|HK CH,K − CCK |t − s|2−HK .

Therefore the same arguments used in the proof of case 1 give case 2.
This completes the proof of Lemma 4.2.

Proof of Theorem 4.1. It is well known by Berman  that, for a jointly measurable zero-mean Gaussian process X := {X(t) ; t ∈ [0, T ]} with bounded variance,
the variance condition
Z TZ T
−1/2
E[X(t) − X(t)]2
dsdt &lt; ∞
0

0

is sufficient for the local time L(t, u) of X to exist on [0, T ] a.s. and to be square
integrable as a function of u. For any [a, b] ⊂ [0, ∞), and for I = [a0 , b0 ] ⊂ [a, b]
such that |b0 − a0 | &lt; δ, according to (4.1), we have,
Z Z
Z Z

HK
H,K
H,K
2 −1/2
E[S
(t) − S
(s)]
dsdt &lt; C
|t − s|− 2 dsdt.
I

I

I

I

The last integral is finite because 0 &lt; HK &lt; 2. Then the gsfBm possesses, on any
interval I ⊂ [a, b] with length |I| &lt; δ, a local time which is square integrable as
function of u. Finally, since [a, b] is a finite interval, we can obtain the local time
n
on [a, b] by a patch-up procedure, i.e. we partition
Pn [a, b] into ∪i=1 [ai−1 , ai ], such
that |ai − ai−1 | &lt; δ, and define L([a, b], x) = i=1 L([ai−1 , ai ], x), where a0 = a
and an = b.

Proposition 4.3. Assume H ∈ (0, 1) and K ∈ (1, 2) such that HK ∈ (0, 1).
Then the gsfBm is LND on [0, T ].

THE GENERALIZED SUB-FRACTIONAL BROWNIAN MOTION

379

Proof. By virtue of (2.1), we have
[StH,K − SsH,K ] = [StHK − SsHK ] + C3 (K)[XtKH − XsKH ].
Therefore, the elementary inequality (a + b)2 ≥ 21 a2 − b2 implies that

p
p
X
X
1
− StH,K
] ≥ V ar 
V ar 
uj [StH,K
uj [StHK
− StHK
]
j
j−1
j
j−1
2
j=1
j=1

p
X
−C32 (K)V ar 
uj [XtKH − XtKH ] .
j

j−1

j=1

According to Remark 3.2, the sfBm S HK is LND on [0, T ], then there exists two
constants δ and Cp such that for any t0 = 0 &lt; t1 &lt; t2 &lt; ... &lt; tp &lt; T with
tp − t1 &lt; δ, we have

p
p

X
X
H,K 
V ar 
uj [StH,K

S
]

C
u2j V ar StHK
− StHK
p
tj−1
j
j
j−1
j=1

j=1

−pC32 (K)

p
X

.
u2j V ar XtKH − XtKH
j

j−1

j=1

Moreover (2.3) and the fact that H ∈ (0, 1), implies that

p
X
V ar 
uj [StH,K
− StH,K
]
j
j−1
j=1

≥ Cp

p
X

u2j |tj − tj−1 |HK − pC32 (K)CK

j=1

p
X

u2j |tj − tj−1 |2H

j=1

p
h
iX
2
H(2−K)
u2j |tj − tj−1 |HK .
≥ Cp − pC3 (K)CK δ
j=1

E[StH,K − SsH,K ]2 ≤ 2E[StHK − SsHK ]2 + 2C32 (K)E[XtKH − XsKH ]2
≤ CH,K |t − s|HK + 2C32 (K)CK |tH − sH |2
≤ [CH,K + 2C32 (K)CK δ H(2−K) ]|t − s|HK
≤ C(H, K, δ)|t − s|HK .
Therefore, it suffices now to choose
1

H(2−K)
Cp
˜
δ&lt;
∧δ
pC32 (K)CK
and to consider
C=

1
[Cp − pC32 (K)CK δ˜H(2−K) ].
C(H, K, δ)

This with Proposition 3.1 complete the proof of Proposition 4.3.

380

AISSA SGHIR

Now, we are in position to give the main result of this section.
Theorem 4.4. Assume H ∈ (0, 1) and K ∈ (1, 2) such that HK ∈ (0, 1) and
let δ the constant appearing in Lemma 4.2. For any integer p ≥ 2 there exists a
constant 0 &lt; Cp &lt; ∞ such that, for any t ≥ 0, any h ∈ (0, δ), all x, y ∈ R, and
any 0 &lt; ξ &lt; 1−HK
2HK ,
E[L(t + h, x) − L(t, x)]p ≤ Cp

hp(1−HK)
,
Γ(1 + p(1 − HK))

(4.2)

E[L(t + h, y) − L(t, y) − L(t + h, x) + L(t, x)]p
hp(1−HK(1+ξ))
≤ Cp |y − x|pξ
.
Γ(1 + p(1 − HK(1 + ξ)))

(4.3)

Proof. We will prove only (4.3), the proof of (4.2) is similar. It follows from (3.1)
that for any x, y ∈ R, t, t + h ≥ 0 and for any integer p ≥ 2,
E[L(t + h, y) − L(t, y) − L(t + h, x) + L(t, x)]p
Z
Z Y
p
p
p
Pp
Y
Y
1
i j=1 uj SsH,K
−iyuj
−ixuj
j
=
[e
−e
]×E e
duj
dsj .
(2π)p [t,t+h]p Rp j=1
j=1
j=1
Using the elementary inequality |1 − eiθ | ≤ 21−ξ |θ|ξ for all 0 &lt; ξ &lt; 1 and any
θ ∈ R, we obtain
E[L(t + h, y) − L(t, y) − L(t + h, x) + L(t, x)]p ≤ (2ξ π)−p p!|x − y|pξ
Z
t&lt;t1 &lt;...&lt;tp &lt;t+h

Z

p
Y
Rp j=1

p
p
p
X
Y
Y
H,K
|uj | E exp i
uj Stj
duj
tj ,
ξ

j=1

j=1

(4.4)

j=1

where in order to apply the LND property of gsfBm, we replaced the integration
over the domain [t, t + h] by over the subset t &lt; t1 &lt; ... &lt; tp &lt; t + h. We deal now
with the inner multiple integral over the u0 s. Change the variables of integration
by mean of the transformation
uj = vj − vj+1 , j = 1, ..., p − 1; up = vp .
Then the linear combination in the exponent in (4.4) is transformed according to
p
X

uj StH,K
=
j

j=1

where t0 = 0. Since S
(4.4) has the form

p
X

vj (StH,K
− StH,K
),
j
j−1

j=1

H,K

is a Gaussian process, the characteristic function in

p
X



1
exp − V ar 
vj (StH,K
− StH,K
) .
j
j−1
2
j=1

(4.5)

Since |x − y|ξ ≤ |x|ξ + |y|ξ for all 0 &lt; ξ &lt; 1, it follows that
p
Y
j=1

|uj |ξ ≤

p−1
Y
j=1

(|vj |ξ + |vj+1 |ξ )|vp |ξ .

(4.6)

THE GENERALIZED SUB-FRACTIONAL BROWNIAN MOTION

381

Moreover,
last product is at most equalPto a finite sum of 2p−1 terms of the
Qp the ξε
p
form j=1 |xj | j , where j = 0, 1 or 2 and j=1 j = p.

2
H,K
Let us write for simply σj2 = E StH,K

S
. Combining the result of
t
j
j−1
Proposition 4.3, (4.5) and (4.6), we get that the integral in (4.4) is dominated by
the sum over all possible choices of ( 1 , ..., m ) ∈ {0, 1, 2}m of the following terms

Z
Z Y
p
p
p
X
Y
C
p
ξ j
2 2

|vj | exp −
v σ
dtj dvj ,
2 j=1 j j j=1
t&lt;t1 &lt;...&lt;tp &lt;t+h Rp j=1
where Cp is the constant given in Proposition 4.3. The change of variable xj = σj vj
converts the last integral to

Z
Z Y
p
p
p
p
Y
X
Y
C
p
σ −1−ξ j dt1 ....dtp ×
|xj |ξ j exp −
x2j 
dxj .
2 j=1
t&lt;t1 &lt;...&lt;tp &lt;t+h j=1
Rp j=1
j=1
Let us denote

p
p
Y
X
C
p
|xj |ξ j exp −
x2j 
dxj .
J(p, ξ) =
2 j=1
Rp j=1
j=1
Z

p
Y

Consequently
E[L(t + h, y) − L(t, y) − L(t + h, x) + L(t, x)]p
Z
m
Y
−1−ξ j
≤ J(p, ξ)Cp |y − x|pξ
σj
dt1 ....dtp .

(4.7)

t&lt;t1 &lt;...&lt;tp &lt;t+h j=1

According to (4.1), for h sufficiently small, namely 0 &lt; h &lt; inf (δ, 1), we have
E[StH,k
− StH,K
]2 ≥ C|ti − tj |HK ,
i
j

∀ ti , tj ∈ [t, t + h].

It follows that the integral on the right hand side of (4.7) is bounded, up to a
constant, by
Z
p
Y
(tj − tj−1 )−HK(1+ξ j ) dt1 ...dtp .
(4.8)
t&lt;t1 &lt;...&lt;tp &lt;t+h j=1

Since, (tj − tj−1 ) &lt; 1, for all j ∈ {2, ..., p}, we have
(tj − tj−1 )−HK(1+ξ j ) ≤ (tj − tj−1 )−HK(1+2ξ) ,

∀ j ∈ {0, 1, 2}.

1
Since by Hypothesis 0 &lt; ξ &lt; 2HK
− 12 , the integral in (4.8) is finite. Moreover, by
an elementary calculation, for all p ≥ 1, h &gt; 0 and bj &lt; 1,
Qp
Z
p
P
Y
j=1 Γ(1 − bj )
p− p
b
−bj
j
j=1
Pp
(sj − sj−1 ) ds1 ...dsp = h
,
Γ(1 + h − j=1 bj )
t&lt;s1 &lt;...&lt;sp &lt;t+h j=1

where s0 = t. It follows that (4.8) is dominated by
Cp

hp(1−HK(1+ξ))
,
Γ(1 + p(1 − HK(1 + ξ))

382

where

AISSA SGHIR

Pp

j=1 j

= p. Consequently

E[L(t+h, y)−L(t, y)−L(t+h, x)+L(t, x)]p ≤ Cp |y−x|pξ

hp(1−HK(1+ξ))
.
Γ(1 + p(1 − HK(1 + ξ)))

Remark 4.5. Using the fact that L(0, x) = 0 a.s for any x ∈ R and (4.3) by
changing t + h by t and t by 0, we get
E[L(t, x) − L(t, y)]p ≤ Cp

|x − y|pξ
.
Γ(1 + p(1 − HK(1 + ξ)))

Acknowledgment. The author would like to thank the anonymous referee for
References
1. Bardina, X and Bascompte, D.: A decomposition and weak approximation of the subfractional Brownian motion, Arxiv: 0905-4360. (2009).
2. Bardina, X and Es-Sebaiy, K.: An extension of bifractional Brownian motion, Comm. Stoch.
Analy. 5 (2011), no. 2, 333–340.
3. Berman, S. M.: Local times and sample function properties of stationary Gaussian processes,
Trans. Amer. Math. Soc. 137 (1969), 277–299.
4. Berman, S. M.: Local nondeterminism and local times of Gaussian processes, Indiana Univ.
Math. J. 23 (1973), 69–94.
5. Bojdecki, T, Gorostiza, L and Talarczyk, A.: Sub-fractional Brownian motion and its relation
to occupation times, Statist. Probab. Lett. 69 (2004), 405–419.
6. Geman, D and Horowitz, J.: Occupation densities, Ann. Prob. 8 (1980), no. 1, 1–67.
7. Mandelbrot, B and Ness, J. W. V.: Fractional Brownian motion, fractional Gaussien noises
and applications, SIAM Reviews. 10 (1968), no. 4, 422–437.
8. Mendy, I.: On the local time of sub-fractional Brownian motion, Ann. math. Blaise. Pascal.
17 (2010), no. 2, 357-374.
9. Ruiz de Ch´
avez, J and Tudor, C.: A decomposition of sub-fractional Brownian motion, Math.
Rep. (Bucur). 11(61) (2009), no. 1, 67–74.
´partement de Mathe
´matiques et Informatique, Faculte
´ des Sciences,
Aissa Sghir: De
´lisation Stochastique et De
´terministe et URAC 04, B.P. 717, OuLaboratoire de Mode
jda 60000, Maroc