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Serials Publications

Communications on Stochastic Analysis
Vol. 7, No. 4 (2013) 523-533

www.serialspublications.com

LOCAL TIME OF A MULTIFRACTIONAL GAUSSIAN PROCESS
AISSA SGHIR*

Abstract. In this paper, a new multifractional Gaussian process is defined
by a integral representation. We prove an approximation in law of this process
and we prove that the covariance function of this process permits us to obtain
a generalization of the sub-fractional Brownian motion [5] by a decomposition
in law into the sum of our process and the standard multifractional Brownian
motion [1]. We prove also the existence and the joint continuity of the local
time of our process.

1. Introduction
Let X

H

:=

{XtH

; t ≥ 0} be the Gaussian process defined by:
Z +∞
H+1
XtH =
(1 − e−θt )θ− 2 dWθ ,
0

where H ∈ (0, 2) and W := {Wt ; t ≥ 0} is a standard Brownian motion. This
process was introduced by Lei and Nualart [10] in order to obtain a decomposition in law of the bifractional Brownian motion [9] and was used later by Bardina
and Bascompte [2] and Ruiz de Chavez and Tudor [11] in order to obtain a decomposition in law of the sub-fractional Brownian motion S H := {StH ; t ≥ 0}
with parameter H ∈ (0, 2). This process was introduced by Bojdecki et al. [5].
It is a continuous centered Gaussian process, starting from zero, with covariance
function:
1
E(StH SsH ) = tH + sH − [(t + s)H + |t − s|H ].
2
In this paper, firstly, we introduce a new multifractional Gaussian process which
generalize the process X H by substituting to the parameter H a function H(.)
such that H(t) ∈ (0, 2) and we prove an approximation in law of this process.
Secondly, we prove that the covariance function of this process permits us to obtain
a generalization of the sub-fractional Brownian motion by a decomposition in law
into the sum of our process and the standard multifractional Brownian motion [1].
We prove also under some assumptions on H(.), the existence, the joint continuity
and the H¨older regularity of the local time of our process. Our idea is inspired
from the work of Boufoussi et al. [6] in case of multifractional Brownian motion.
We use the concept of local nondeterminism for Gaussian process introduced by
Received 2012-8-10; Communicated by the editors.
2010 Mathematics Subject Classification. Primary 60G18; Secondary 60J55.
Key words and phrases. Multifractional Gaussian process, sub-fractional Brownian motion,
decomposition in law, local time, local nondeterminism.
* This research is supported by the Doe Foundation.
523

524

AISSA SGHIR

Berman [4] and the analytic method used by Berman [3] for the calculation of the
moments of local time.
2. The Multifractional Gaussian Process
Definition 2.1. Let H : [0, +∞[→ (0, 2) be a function satisfying: there exists
finite and positive constants β, C2 and C3 such that
C2 |t − s|β ≤ |H(t) − H(s)| ≤ C3 |t − s|β , f or all s, t ≥ 0.

(2.1)

The right hand side of (2.1) means that H(.) is β-H¨
older continuous.
H(t)

Definition 2.2. For t ≥ 0, we denote by X H(t) := {Xt
; t ≥ 0} the multifractional Gaussian process with the multifractional function H(t) defined by:
Z +∞
H(t)+1
H(t)
Xt
=
(1 − e−θt )θ− 2 dWθ ,
0

where the integration is taken in the mean square sense.
An approximation in law of our process can be obtained by the same argument
used by Bardina and Bascompte [2] in case of the process X H .
Proposition 2.3. Let {Ns ; s ≥ 0 } be a standard Poisson process and let θ ∈
(0, π) ∪ (π, 2π). Then the processes,
Z +∞

H(t)+1
2
XεH(.) =
(1 − e−st )s− 2 cos θN 2s2 ds, t ∈ [0, T ]
ε
ε 0
and

Z +∞

H(t)+1
2
(1 − e−st )s− 2 sin θN 2s2 ds, t ∈ [0, T ]
ε
ε 0
converge in law, in the sense of the finite dimensional distributions, toward two
independent processes with the same law that X H(.) .
˜ εH(.) =
X

The following result concerns the regularity property of X H(.) .
Lemma 2.4. Let T &gt; 0 fixed. Assume that H(.) is H¨
older continuous with exponent β ∈ (0, 1). There exists a finite and positive constant C, such that for all
t, s ∈ [0, T ], we have
H(t)

E|Xt

− XsH(s) |2 ≤ C|t − s|2β .

To prove Lemma 2.4, we need the following result.
Lemma 2.5. Let T &gt; 0 fixed and [µ, ν] ⊂ (0, 2). There exists two positive and
1
2
finite constants Cµ,ν
and Cµ,ν
such that for all λ, λ0 ∈ [µ, ν],
0

1
2
Cµ,ν
|λ − λ0 |2 ≤ sup E[Xtλ − Xtλ ]2 ≤ Cµ,ν
|λ − λ0 |2 .
t∈[0,T ]

Proof. We have
0

Z

E[Xtλ − Xtλ ]2 =
0

+∞

(1 − e−θt )2 (θ−

λ+1
2

− θ−

λ0 +1
2

)2 dθ.

(2.2)

MULTIFRACTIONAL GAUSSIAN PROCESS

525

Using the theorem on finite increments, ([7], Corollary 2.6.1), for the function
x+1
x → θ− 2 ; x ∈ (λ, λ0 ), there exists ξ ∈ (λ, λ0 ) such that
λ+1
1
ξ+1
λ0 +1
− 2
− θ− 2 = | ln(θ)|θ− 2 |λ − λ0 |,
θ
2
therefore
Z +∞
1
0 2
λ
λ0 2
(1 − e−θt )2 ln2 (θ)θ−(ξ+1) dθ
E[Xt − Xt ] = |λ − λ |
4
0
nZ 1
1
≤ |λ − λ0 |2 sup
(1 − e−θt )2 ln2 (θ)θ−(ξ+1) dθ
4
t∈[0,T ]
0
Z +∞
o
+
(1 − e−θt )2 ln2 (θ)θ−(ξ+1) dθ

1
|λ − λ0 |2 sup
4
t∈[0,T ]

nZ

1
1

(1 − e−θt )2 ln2 (θ)θ−(ν+1) dθ

0

Z

+∞

+

o
(1 − e−θt )2 ln2 (θ)θ−(µ+1) dθ .

1

Consequently

0

2
sup E[Xtλ − Xtλ ]2 ≤ Cµ,ν
|λ − λ0 |2 .
t∈[0,T ]

Changing the sup by the inf, we complete the proof of Lemma 2.5.

Remark 2.6. The theorem on finite increments for the function x → ex , implies
that
ey − ex &lt; ey (y − x),
∀y &gt; x,
therefore, there exists a finite and positive constant C1 such that for any H ∈ (0, 1)
and all s, t ∈ [a, b] ⊂ (0, +∞), we have
E[XtH − XsH ]2 ≤ C1 |t − s|2 ,
where

Z

+∞

C1 =

(2.3)

e−2aθ θ1−H dθ.

0

Now we are ready to prove Lemma 2.4.
Proof of Lemma 2.4. Using the elementary inequality (a + b)2 ≤ 2(a2 + b2 ), we
obtain
i
h
H(t)
H(t)
− XsH(t) |2 + E|XsH(t) − XsH(s) |2 .
E|Xt
− XsH(s) |2 ≤ 2 E|Xt
The first term on the right hand side of the previous expression is the variance of
the increments of the process X H of parameter H(t). Therefore (2.3) implies that
H(t)

E|Xt

− XsH(t) |2 ≤ C|t − s|2 .

Moreover, by virtue of (2.2), we have
E|XsH(t) − XsH(s) |2 ≤ C|H(t) − H(s)|2
≤ C|t − s|2β .

526

AISSA SGHIR

Consequently
H(t)

E|Xt

− XsH(s) |2 ≤ C|t − s|2β [1 + |t − s|2(1−β) ]
≤ C|t − s|2β .

The proof of Lemma 2.4 is done.

In the next result, we give the formula of the covariance function our process.
The proof follows the lines of that given by Bardina and Bascompte [2] in case of
the process X H .
Proposition 2.7. The process X H(.) is Gaussian, centered, and its covariance
function is
(
0
0
Γ(1−H 0 ) H 0
[t + sH − (t + s)H ], if H(.) ∈ (0, 1),
0
H(t) H(s)
H
0
E(Xt Xs ) =
0
0
Γ(2−H )
H0
− tH − sH ], if H(.) ∈ (1, 2).
H 0 (H 0 −1) [(t + s)
where H 0 =

H(t)+H(s)
.
2

As application, we obtain a generalization of the sub-fractional Brownian motion S H by a decomposition in law into the sum of our process and the standard
multifractional Brownian motion [1]. The covariance function of this last process
was given by Ayyache et al. [1] as follows:
H(t)

E(Bt

0

0

0

BsH(s) ) = D(H(t), H(s))[tH + sH − |t − s|H ],

where

p
D(x, y) =

Γ(x + 1)Γ(y + 1) sin(πx/2) sin(πy/2)
.
2Γ((x + y)/2 + 1) sin(π(x + y)/4)

Theorem 2.8. The centered multifractional Gaussian S H(.) process with a multifrcational function H(t) ∈ (0, 1) defined by the decomposition in law:
H(t)

q
where C(H(t)) =

St

H(t)
2Γ(1−H(t))

H(t)

= Bt

H(t)

+ C(H(.))Xt

,

and B H(.) and X H(.) are independent, is a gener-

alization of the sub-fractional Brownian motion S H with parameter H ∈ (0, 1).
3. Local Time and Local Nondeterminism
We begin this section by the definition of local time. For a complete survey on
local time, we refer to Geman and Horowitz [8] 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

MULTIFRACTIONAL GAUSSIAN PROCESS

527

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.
R

B

Sometimes, we write L(t, x) instead of L([0, t], x).
Here is the outline of the analytic method used by Berman [3] 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
iu(Xs −x)
L(t, x) =
e
ds du.
(3.1)
2π −∞
0
We end 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 )2 &gt; 0, f or t ∈ J.
(P) :
E[Xt − Xs ]2 &gt; 0, whenever 0 &lt; |t − s| &lt; δ,
To introduce the concept of LND, Berman [4] 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.

(3.2)

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 [4] 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.

528

AISSA SGHIR

4. Existence and Joint Continuity of Local Time
The purpose of this section is to present sufficient conditions for the existence
of the local time of X H(.) . Furthermore, using the LND approach, we show that
the local time of X H(.) have a jointly continuous version.
Theorem 4.1. Assume H(.) satisfying (2.1) with β ∈ (0, 1). On each timeinterval [a, b] ⊂ [0, ∞), X H(.) 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 X H(.) will be the key for the existence and the joint
continuity of local times.
Lemma 4.2. Assume H(.) satisfying (2.1) with β ∈ (0, 1). There exists two
positive and finite constants δ &gt; 0 and C such that
H(t)

E[Xt

− XsH(s) ]2 ≥ C|t − s|2β ,

(4.1)

for all s, t ≥ 0 such that |t − s| &lt; δ.
Proof. By virtue of the elementary inequality (a + b)2 ≥ 21 a2 − b2 , we get
H(t)

E|Xt

− XsH(s) |2 ≥

1
H(t)
E|XsH(t) − XsH(s) |2 − E|Xt
− XsH(t) |2 .
2

Therefore (2.1), (2.2) and (2.3) implies that
"
#
1
Cµ,ν
C2
H(t)

H(s) 2
2(1−β)
E|Xt
− Xs | ≥ |t − s|
− C1 |t − s|
.
2
Since β &lt; 1, we can choose δ &gt; 0 small enough such that for all t, s ≥ 0 with
|t − s| &lt; δ, we have
"
#
1
C2
Cµ,ν
2(1−β)
− C1 |t − s|
&gt; 0.
2
Indeed, it suffices to choose
!1/2(1−β)
1
Cµ,ν
C2
∧1
.
2C1

δ&lt;
Finally,

H(t)

E[Xt
with |t − s| &lt; δ and

− XsH(s) ]2 ≥ C|t − s|2 ,

#
1
Cµ,ν
C2
2(1−β)
− C1 δ
.
C=
2
"

The proof of Lemma 4.2 is done.

MULTIFRACTIONAL GAUSSIAN PROCESS

529

Proof of Theorem 4.1. It is well known by Berman [3] 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
−1/2
H(t)
H(s) 2
E[Xt
− Xs ]
dsdt &lt; C
|t − s|−β dsdt.
I

I

I

I

The last integral is finite, then X
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 on [a, b] by a patch-up
procedure, i.e. we partition
[a, b] into ∪ni=1 [ai−1 , ai ], such that |ai − ai−1 | &lt; δ, and
Pn
define L([a, b], x) = i=1 L([ai−1 , ai ], x), where a0 = a and an = b.

H(.)

Proposition 4.3. Assume H(.) is derivable and H¨
older continuous with β ∈
(0, 1). Then for every ε &gt; 0, and any T &gt; ε, X H(.) is locally LND on [ε, T ].
H(t)

Proof. Notice that X H(.) is a zero mean Gaussian process, and E(Xt )2 &gt; 0 for
all t ∈ [ε, T ]. Moreover, thanks to (4.1) also the second point in (P) is satisfied on
[ε, T ]. It remains to show that X H(.) satisfies (3.2).
We have
H(θ)
σ(Xθ
, θ ≤ s) ⊂ σ(Wθ , θ ≤ s), f or all s ≥ 0.
Then, for any t &gt; s,

H(θ)
H(t)
H(t)
, θ ≤ s ≥ V ar Xt
− XsH(s) /Wθ , θ ≤ s .
V ar Xt
− XsH(s) /Xθ
Rs
H(t)+1
The measurability of 0 (1 − e−θt )θ− 2 dWθ with respect to σ(Wθ , θ ≤ s) and
R +∞
H(t)+1
the fact that s (1 − e−θt )θ− 2 dWθ is independent of σ(Wθ , θ ≤ s), (by the
independence of the increments of the Brownian motion), implies that

H(θ)
H(t)
V ar Xt
− XsH(s) /Xθ
, θ≤s
Z +∞ h
i
H(t)+1
H(s)+1 2

(1 − e−θt )θ− 2 − (1 − e−θs )θ− 2
dθ.
s

Making use of the theorem on finite increments for the function: x → fθ (x) =
H(x)+1
(1 − e−θx )θ− 2
for x ∈ (s, t), there exists ξ ∈ (s, t) such that
Z +∞

H(t)
(fθ0 (x))2 dθ .
V ar Xt
− XsH(s) /XuH(u) , u ≤ s ≥ |t − s|2 inf
x∈[ε,T ]

T

Therefore the relative prediction error Vp given by (3.2) is at least equal to
R

+∞
inf x∈[ε,T ] T (fθ0 (x))2 dθ
R
,
+∞
supx∈[ε,T ] 0 (fθ0 (x))2 dθ

530

AISSA SGHIR

which is bounded away from 0, as t tends to s, and the proof is complete.

Remark 4.4. When applying the LND to the estimation of the moments of local
times of X H(.) , the condition t &gt; ε can be circumvented by slightly adjusting the
arguments. Indeed, we can consider X = X H(.) +Y , where Y is a standard normal
random variable independent of X H(.) . Since the occupation density of X has the
H(t)
same local properties as that of X H(.) and E(Xt )2 = E(Xt )2 + E(Y )2 &gt; 0 for
all t ∈ [0, T ], we can use the LND on the whole interval [0, T ] for X H(.) .
We end this section by the following main result.
Theorem 4.5. Assume H(.) is derivable and satisfying (2.1) with β ∈ (0, 1) and
let δ the constant appearing in Lemma 4.2. For any integer p ≥ 2 there exists a
finite positive constant Cp such that, for any t ≥ 0, any h ∈ (0, δ), all x, y ∈ R,
and any 0 &lt; ξ &lt; 1−β
2β ,
E[L(t + h, x) − L(t, x)]p ≤ Cp

hp(1−β)
,
Γ(1 + p(1 − β))

(4.2)

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

hp(1−β(1+ξ))
.
Γ(1 + p(1 − β(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
P
Y
Z
Z Y
p
p
p
H(s )
Y
1
i p
uj Xsj j
−ixuj
−iyuj
j=1
=
du
dsj .

e
]
×
E
e
[e
j
(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

Z
t&lt;t1 &lt;...&lt;tp &lt;t+h

p
Y
Rp j=1

|uj |ξ E[exp(i

p
X

H(tj )

uj Xtj

j=1

)]

p
Y

duj

j=1

p
Y

tj ,

(4.4)

j=1

where in order to apply the LND property of X H(.) , 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
j=1

H(tj )

uj Xtj

=

p
X
j=1

H(tj )

vj (Xtj

H(t )

− Xtj −1j ),

MULTIFRACTIONAL GAUSSIAN PROCESS

531

where t0 = 0. Since X H(.) is a Gaussian process, the characteristic function in
(4.4) has the form


p
X
1
H(t )
H(t )
(4.5)
exp − V ar 
vj (Xtj j − Xtj −1j ) .
2
j=1
Since |x − y|ξ ≤ |x|ξ + |y|ξ for all 0 &lt; ξ &lt; 1, it follows that
p
Y

|uj |ξ ≤

j=1

p−1
Y

(|vj |ξ + |vj+1 |ξ )|vp |ξ .

(4.6)

j=1

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(t )
H(t )
Let us write for simply σj2 = E Xtj j − Xtj −1j
. Combining the result of
Proposition 3.1, (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
Y
X
C
p
dtj dvj ,
vj2 σj2 
|vj |ξ j exp −
2 j=1
t&lt;t1 &lt;...&lt;tp &lt;t+h Rp j=1
j=1
where Cp is the constant given in Proposition 3.1. The change of variable xj = σj vj
converts the last integral to

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

p
p
X
Y
C
p
ξ j
2

|xj | exp −
J(p, ξ) =
x
dxj .
2 j=1 j j=1
Rp j=1
Z

p
Y

Consequently
E[L(t + h, y) − L(t, y) − L(t + h, x) + L(t, x)]p
Z
p
Y
−1−ξ j
σj
≤ J(p, ξ)Cp |y − x|pξ
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; δ ∧ 1, we have
H(tj )

E[Xtj

H(t )

− Xtj −1j ]2 ≥ C|ti − tj |2β ,

∀ 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 )−β(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 )−β(1+ξ j ) ≤ (tj − tj−1 )−β(1+2ξ) ,

∀ j ∈ {0, 1, 2}.

532

AISSA SGHIR

1
Since by Hypothesis 0 &lt; ξ &lt; 2β
− 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
Pp
Y
j=1 Γ(1 − bj )
Pp
(sj − sj−1 )−bj ds1 ...dsp = hp− j=1 bj
,
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
where

Pp

j=1 j

hp(1−β(1+ξ))
,
Γ(1 + p(1 − β(1 + ξ))

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

hp(1−β(1+ξ))
.
Γ(1 + p(1 − β(1 + ξ)))

Remark 4.6. (1) The process X H has infinitely differentiable trajectories and it is
well-known that in this case the local time does not exist because the occupation
measure is singular.
(2) We believe that the same arguments used in this paper can be used for the
bifractional Brownian motion and the Gaussian process introduced in Sghir [12].
Acknowledgment. The author would like to thank the anonymous referee for
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MULTIFRACTIONAL GAUSSIAN PROCESS

533

´lisation Stochastique et De
´terministe et URAC
AISSA SGHIR: Laboratoire de Mode
´ des Sciences, Oujda, 60000, MAROC
04, Faculte