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DOI 10.1515/rose-XXXX

Random Oper. Stoch. Equ. 2014; 22 (2):95–105

Research Article
Aissa Sghir

A self-similar Gaussian process
Abstract: In this paper we introduce and study a self-similar Gaussian process denoted by S H,K with
parameters H ∈ (0, 1) and K ∈ [0, 1]. This process generalize the well known fractional Brownian motion
introduced by Mandelbrot and Van Ness (10), sub-fractional Brownian motion introduced by Bojdecki
et al. (2) and the Gaussian process introduced by Lei and Nualart (9) in order to obtain a decomposition
in law of Bifractional Brownian motion.
Keywords: Besov spaces, Sub-fractional Brownian motion, Fractional Brownian motion, P-variation,
Invariance principle, Chung’s type law of iterated logarithm.
MSC 2010: 60B12, 60G18.
Aissa Sghir: Faculté des Sciences Oujda, Laboratoire de Modélisation Stochastique, et Déterministe et URAC 04, B.P.
717. Maroc, e-mail: semastai@hotmail.fr
Communicated by: V. Girko

1 Introduction
The fractional Brownian motion B H := {BtH ; t ≥ 0}, (fBm for short), was introduced by Mandelbrot
and Van Ness (10). It is the unique continuous centered Gaussian process, starting from zero, with
covariance function
1
R(t, s) = [tH + sH − |t − s|H ],
(1)
2
where H ∈ (0, 2). The self-similarity and stationarity of the increments are two main properties for
which fBm enjoyed success as modeling tool in finance and telecommunications. An extension of Brownian motion, (Bm for short), which preserves many properties of fBm, 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. (2). 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)
2
where H ∈ (0, 2). Bojdecki et al. (2) have showed by using (2) that sfBm is a self-similar process with
index H
2 and its increments satisfy for all s ≤ t:
E(StH − SsH )2 = −2H−1 (tH + sH ) + (t + s)H + (t − s)H ,

(3)

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

∀ H ∈ (0, 1]

(4)

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

∀ H ∈ [1, 2).

(5)

On the other hand, Ruiz de Chavez and Tudor (14) have obtained the following equality in law
StH = BtH + C(H)XtH ,
where H ∈]0, 1[, C(H) =



H
2Γ(1−H) ,

XtH =

∫ +∞
0

(1 − e−θt )θ−

H+1
2

dWθ and Bm W and fBm B H are

independent. The Gaussian process X H was introduced by Lei and Nualart (9) in order to obtain a

96

Aissa Sghir, A self-similar Gaussian process

decomposition of Bifractional Brownian motion, (bfBm for short), into the sum of a transformation of
X H and a fBm. Notice that bfBm is a continuous centered Gaussian process, starting from zero, with
covariance function
1
s(t, s) = K [(tH + sH )K − |t − s|HK ],
2
where H ∈ (0, 2) and K ∈ (0, 1]. The case K = 1 corresponds to fBm.
Remark 1.1. i)− Notice that for H = 1, both fBm and sfBm are Bm with covariance function
r(t, s) =

1
[t + s − |t − s|].
2

ii)− H
2 is called the Hurst parameter of fBm and sfBm.
iii)− The increments of fBm and sfBm are not independent except in the Bm case. The dependence
structure of the increments is modeled by parameter H.
iv)− sfBm does not have stationary increments, but this property is replaced by (4) and (5).
v)− Lei and Nualart (9) have proved that the trajectories of X H are infinitely differentiable on (0, ∞)
and absolutely continuous on [0, ∞).
Remark 1.2. Using integration by parts, we get the following form of the covariance function of X H
T (t, s) =

Γ(1 − H) H
[t + sH − (t + s)H ].
H

Now we are ready to introduce and justify the definition and the existence of our process.
Definition 1.3. The process S H,K := {StH,K ; t ≥ 0}, with parameters H ∈ (0, 1) and K ∈ [0, 1] is a
centered Gaussian process, starting from zero, with covariance function
U (t, s) =

1 H
[t + sH − K(t + s)H − (1 − K)|t − s|H ].
2

Theorem 1.4. The covariance function U is symmetric and positive-definite. Moreover, S H,K has the
following equality in law


StH,K = 1 − KBtH + C(H) KXtH ,
(6)
where H ∈ (0, 1), K ∈ (0, 1) and Bm W and fBm B H are independents.
Proof. It suffices to see that
U (t, s) = (1 − K)R(t, s) + KC 2 (H)T (t, s).

Remark 1.5. i)− S H,K is self-similar process with index H
2.
ii)− For H = 1, U (t, s) = (1 − K)r(t, s).
iii)− For K = 0, U (t, s) = R(t, s).
iv)− For K = 21 , U (t, s) = 12 S(t, s).
v)− For K = 1, U (t, s) = C 2 (H)T (t, s).
vi)− The above mentioned properties make S H,K a possible candidate for models which involve long
dependence, self-similarity and non stationarity.

Lemma 1.6. Let T > 0 fixed. For all 0 ≤ t, s ≤ T , there exists two constants 0 < CH,K < CH,K
< ∞,
such that

CH,K |t − s|H ≤ E(StH,K − SsH,K )2 ≤ CH,K
|t − s|H .
(7)

Proof. This lemma is an immediate consequence of (6) and the elementary inequalities (a + b)2 ≤
2(a2 + b2 ), and (a + b)2 ≥ 12 a2 − b2 .

Aissa Sghir, A self-similar Gaussian process

97

Now, we look for the α-variations of S H,K . As consequence of the ergodic theorem and the scaling
2
property of the fBm, it is easy to show, (see Rogers (13)), that the fBm has a H
-variations equals to
H
CH t, where CH = E(|ξ| ) and ξ is a standard normal random variable. Then, (6) allows us to obtain
2
the H
-variations of S H,K . The proof is similar to that given by Lei and Nualart (9) in case of bfBm.
2
Proposition 1.7. The S H,K with parameters H ∈ (0, 1) and K ∈ (0, 1) has a H
-variations equals to
1
H
(1 − K) H CH t, where CH = E(|ξ| ) and ξ is a standard normal random variable.

The following result shows that S H,K satisfies the same form of Chung’s law for the fBm and sfBm. It
follows by the same arguments used by Mendy (11) in case of sfBm.
Proposition 1.8. Assume that H ∈ (0, 1) and K ∈ (0, 1). Then the following Chung’s law of iterated
logarithm hold almost surely for the S H,K
lim inf

δ→0


|StH,K − SsH,K |
= 1 − KCH ,
H
(δ/
log
|
log(δ)|)
s∈[t,t+δ]
sup

where CH is the constant appearing in the Chung’s law of fBm , (see Monrad and Rootzen (12)).
The remainder of this paper is organized as follows: In the next section, we will present some basic fact
about Besov spaces and some applications. In the last section, we will prove the existence, the joint
continuity and the Hölder regularity of the local time of S H,K . We will also give the pointwise Hölder
exponent of local time of S H,K .

2 The Functional Framework
In this section, we will present a brief survey of Besov spaces. For more details, we refer the reader to
Boufoussi (4) and Ciesielski et al. (7).
We denote by Lp ([0, 1]), 1 ≤ p < +∞, the space of Lebesgue integrable real-valued functions defined
on [0, 1] with exponent p. The modulus of continuity of a Borel function f : [0, 1] → R in Lp ([0, 1]) norm
is defined by
ωp (f, t) = sup ∥∆h f ∥p ,
0≤h≤t

where
∆h f (x) = 1[0,1−h] (x)[f (x + h) − f (x)].
ωµ,ν
Bp,∞
,

The Besov space, denoted by
is a non-separable Banach space of real-valued continuous functions
f on [0, 1], endowed with the norm
ωp (f, t)
,
0<t≤1 ωµ,ν (t)

ω

µ,ν
∥f ∥p,∞
= ∥f ∥p + sup

where

1
ωµ,ν (t) = tµ (1 + log( ))ν ,
t

for any 0 < µ < 1 and ν > 0.
Let {φn = φj,k , j ≥ 0, k = 1, ..., 2j } be the Schauder basis. The decomposition and the coefficients
of continuous functions f on [0, 1], in this basis are respectively given as follows
f (t) =




Cn (f )φn (t),

n=0

and



 C0 (f ) = f (0), C1 (f ) = f (1) − f (0),
n = 2j + k, j ≥ 0 , k = 1, ..., 2j ,

 C (f ) = f = 2 2j (2f ( 2k−1 ) − f ( 2k−2 ) − f ( 2k )).
n
j,k
2j+1
2j+1
2j+1

98

Aissa Sghir, A self-similar Gaussian process
ω

µ,ν
We consider the separable Banach subspace of Bp,∞
defined as follows

ω

µ,ν
Bp,∞

,0

ω

µ,ν
= {f ∈ Bp,∞

/ ωp (f, t) = o(ωµ,ν (t)) (t ↓ 0)}.
ω

,0

µ,ν
It is known from Ciesielski et al. (7) that the subspace Bp,∞
corresponds to sequences (fj,k )j,k such
that
 j
 p1
1
1
2
2−j( 2 −µ+ p ) ∑
|fj,k |p  = 0.
lim
j→+∞
(1 + j)ν

k=1

For the proof of our results, we need the following tightness and Kolmogorov criteria in the subspace
ωµ,ν ,0
Bp,∞
, proved by Ait Ouahra et al. (1) and Boufoussi et al. (6).
Theorem 2.1. Let {Xtn : t ∈ [0, 1]}n≥1 be a sequence of stochastic processes satisfying:
(i) X0n = for all n ≥ 1.
(ii) For all p ≥ 2, there exists a constant 0 < Cp < ∞ such that
E|Xtn − Xsn |p ≤ Cp |t − s|pµ ,
ω

µ,ν
Then, the sequence {Xtn : t ∈ [0, 1]} is tight in Bp,∞

,0

∀ t, s ∈ [0, 1].

for all 0 < µ < 1 and p > max(µ−1 , ν −1 ).

Theorem 2.2. If the increments of a process {Xt : t ∈ [0, 1]} satisfy for all p ≥ 2, there exists a constant
0 < Cp < ∞ such that
E|Xtn − Xsn |p ≤ Cp |t − s|pµ ,
∀ t, s ∈ [0, 1].
ω

µ,ν
Then, the trajectories of X belong almost surely to the subspace Bp,∞
−1
−1
max(µ , ν ).

,0

for all 0 < µ < 1 and p >

As applications of (7) and the fact that S H,K is a Gaussian process, we get the following regularity of
S H,K .
ω H ,ν ,0

2
Lemma 2.3. The trajectory t → StH,K belongs almost surely to Besov space Bp,∞

for any ν > p1 .

We end this section by an invariance principle of the S H,K in a class of Besov space. Let (Yn )n≥1 be a
stationary sequence of centered Gaussian random variables. It is known from Boufoussi and Lakhel (5)
that if
n ∑
n

E(Yk Yl ) ∼ (1 − K)nH , as n → ∞,
(8)
k=1 l=1

then the sequence of processes
Ztn =

[nt]
1 ∑
H

n2

Yk ,

k=1

ω H ,ν ,0

2
converge in law, as n → ∞, to 1 − KBtH in the Besov space Bp,∞
, for all ν > p1 . On the other hand
H
H
1
it is easy to see by using the self-similarity of the process X that H X[nt]
= XH
[nt] in law. Then, the
n

sequence of processes
Wtn =

2

n

√ [nt]
C(H) K ∑ H
H
(Xk − Xk−1
),
H
n2
k=1

ω H ,ν ,0

2
, for all ν >
converge in law, as n → ∞, to C(H) KXtH in the Besov space Bp,∞
get the following result.

1
p.

Therefore, we

Theorem 2.4. Let H ∈ (0, 1), K ∈ (0, 1) and let (Yn )n≥1 be a stationary sequence of centered Gaussian
random variables satisfying (8). If Z n and W n are independents, then the sequence of processes Ztn +Wtn
ω H ,ν ,0

2
converge in law, as n → ∞, to a S H,K of parameters H and K, in Bp,∞

, for all ν > p1 .

Aissa Sghir, A self-similar Gaussian process

99

3 Local time of S H,K
For a complete survey on local time and fractional derivative, we refer the reader 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 measurable modification
that satisfies the occupation density formula : for every Borel set B ⊂ R+ and every measurable function
f : R → R+ ,


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 times.
For fixed sample function at fixed t, the Fourier transform on x of L(t, x) is the function

F (u) =
eiux L(t, x)dx.
R

Using the density of occupation formula, we get

∫t
eiuXs ds.

F (u) =
0

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


+∞ ∫ t

1
 eiu(Xs −x) ds du.
L(t, x) =

−∞

0

Let LH (t, x) be the local time of S H,K . Following the same method used by I. Mendy (11) in case of
sfBm by using the concept of local nondeterminism, (LND for short), for Gaussian process introduced
by Berman (3), we get the following Hölder continuities of LH (t, x).
Lemma 3.1. Assume that H ∈ (0, 1) and K ∈ (0, 1). For any integer p ≥ 1, there exists a constant
0 < Cp,H,K < ∞ such that, for any 0 ≤ t ≤ T, any h ∈]0, δ[, any ξ < 1 ∧ 1−H
2H and all x, y ∈ R,
E[LH (t + h, x) − LH (t, x)]2p ≤ Cp,H,K h2p(1−H) ,

(9)

E[LH (t + h, x) − LH (t, x) − LH (t + h, y) + LH (t, y)]2p ≤ Cp,H,K |x − y|2pξ h2p(1−H(1+ξ)) .

(10)

Remark 3.2. i)− Using (10) and the fact that LH (0, x) = 0, we get the spacial Hölder regularity of
LH (t, x).
sup E[LH (t, x) − LH (t, y)]2p ≤ Cp,H,K |x − y|2pξ .
0≤t≤T
H

ii)− The local time L (t, x) satisfies the scaling property
{
}
{
}
H
H
(d) λ1− 2 LH (t, x)
LH (λt, xλ 2 )
t≥0

t≥0

,

∀λ > 0,

100

Aissa Sghir, A self-similar Gaussian process

and the occupation density formula

∫t


f (SsH,K )ds

f (x)LH (t, x)dx,

=
R

0

for any bounded or nonnegative Borel function f .
The following result concerns a limit theorem for occupation times of S H,K .
Theorem 3.3. Let f : R → R a bounded Borel function with compact support. Then the sequence of
processes


∫nt


1
H,K
f
(S
)ds
,
s
 n1− H2

0

t≥0

converge in law, as n → +∞, to the process


∫

f (x)dxLH (t, 0)


R

ω

1−H,ν
in the Besov space Bp,∞

,0

,

t≥0

for all ν > p1 .

Proof. We set
Ant

=

∫nt

1

f (SsH,K )ds.

H

n1− 2

0

By virtue of the the occupation density formula and the scaling property of the local time of S H,K , we
get the following equality in law

x
Ant =
f (x)LH (t, H )dx.
n2
R

Then we have the convergence of the finite-dimensional distribution. For the tightness, applying the
Hölder inequality, there exists a constant 0 < C < ∞ such that

E|Ant



Ans |2p

 2p

p′ ∫

H

p
L (t, xH ) − LH (s, xH


|f (x)| dx
≤CE

n2
n2
S


≤C



S


p′

|f (x)| dx

S
1
2p

2p

) dx

2p
p′





x
x
E LH (t, H ) − LH (s, H
2
n
n2

2p

) dx,

S

1
p′

where
+ = 1 and S = supp(f ).
Thanks to (9), there exists a constant 0 < Cp,H,K < ∞ such that
E|Ant − Ans |2p ≤ Cp,H,K |t − s|1−H .
This completes the proof of Theorem 3.3.
The last result of this work is the following lemma. Its proof is similar to that given by Mendy (11).
Lemma 3.4. The following lower bounds for the moduli of continuity of local time of S H,K holds almost
surely
LH (t + δ, x) − LH (t, x)
1
≤ lim sup sup 1−H
.
δ→0
2CH,K
(log log(δ −1 ))H
x∈R δ

Aissa Sghir, A self-similar Gaussian process

101

References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]

M. Ait Ouahra, B. Boufoussi and E. Lakhel (2002). Théorèmes limites pour certaines fonctionnelles associées aux
processus stables dans une classe d’espaces de Besov Standard, Stoch. Stoch. Rep. 74, 411–427.
T. Bojdecki, L. Gorostiza, A. Talarczyk (2004). Sub-fractional Brownian motion and its relation to occupation times,
Statist. Probab. Lett. 69, 405–419.
S. M. Berman (1973). Local nondeterminism and local times of Gaussian processes. Indiana University Mathematical
Journal 23, 69–94.
B. Boufoussi (1994). Espaces de Besov, caractérisations et applications. Thèse de l’Université de Henri Poincaré,
Nancy, France.
B. Boufoussi and E. Lakhel (2001). Weak convergence in Besov spaces to fractional Brownian motion. Prob theo.
Stat. C. R. Acad. Sci. Paris, 333, Serie I, p. 39–44.
B. Boufoussi, E. Lakhel and M. Dozzi (2005). A Kolmogorov and Tightness Criterion in Modular Besov Spaces and
an Application to a Class of Gaussian Processes, Stochastic Analysis and Applications. 23 (4), 665–685.
Z. Ciesielski, G. Kerkyacharian and B. Roynette (1993). Quelques espaces fonctionnels associés à des processus
Gaussiens, Studia Math. 107 (2), 171–204.
D. Geman and J. Horowitz (1980). Occupation densities, Ann. Prob. 8 (1), 1–67.
P. Lei and D. Nualart (2009). A decomposition of the bifractional Brownian motion and some applictions, Statist.
probab. Lett. 779, 619–624.
B. Mandelbrot and J. W. V. Ness (1968). Fractional Brownian motion, fractional Gaussien noises and applications.
SIAM Reviews. 10 (4), 422–437.
I. Mendy (2010). On the local time of sub-fractional Brownian motion. Ann. math. Blaise. Pascal. 17 (2), 357–374.
D. Monrad and H. Rootzen (1995). Small values of Gaussian processes and functional laws of the iterated logarithm.
Probab. Th. Rel. Fields. 101, 173–192.
L. C. G. Rogers (1997). Arbitrage with fractional Brownian motion, Math. Finance. 7, 95–105.
J. Ruiz de Chávez and C. Tudor (2009). A decomposition of sub-fractional Brownian motion, Math. Rep. (Bucur). 11
(61) (1), 67–74.

Received June, 2012; accepted December 12, 2013.

102

Aissa Sghir, A self-similar Gaussian process

−0.5
−2.0

−1.5

−1.0

fBm

0.0

0.5

1.0

Path of a fractional Brownian motion : N= 1000, H=0.4

0.0

0.2

0.4

0.6

0.8

1.0

time

Fig. 1. Path of fBm: H = 0.4.

0.4
0.0

0.2

fBm

0.6

0.8

Path of a fractional Brownian motion : N= 1000, H=1.6

0.0

0.2

0.4

0.6
time

Fig. 2. Path of fBm: H = 1.6.

0.8

1.0

Aissa Sghir, A self-similar Gaussian process

0.0
−0.6 −0.4 −0.2

incrfBm

0.2

0.4

0.6

Path of increments of fBm : N= 5000, H=0.4

0.0

0.2

0.4

0.6

0.8

1.0

time

Fig. 3. Increments of fBm: H = 0.4.

0.0
−0.5

sfBm

0.5

Path of sfBm with parameters : N= 1000, H=1

0.0

0.2

0.4

0.6
time

Fig. 4. Path of Bm: H = 1.

0.8

1.0

103

104

Aissa Sghir, A self-similar Gaussian process

0.0
−1.0

−0.5

sfBm

0.5

1.0

1.5

Path of sfBm with parameters : N= 1000, H=0.4

0.0

0.2

0.4

0.6

0.8

1.0

time

Fig. 5. Path of sfBm: H = 0.4.

−0.3
−0.6

−0.5

−0.4

sfBm

−0.2

−0.1

0.0

Path of sfBm with parameters : N= 1000, H=1.6

0.0

0.2

0.4

0.6
time

Fig. 6. Path of sfBm: H = 1.6.

0.8

1.0

Aissa Sghir, A self-similar Gaussian process

−0.0005
−0.0015

incrfBm

0.0005 0.0010

Path of increments of sfBm : N= 5000, H=1.8

0.0

0.2

0.4

0.6
time

Fig. 7. Increments of sfBm: H = 1.8.

0.8

1.0

105


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