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Stochastic Analysis and Applications

ISSN: 0736-2994 (Print) 1532-9356 (Online) Journal homepage: http://www.tandfonline.com/loi/lsaa20

Continuity in law with respect to the Hurst index
of some additive functionals of sub-fractional
Brownian motion
M. Ait Ouahra & A. Sghir
To cite this article: M. Ait Ouahra & A. Sghir (2017): Continuity in law with respect to the Hurst
index of some additive functionals of sub-fractional Brownian motion, Stochastic Analysis and
Applications, DOI: 10.1080/07362994.2017.1309979
To link to this article: http://dx.doi.org/10.1080/07362994.2017.1309979

Published online: 13 Apr 2017.

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Date: 23 April 2017, At: 22:17

STOCHASTIC ANALYSIS AND APPLICATIONS
, VOL. , NO. , –
http://dx.doi.org/./..

Continuity in law with respect to the Hurst index of some
additive functionals of sub-fractional Brownian motion
M. Ait Ouahraa and A. Sghirb
a

Faculté des Sciences, Laboratoire de Modélisation Stochastique et Déterministe et URAC , Université
Mohammed Premier, Oujda, Maroc; b Faculté des Sciences Meknès, Équipe EDP et Calcul Scientifique, Université
Moulay Ismail, Zitoune, Maroc

ABSTRACT

ARTICLE HISTORY

In this article, first, we prove some properties of the sub-fractional
Brownian motion introduced by Bojdecki et al. [Statist. Probab. Lett.
69(2004):405–419]. Second, we prove the continuity in law, with respect
to small perturbations of the Hurst index, in some anisotropic Besov
spaces, of some continuous additive functionals of the sub-fractional
Brownian motion. We prove that our result can be obtained easily, by
using the decomposition in law of the sub-fractional Brownian motion
given by Bardina and Bascompte [Collect. Math. 61(2010):191–204] and
Ruiz de Chavez and Tudor [Math. Rep. 11(2009):67–74], without using the
result of Wu and Xiao [Stoch. Proc. Appl. 119(2009):1823–1844] by connecting the sub-fractional Brownian motion to its stationary Gaussian process through Lamperti’s transform. This decomposition in law leads to a
better understanding and simple proof of our result.

Received  February 
Accepted  March 
KEYWORDS

Anisotropic Besov spaces;
limit theorem; tightness;
continuity in law; fractional
Brownian motion;
sub-fractional Brownian
motion; local time; fractional
derivative; slowly varying
function
MATHEMATICS SUBJECT
CLASSIFICATION

; E; B

1. Introduction
Let {SH ; H ∈ (0, 2)} be a family of sub-fractional Brownian motions (sfBm). We prove in
Theorem 5.1 that the laws of this family converge weakly, in some Besov spaces, to that of SH0
when H tends to H0 ∈ (0, 2). These results justify the use of SHˆ as a model in applied situations
when the true value of the parameter H is unknown and Hˆ is some estimation of H. Motivated by this purpose, Jolis and Viles [14] proved that the law of the local times of fractional
Brownian motions (fBm), with Hurst parameter H ∈ (0, 2) converges weakly, in the continuous functions space, to that of the local time of the fBm with Hurst parameter H0 ∈ (0, 2),
when H tends to H0 . They mainly used the techniques based on Fourier transforms developed by Berman [3], jointly with a study of the covariance matrix and the correlation of the
increments of fBm, when H belongs to a neighborhood of H0 . The result of Jolis and Viles [14]
was generalized by Wu and Xiao [26] for some anisotropic Gaussian random fields satisfying
some condition A based on the recently developed properties of sectorial local nondeterminism for anisotropic Gaussian random fields, see [25]. Their result include fractional Brownian
sheets and some anisotropic Gaussian fields with stationary increments whose spectral densities satisfy the spectral condition in [26], Proposition 2.2, and in particular, the result of Jolis
and Viles [14] for fBm (see [26, Remark 2.3]).
CONTACT M. Ait Ouahra
©  Taylor & Francis Group, LLC

ouahra@gmail.com

Mathematics, BV Mohammed VI, B.P. , Oujda , Morocco.

2

M. AIT OUAHRA AND A. SGHIR

In this article, we prove that our result can be obtained easily, by using the decomposition
in law of the sub-fractional Brownian motion given by Bardina and Bascompte [2] and Ruiz de
Chavez and Tudor [22], without using the result of Wu and Xiao [26] by connecting the subfractional Brownian motion to its stationary Gaussian process through Lamperti’s transform
[16]. This decomposition in law leads to a better understanding and simple proofs of our
results.
In Theorem 5.2 and Theorem 5.3, we prove that the local time and the generalized fractional derivative with kernel depending on slowly varying function of the local time of sfBm
satisfy the same property of continuity in law of sfBm.
Our result is new, in the continuous functions space, for the generalized fractional derivatives with kernel depending on slowly varying function of the local time of the fBm. The fBm
BH := {BtH ; t ≥ 0} was introduced by Mandelbrot and Van Ness [19]. It is the unique continuous centered Gaussian process, starting from zero, with covariance function
1

E BtH BHs = [t H + sH − |t − s|H ],
2
where H ∈ (0, 2). H2 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. An extension of
Brownian motion (Bm), which preserves many properties of fBm, but not the stationarity of
the increments, is the so called sfBm SH := {StH ; t ≥ 0}. It was introduced by Bojdecki et al.
[5] as a continuous centered Gaussian process, starting from zero, with covariance function:


1
E StH SHs = t H + sH − [(t + s)H + |t − s|H ],
2

(1)

where H ∈ (0, 2).
They have showed, by using (1), that sfBm satisfies the following properties:
r Self-similarity:
H


H
∀c > 0.
Sct ; t ≥ 0 d c 2 StH ; t ≥ 0 ,

r Second moment of increments: For all s ≤ t

2

∀ H ∈ (0, 1] (2)
(t − s)H ≤ E StH − SHs ≤ (2 − 2H−1 )(t − s)H ,


2
∀ H ∈ [1, 2).
(3)
(2 − 2H−1 )(t − s)H ≤ E StH − SHs ≤ (t − s)H ,

On the other hand, Bardina and Bascompte [2] and Ruiz de Chavez and Tudor [22]
obtained for H ∈ (0, 1), the following decomposition in law of sfBm:
StH d BtH + C1 (H )XtH ,
where C1 (H ) =
H




H
, XtH
2 (1−H )

=

+∞
0

(1 − e−θt )θ −

H+1
2

(4)

dWθ , the standard Bm W and the fBm

B are independent.
The process X H was introduced by Lei and Nualart [17] in order to obtain a decomposition in law of the bifractional Brownian motion with parameters H ∈ (0, 1) and K ∈ (0, 1).
Moreover, Bardina and Bascompte [2] proved that the process X H is Gaussian, centered, and
its covariance function is
(1−H ) H
[t + sH − (t + s)H ], ∀H ∈ (0, 1),
H
H H
E(Xt Xs ) = (2−H )
[(t + s)H − t H − sH ], ∀H ∈ (1, 2).
H(H−1)

STOCHASTIC ANALYSIS AND APPLICATIONS

3

Remark 1.1.
i) Notice that for H = 1 both processes fBm and sfBm coincide with Bm.
ii) sfBm is neither a semimartingale, nor a Markov process unless H = 1.
iii) sfBm does not have stationary increments, but this property is replaced by (2) and (3).
iv) The above mentioned properties make sfBm a possible candidate for models which
involve long range dependence, self-similarity and non-stationarity of increments.
We end this section by the definition of the generalized fractional derivative with kernel
depending on slowly varying function, refer to [4].
Definition 1.2. We say that a measurable function l : R+ → R+ is slowly varying at infinity,
if for all positive t, we have
lim

x→+∞

l(tx)
= 1.
l(x)

We are interested in the behavior of l at +∞, then we can assume for example that l is
bounded on each interval of the form [0, a], where a > 0.
For any γ ∈]0, β[ and g ∈ C β ∩ L1 (R), (C β is the Hölder space with order β > 0), we
define
+∞
1
g(x ± y) − g(x)
l(y)
dy.
K±l,γ g(x) :=
(−γ ) 0
y1+γ
Note that
l(x) = o(xβ ) as x → +∞ for any β > 0, (see [4, Proposition 1.3.7]), so when γ > 0,
+∞
l,γ
β
1
we get 1 yl(y)
1+γ < +∞. Consequently, if g ∈ C ∩ L (R) for some γ ∈]0, β[, then K± g(x)
is a defined bounded continuous function.
We put
K l,γ := K+l,γ − K−l,γ ,
and is called the generalized fractional derivative.
Remark 1.3. The classical fractional derivative Dγ corresponds to l ≡ 1. These additive functionals appeared in some limit theorems discussed in [27] for Bm, in [24] for fBm, and in [12]
for symmetric stable process. Moreover, according to Samko et al. [23], if f ∈ C β ∩ L1 (R)
then D f ∈ C β−γ , where D ∈ {Dγ± , Dγ }.
The remainder of this article is organized as follows: In the next section, we present some
basic facts about anisotropic Besov spaces. In Section 3, we prove some properties of sfBm. In
Section 4, we prove some basic facts about local time of sfBm. Finally, the last section contains
the statement and proofs of our theorems of continuity in law.
In the sequel C, Ca,b , C(η, H0 ) and Cm denotes finite and positive constants which be different even when they vary from one line to the next.

2. The functional framework
We first present a brief survey of Besov spaces. For more details, refer to [6, 9].
Let I = [0, 1]. We denote by L p (I), 1 ≤ p < +∞, the space of Lebesgue integrable realvalued functions defined on I with exponent p. The modulus of continuity of a function
f : I → R in L p (I) norm is defined for all h ∈ R by

4

M. AIT OUAHRA AND A. SGHIR

ω p ( f , t ) = sup
h f
p ,
0≤h≤t

where
h f (t ) = 1[0,1−h] (t )[ f (t + h) − f (t )].
ω

μ,ν
Definition 2.1. The Besov space denoted by B p,∞
, 1 ≤ p < +∞, is a non-separable Banach
space of real-valued continuous functions f on I, endowed with the norm

ωp( f , t )
,
0<t≤1 ωμ,ν (t )

ω

μ,ν
=
f
p + sup

f
p,∞

where


ν
1
ωμ,ν (t ) = t μ 1 + log
,
t

for any 0 < μ < 1 and ν > 0.
Ciesielski et al. [9] have shown by using the techniques of constructive approximation of
functions, that Besov spaces are isomorphic to spaces of real sequences. This characterization
allows to prove some regularities of the generalized fractional derivative of local time of αsymmetric stable process with index 1 < α ≤ 2, in some Besov spaces, (see [21, Lemma 2.4]).
The Schauder basis on I is defined by

ϕ1 (t ) = t1[0,1] (t ),
⎨ ϕ0 (t ) = 1[0,1] (t ),
n = 2 j + k, j ≥ 0, k = 1, . . . , 2 j ,
j

ϕ j,k (t ) = ϕn (t ) = 21− 2 (2 j t − k),
where (u) = u1[0, 1 ] (u) + (1 − u)1] 1 ,1] (u).
2
2
The decomposition and the coefficients of continuous functions f on I, in this basis are
respectively given as follows
f (t ) =




Cn ( f )ϕn (t ).

n=0

and


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






j

Cn ( f ) = 2 2 2 f 2k−1
− f 2k−2
− f 2 2k
.
j+1
2 j+1
2 j+1
ω

μ,ν
, 1 ≤ p < +∞, defined as follows
We consider the separable Banach subspace of B p,∞

ω

,0

ω

μ,ν
μ,ν
B p,∞
= { f ∈ B p,∞
/ ω p ( f , t ) = o(ωμ,ν (t )) (t ↓ 0)}.

The following characterization theorem is due to Ciesielski et al. [9, Theorem III.2].
ω

,0

μ,ν
, 1 ≤ p < +∞, corresponds to the sequences (Cn ( f ))n such
Theorem 2.2. The subspace B p,∞
that

⎤ 1p
1
1
2 j+1
2− j( 2 −μ+ p ) ⎣
lim
|Cn ( f )| p ⎦ = 0.
j→+∞ (1 + j)ν
j

n=2 +1

ω

,0

μ,ν
We need the following tightness criterion in B p,∞
, 2 ≤ p < +∞, (see [1, Lemma 4.3]).

STOCHASTIC ANALYSIS AND APPLICATIONS

5

Theorem 2.3. Let {Xtn ; t ∈ [0, 1]}n≥1 be a sequence of stochastic processes satisfying:
(i) X0n = 0 for all n ≥ 1.
(ii) For all 2 ≤ p < +∞, there exists a constant 0 < Cp < +∞ such that
E|Xtn − Xsn | p ≤ Cp |t − s| pμ ,

∀ t, s ∈ [0, 1],
ω

,0

μ,ν
for all ν > 0 and p >
where 0 < μ < 1. Then, the sequence {X n }n≥1 is tight in B p,∞
−1
−1
max(μ , ν ).

Next, we give the definition of anisotropic Besov spaces. For this, we denote by L p (I 2 ) the
space of Lebesgue integrable functions with exponent 1 ≤ p < ∞. For any function f : I 2 →
R, any h ∈ R, the progressive difference in direction x1 (resp. x2 ), is defined by
h,1 f (x1 , x2 ) = f (x1 + h, x2 ) − f (x1 , x2 ),
h,2 f (x1 , x2 ) = f (x1 , x2 + h) − f (x1 , x2 ).
For any (h1 , h2 ) ∈ R2 , we set
h1 ,h2 f = h1 ,1 ◦ h2 ,2 f ,
2h,i f = h,i ◦ h,i f ,

i = 1.2

For any Borel function f : I 2 → R, such that f ∈ L p (I 2 ), one can measure its smoothness by
its modulus of continuity computed in L p (I 2 ) norm.
For this end, let us define for any (t1 , t2 ) ∈ I 2
ω(1,0).p ( f , t1 ) = sup
h1 ,1 f
p ,
|h1 |≤t1

ω(0,1).p ( f , t2 ) = sup
h2 ,2 f
p ,
|h2 |≤t2

ω(1,1).p ( f , t1 , t2 ) =

sup

|h1 |≤t1 ,|h2 |≤t2


h1 ,h2 f
p .

Definition 2.4. Let 0 < α1 , α2 < 1 and ν ∈ R. The anisotropic Besov space, denoted by
Lip p (α1 , α2 , ν ), 1 ≤ p < +∞, is a non-separable Banach space of real-valued continuous
functions f on I 2 , endowed with the norm
α1 ,α2


f
ωp ν

ω(1,0).p ( f , t1 )
ω(1,0).p ( f , t2 )
ω(1,1).p ( f , t1 , t2 )
+ sup
+ sup
,
α1 ,α2
α1 ,α2
α1 ,α2
(t1 , 1)
(1, t2 )
(t1 , t2 )
0<t1 ≤1 ων
0<t2 ≤1 ων
0<t1 ,t2 ≤1 ων

:=
f
p + sup

where
ωνα1 ,α2 (t1 , t2 )

=

t1α1 t2α2





1
1 + log
t1 t2

ν
.

We consider the separable Banach subspace of Lip p (α1 , α2 , ν ), 1 ≤ p < +∞, defined as
Lip∗p (α1 , α2 , ν ) := { f ∈ Lip p (α1 , α2 , ν ) / ω(1,0).p ( f , t1 ) = o(ωνα1 ,α2 (t1 , 1)) as t1 → 0,
ω(0,1).p ( f , t2 ) = o(ωνα1 ,α2 (1, t2 )) as t2 → 0,

ω(1,1).p ( f , t1 , t2 ) = o(ωνα1 ,α2 (t1 , t2 )) as t1 ∧ t2 → 0},
where t1 ∧ t2 := min(t1 , t2 ).
Now, for any continuous functions f on I 2 , we have the following decomposition
f (t1 , t2 ) =






m=0 max(n,n )=m

Cn,n ( f )ϕn ⊗ ϕn (t1 , t2 ),

6

M. AIT OUAHRA AND A. SGHIR

where Cn,n ( f ) = Cn1 ◦ Cn2 ( f ), with
1
Cn ( f )(t ) = Cn ( f (., t )),
Cn2 ( f )(t ) = Cn ( f (t, .)).
We have the following characterization of anisotropic Besov spaces in terms of the coefficients
of the expansion of a continuous function with respect to a basis which consists of tensor
products of Schauder functions, and we refer to [15, Theorem A.2]. This characterization was
used by Ouahhabi and Sghir [21, Theorem 3.3], to prove some regularities of the generalized
fractional derivative of local time of α-symmetric stable process with 1 < α ≤ 2, in some
anisotropic Besov spaces.
Theorem 2.5. The subspace Lip∗p (α1 , α2 , ν ), 1 ≤ p < +∞, corresponds to the sequences
(Cn,n ( f )) such that

⎤ 1p
2 j+1
− j( 12 −α1 + 1p )

2

|Cn,l ( f )| p ⎦ = 0,
l = 0, 1
lim
j→+∞ (1 + j)ν
j
n=2 +1

lim

j→+∞

lim

2

j, j →+∞

2

− j( 12 −α2 + 1p )

(1 + j)ν

− j( 12 −α1 + 1p ) − j ( 12 −α2 + 1p )

2
(1 + j + j )ν




j+1

2





⎤ 1p

j+1

2


|Cl,n ( f )| p ⎦ = 0,

n=2 j +1


j +1
2

l = 0, 1

⎤ 1p

|Cn,n ( f )| p ⎦ = 0.

n=2 j +1 n =2 j +1

We need also the following tightness criterion in the subspace Lip∗p (α1 , α2 , ν ), 2 ≤ p <
+∞, (see [8, Lemma 2.5]).
n
; (s, t ) ∈ [0, 1]2 }n≥1 be a sequence of random fields satisfying:
Theorem 2.6. Let {Xs,t
n
n
(i) X.,0 = X0,. = x for some x ∈ R,
(ii) For all 2 ≤ p < +∞, there exists a constant 0 < Cp < +∞ such that
n
n
n
p
α1 p
E|Xs,t
− Xsn ,t − Xs,t
|t − t |α2 p ,
+ Xs ,t | ≤ C p |s − s |

∀ t, s ∈ [0, 1],

where 0 < α1 , α2 < 1. Then, the sequence {X n }n≥1 is tight in Lip∗p (α1 , α2 , ν ), for all ν >
2
.
p

3. On the sub-fractional Brownian motion
The first main property of sfBm given in this section is inspired from the well known
local asymptotic self-similarity property, (see [10, Proposition 1], for some multifractional
Gaussian processes and [18, Proposition 5], for the multifractional Brownian motion).
Proposition 3.1. Let H0 ∈ (0, 1). For any H ∈ [H0 , 1) and every t ≥ 0, as ε → 0, the sequence
of tangent processes of sfBm
H

St+εu − StH
; u ∈ [0, 1] ,
H
ε2
ωH



2
converge in law, to the fBm {BHu ; u ∈ [0, 1]}, in the Besov space B p,∞
−1
−1
max(μ , ν ).

,0

for all ν > 0 and p >

STOCHASTIC ANALYSIS AND APPLICATIONS

7

Proof. Let ZεH denotes the tangent process of the sfBm StH , that is,
ZεH (u) =

H
− StH
St+εu
H

ε2

.

The tightness of {ZεH } follows easily from (2), ZεH (0) = 0 and Theorem 2.3. Indeed, by using
(2) and that ZεH is a Gaussian process, for all p ≥ 2, there exists Cp > 0 such that
p

H
E ZεH (u) − ZεH (v ) ≤ Cp |u − v| p 2 .
Therefore, by choosing p such that p H2 > 1, Theorem 2.3 gives the tightness.
It remains to show the convergence of the finite-dimensional distributions. The Gaussian
process is completely determined by its mean and the covariance function. For this, for any
u, v ∈ [0, 1], put


Aε (u, v ) = E ZεH (u)ZεH (v ) .
It is enough to show that
1
lim Aε (u, v ) = R(u, v ) = [uH + v H − |u − v|H ].
2

ε→0

Clearly, (1) gives



1 −(2t + ε(u + v ))H + (2t + εu)H + (2t + εv )H − (2t )H
.
Aε (u, v ) = R(u, v ) +
2
εH

The l’Hôpital’s rule and the fact that H ∈ (0, 1) implies that
−(2t + ε(u + v ))H + (2t + εu)H + (2t + εv )H − (2t )H
= 0.
ε→0
εH
So the desired estimate is obtained, which completes the proof of Proposition 3.1.
lim



The second main result of this section is a consequence of (4).
Lemma 3.2. Let [a, b] ⊂ (0, 1). There exists a constant Ca,b > 0, such that for all H, H0 ∈
[a, b],

2
sup E StH − StH0 ≤ Ca,b |H − H0 |2 .
(5)
t∈[0,1]

Proof. The proof of this lemma will be decomposed in three steps. Without loss of generality,
we assume H < H0 .
Step 1. According to [7, Lemma 3.1] where they used the harmonisable representation of
fBm or [18, Theorem 4 ] where they used the moving average representation of fBm, there
exists a constant Ca,b > 0, such that
2

sup E BtH − BtH0 ≤ Ca,b |H − H0 |2 .
(6)
t∈[0,1]

Step 2. We show that there exists a constant Ca,b > 0, such that
2

sup E XtH − XtH0 ≤ Ca,b |H − H0 |2 .

(7)

t∈[0,1]

Indeed


E[XtH − XtH0 ]2 =

+∞
0

(1 − e−θt )2 (θ −

H+1
2

− θ−

H0 +1
2

)2 dθ.

8

M. AIT OUAHRA AND A. SGHIR

Using the fact that H ∈ (0, 1) and the theorem on finite increments for the function x →
x+1
θ − 2 for x ∈ (H, H0 ), there exists ξ ∈ (H, H0 ) such that
2

E XtH − XtH0

1
|H − H0 |2
4



+∞

(1 − e−θt )2 ln2 (θ )θ −(ξ +1) dθ
1

2
≤ C|H − H0 | sup
(1 − e−θt )2 ln2 (θ )θ −(ξ +1) dθ +
=

0

t∈[0,1]



0
1

≤ C|H − H0 |2 sup
t∈[0,1]

(1 − e−θt )2 ln2 (θ )θ −(b+1) dθ +

0



+∞

−θt 2

(1 − e

) ln (θ )θ
2

−(ξ +1)




1
+∞

(1 − e−θt )2 ln2 (θ )θ −(a+1) dθ



1

≤ C|H − H0 |2 {I1 + I2 },

where



1

I1 = sup
t∈[0,1]

and

0



+∞

I2 = sup
t∈[0,1]


(1 − e−θt )2 ln2 (θ )θ −(b+1) dθ ,

(1 − e

−θt 2

) ln (θ )θ
2

−(a+1)

dθ .

1

Let us deal with I1 .
Using the elementary inequalities: 0 ≤ 1 − e−x ≤ x and |x ln(x)| ≤ 1e for any x ∈]0, 1], and
the fact that 0 < b < 1, we get
2 1

1
1−b 2
2
1−b
1−b 2
I1 ≤
θ 2 ln θ 2
θ ln (θ )dθ ≤
dθ < ∞.
1−b
0
0
Now, we deal with I2 .
Using the fact that |1 − e−x | ≤ 1 for any x ∈ R+ , 0 < a < 1 and integrating twice by parts
we obtain I2 < ∞.
Consequently,

2
sup E XtH − XtH0 ≤ Ca,b |H − H0 |2 .
t∈[0,1]

Step 3. We show that there exists a constant 0 < Ca,b < ∞ such that
|C1 (H ) − C1 (H0 )|2 ≤ Ca,b |H − H0 |2 .

(8)

Recall that C1 (.) appears in (4).
It is well known that the function x → (x) satisfies the Euler’s reflection formula:
π
∀H ∈ (0, 1),
(H ) (1 − H ) =
sin(πH )
then


C1 (H ) =

H (H ) sin(πH )
.


(9)

Using the theorem on finite increments for the function x → C1 (x) in (9) for x ∈ (H, H0 ),
and the fact that x → (x) has continuous derivatives of all order for x > 0, gives the desired
estimate.
Now we return to the proof of Lemma 3.2.

STOCHASTIC ANALYSIS AND APPLICATIONS

9

In view of (4) and the elementary inequality (a + b)2 ≤ 2(a2 + b2 ), we obtain
2


2
E StH − StH0 ≤ 2E[BtH − BtH0 ]2 + 2E C1 (H )XtH − C1 (H0 )XtH0


2
2
≤ 2E BtH − BtH0 + 4C12 (H )E[XtH − XtH0 ]2 + 4|C1 (H ) − C1 (H0 )|2 E XtH0 .
Since x → (x) is continuous on [a, b], then, with (9), x → C1 (x) is continuous on [a, b],
and there exists a constant Ca,b > 0 such that
C12 (H ) ≤ Ca,b

(10)

Therefore, combining (6), (7), (8), (10), and the fact that X H0 is a Gaussian process, the proof

of Lemma 3.2 is completed.

4. On the local time of sfBm
For a complete survey on local time of Gaussian processes, we refer to [13] and the references
therein.
We denote by LH (t, x) the local time of sfBm. It is the Radon–Nikodym derivative of the
occupation measure of sfBm on B defined as
μB (A) = λ{s ∈ B : SHs ∈ A},

∀A ∈ B(R),

where λ is the one-dimensional Lebesgue measure on R+ .
Mendy [20] proved by using the concept of local nondeterminism for Gaussian process
introduced by Berman [3], the existence and the joint continuity of the local time LH (t, x),
H ∈ (0, 1). However, to prove the tightness in our Theorem 5.2, we give an adaptation of the
result of Mendy [20] because we will need a precise evaluation of the constants appearing in
it, when H belongs to a neighborhood of H0 .
Theorem 4.1. Let H0 ∈ (0, 1). Then, there exists 0 < η < min{H0 , 1 − H0 } such that the family {SH ; H ∈ (H0 − η, H0 + η) ⊂ (0, 1)} satisfies:
1) For all m ≥ 2, there exist Cm > 0 and δ > 0 (depending on m, η and H0 ) such that for
any h ∈ (0, δ) and any x ∈ R,
E[LH (t + h, x) − LH (t, x)]m ≤ Cm hm(1−(H0 +η)) .

(11)

2) For all even m ≥ 2, there exists Cm > 0 and δ > 0 (depending on m, η and H0 ) such that
for any h ∈ (0, δ), all x, y ∈ R and any 0 < ξ < 1 ∧ ( 2(H01+η) − 12 ),
E[LH (t + h, x) − LH (t, x) − LH (t + h, y) + LH (t, y)]m
≤ Cm hm(1−(H0 +η)(1+ξ )) |x − y|mξ .

(12)

Following the same arguments used by Mendy [20], we see that the key ingredient to prove
Theorem 4.1 is the following lemma.
Lemma 4.2. Let H0 ∈ (0, 1). There exists 0 < η < min{H0 , 1 − H0 } such that, for any H ∈
(H0 − η, H0 + η) and all even m ≥ 2, there exist Cm > 0 and δ > 0 (depending on m, η, and
H0 ) such that


m
m
"
#




(13)
u j StHj − StHj−1 ⎠ ≥ Cm
u2j Var StHj − StHj−1 ,
Var ⎝
j=1

j=1

for any ordered t0 = 0 < t1 < t2 < . . . < tm < 1 with tm − t1 < δ and (u1 , u2 , . . . , um ) ∈ Rm .

10

M. AIT OUAHRA AND A. SGHIR

Proof. According to Mendy [20], by virtue of the decomposition (4) and the elementary
inequality (a + b)2 ≥ 12 a2 − b2 we obtain,


m
"
#

Var ⎝
u j StHj − StHj−1 ⎠
j=1



m






m
"
#

u j [BtHj − BtHj−1 ]⎠ − C12 (H )Var ⎝
u j XtHj − XtHj−1 ⎠

1
Var ⎝
2
j=1
j=1


m
m
"
#




1
H
H
2


u j Bt j − Bt j−1
u2j Var XtHj − XtHj−1 .
− mC1 (H )
≥ Var
2
j=1
j=1


Moreover, we have

2
E XtH − XsH ≤ K(H )|t − s|2 ,
where


K(H ) = sup
s∈[0,1]

+∞

e−2sθ θ 1−H dθ

0

is the constant appeared in Mendy [20] to prove Lemma 3.2.
We will prove that the constants C1 (H ) and K(H ) can be taken bounded independent of
the parameter H for any H ∈ (H0 − η, H0 + η).
Let us deal with C1 (H ): We need the following result concerning the Weierstrass’s form of
the function x → (x), for all x > 0,
x

1 & ek
,
(x) = γ x
xe k=1 1 + xk

where

'
γ = lim

n→+∞

n

1
k=1

k

(
− ln(n) > 0

is the Euler’s constant. This result combined with the elementary inequality 1 + x ≤ ex for all
x ∈ R, implies that
C12 (H ) =

(H0 + η)(1 − H0 + η) γ (1−H0 +η)
H

e
:= C(η, H0 ).
2 (1 − H )
2

Notice that a similar estimation of C12 (H ) can be obtained by using the fact that the function
x → (x) is continuous on [H0 − η, H0 + η].
Now, we deal with K(H ): We have

1
+∞
−2sθ 1−H
−2sθ 1−H
e θ
dθ +
e θ

K(H ) ≤ sup
s∈[0,1]

0



≤ sup
s∈[0,1]

1
1

0

:= C(η, H0 ).

e−2sθ θ 1−H0 −η dθ +



+∞
1


e−2sθ θ 1−H0 +η dθ

STOCHASTIC ANALYSIS AND APPLICATIONS

11

On the other hand, according to [14, Lemma 3.2], for any H ∈ (H0 − η, H0 + η), there exist
Am > 0 and δ˜ > 0 (depending on m, η, and H0 ) such that


m
m




u j [BtHj − BtHj−1 ]⎠ ≥ Am
u2j Var BtHj − BtHj−1 ,
Var ⎝
j=1

j=1

for any ordered t0 = 0 < t1 < t2 < . . . < tm < 1 with tm − t1 < δ˜ and (u1 , u2 , . . . , um ) ∈
Rm .
Finally, we obtain


m
m
m



u j [StHj − StHj−1 ]⎠ ≥ Am
u2j |t j − t j−1 |H − mC(η, H0 )
u2j |t j − t j−1 |2
Var ⎝
j=1

j=1

j=1

m


≥ Am − mC(η, H0 ) max |t j − t j−1 |2−H0 −η
u2j |t j − t j−1 |H ,
j=1

because |t j − t j−1 | < 1.
We choose δ small enough such that for all j ∈ {1, . . . , m} and |t j − t j−1 | < δ, we get
Am − mC(η, H0 ) max |t j − t j−1 |2−H0 −η > 0,
because H0 + η < 1.
For this, it suffices to choose

δ<

Am
mC(η, H0 )

2−H1 −η
0

˜
∧ δ.

We note
Cm = Am − mC(η, H0 )δ 2−H0 −η > 0.
Finally,



Var ⎝

m

j=1


u j [StHj



StHj−1 ]⎠



≥ Am − mC(η, H0 )δ

2−H0 −η

m




u2j Var StHj − StHj−1 ,

j=1

where we have used the last inequality (2). So, the lemma is proved.
Finally, Lemma 4.2 with the same arguments used by Mendy [20], give the proof of

Theorem 4.1.

5. Stability in law of some additive functionals of sfBm
In this section, we state and prove the continuity of the laws of sfBm and of its some additive
functionals, in the index H ∈ (0, 1).
Theorem 5.1. Given a fixed H0 ∈ (0, 2). The family {SH ; H ∈ (0, 2)} of sfBm converges in
ωμ ,ν ,0
law to that of the sfBm SH0 when H tends to H0 , in the Besov space B p,∞1 , 1 ≤ p < +∞, for
any ν > 1p , where μ1 = H02−η .
Theorem 5.2. Given a fixed H0 ∈ (0, 1). The family {LH ; H ∈ (0, 1)} of local times of sfBm
SH converges in law to LH0 , when H tends to H0 , in the anisotropic Besov space Lip∗p (α1 , α2 , ν ),
1 ≤ p < +∞, for any ν > 1p , where α1 = (1 − (H0 + η)(1 + ξ )) and α2 = ξ .

12

M. AIT OUAHRA AND A. SGHIR

Theorem 5.3. Let 0 < γ < ξ < 1 ∧ ( 2(H01+η) − 12 ) and K ∈ {K±l,γ , K l,γ }. The family
{KLH ; H ∈ (0, 1)} of generalized fractional derivatives of local times of sfBm SH converges in law to KLH0 , when H tends to H0 , in the anisotropic Besov space Lip∗p (α1 , α2 − γ , ν ),
1 ≤ p < +∞, for any ν > 1p .
Remark 5.4. The subspace Lip∗p (α1 , α2 − γ , ν ) is appeared in [21, Theorem 3.3] to prove
some regularities of the generalized fractional derivative of local time of α-symmetric stable process with index 1 < α ≤ 2, in some anisotropic Besov spaces. A similar result for sfBm
can be obtained easily by using Theorem 4.1.
Proof of Theorem 5.1. The tightness follows from (2) and (3) and the fact that SH0 = 0. Indeed,
by using (2) and (3) and the fact that sfBm is a Gaussian process, there exists a constant Cp > 0
(depending on p, η and H0 ), such that
p

E[StH − SHs ] p ≤ Cp |t − s| 2 (H0 −η) .
Therefore, by choosing p such that 2p (H0 − η) > 1, Theorem 2.3 gives the tightness of the
ωμ ,ν ,0
laws of the family {SH ; H ∈ (H0 − η, 1)} in B p,∞1 . sfBm SH is a Gaussian process, then the
convergence of the finite dimensional distributions follows from the fact that the covariance

function of the sfBm SH converge to that of the sfBm SH0 when H → H0 .
Proof of Theorem 5.2. By virtue of Theorem 4.1 and Theorem 2.6, the laws of the family
{LH ; H ∈ (H0 − η, H0 + η)} is tight in Lip∗p (α1 , α2 , ν ). The proof of the convergence of the
finite dimensional distributions is similar to that given in [14] in case of the local time of fBm
and was later generalized in [26] for anisotropic Gaussian random fields. The key ingredients

are the occupation density formula of local time and Theorem 5.1.
To prove the convergence of the finite dimensional distributions in Theorem 5.3, we need
the following lemma. It is a generalization of the result of [11] in case of fractional derivative.
Let E be a Banach space endowed with the norm
.
E . C(R, E) denote the space of continuous functions on R endowed with the sup norm
.
∞,E and C β (R, E) the space of β-Hölder
continuous E-valued functions endowed with the norm

f
∞,β,E :=
f
∞,E +
f
β,E
:= sup
f (x)
E + sup
x

x =y


f (x) − f (y)
E
.
|x − y|β

Lemma 5.5. Let K ∈ {K±l,γ , K l,γ }. Then K is a bounded linear operator from C β (R, E) to
C(R, E).
Proof. Let f be a function in the space C β (R, E). Assume that K = K+l,γ . We have
)
) +∞
1 )
f (x + y) − f (x) )
l,γ
)

K+ f (x)
E =
l(y)
dy)
)
(−γ ) ) 0
y1+γ
E
1
1

f (x + y) − f (x)
E

l(y)
dy
(−γ ) 0
y1+γ

+∞

f (x + y) − f (x)
E
l(y)
dy
+
y1+γ
1

STOCHASTIC ANALYSIS AND APPLICATIONS

13

+∞
Using the fact that l is bounded on [0, 1], 1 yl(y)
1+γ < +∞ and 0 < γ < β we get
1
+∞
1
1
l(y)
l,γ
δ−γ −1

K+ f (x)
E ≤

f
β,E

f
∞,E
y
dy +
dy
(−γ )
(−γ )
y1+γ
0
1
≤ C(γ )(
f
β,E +
f
∞,E )
:= C(γ )
f
∞,β,E ,
hence, the proof is complete.



Proof of Theorem 5.3. Clearly, this result follows easily from Theorem 5.2. and

Lemma 5.5.
Remark 5.6. Notice that our results are new, in the continuous functions space, for the generalized fractional derivatives with kernel depending on slowly varying function of the local
time of the fBm.

Acknowledgments
The authors would like to thank the Editor in chief and the anonymous referee for useful comments.
This article was completed while M. Ait Ouahra was visiting the Department of Statistics and Probability
at Michigan State University (DSPMSU). M. Ait Ouahra would like to express his sincere thanks to the
staff of DSPMSU for generous support and hospitality, especially Prof. Yimin Xiao.

Funding
M. Ait Ouahra was supported by a Fulbright Visiting Scholar grant 2012–2013.

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