Real harmonizable multifractional Lévy motions

Real harmonizable multifractional Lévy motions

Céline Lacaux

Université Paul Sabatier UFR MIG, laboratoire de statistique et de probabilités, 118, route de Narbonne, 31062 Toulouse, France Received 23 June 2002; received in revised form 22 July 2003; accepted 5 November 2003

Available online 10 February 2004

Abstract

In this paper, the class of real harmonizable multifractional Lévy motions (in short RHMLMs) is introduced. This class is a generalization of the multifractional Brownian motion (in short MBM) and of the class of real harmonizable fractional Lévy motions. One of its main interest is that it contains some non-Gaussian fields which share many properties with the MBM. RHMLMs have locally Hölder sample paths and their Hölder exponent is allowed to vary along the trajectory. Moreover these fields are locally asymptotically self-similar. The multifractional function can be estimated with the localized generalized quadratic variations.

2004 Elsevier SAS. All rights reserved.

Résumé

Cet article introduit une classe de champs réels appelés champs de Lévy multifractionnaires au moyen d’une représentation harmonisable. Cette classe contient à la fois celle des champs de Lévy fractionnaires et le mouvement Brownien multifractionnaire (MBM en abrégé). Elle fournit notamment des exemples de champs non gaussiens ayant des propriétés semblables à celles du MBM. En particulier, les trajectoires d’un champ de Lévy multifractionnaire sont presque sûrement localement höldériennes. Par ailleurs, l’exposant de Hölder ponctuel peut varier le long d’une trajectoire et est égal à celui trouvé dans le cas du MBM. D’autre part, ces champs sont aussi localement asymptotiquement autosimilaires. Enfin l’étude est complétée par l’identification de la fonction multifractionnaire au moyen des variations quadratiques locales et généralisées.

2004 Elsevier SAS. All rights reserved.

MSC: 60G17; 62G05

Keywords: Local asymptotic self-similarity; Multifractional function; Identification Mots-clés : Autosimilarité locale ; Fonction multifractionnaire ; Identification

1. Introduction

The fractional Brownian motion (in short FBM), introduced by Mandelbrot and Van Ness in [12], provides a powerful model in applied mathematics. The FBM of indexH (0< H <1) is the only centered Gaussian field, vanishing at zero, with stationary increments and which is self-similar with indexH. Its Hurst exponent governs all

E-mail address: lacaux@lsp.ups-tlse.fr (C. Lacaux).

0246-0203/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2003.11.001

its properties. As an example, the pointwise Hölder exponent ofBH is almost surely equal toH. However the field of applications is restricted because the FBM is a Gaussian field. Moreover since the pointwise Hölder exponent ofBH is almost surely constant, the FBM can not be used to model some phenomena for which regularity varies in space.

In order to extend the range of applications, the multifractional Brownian motion (in short MBM) has been introduced independently in [13,8]. This Gaussian field does not have any more stationary increments. Moreover the MBM does not remain self-similar but is locally asymptotically self-similar (in short lass). Let us recall how a lass field is defined in [8]. A field(X(u))_u∈Rd is locally asymptotically self-similar with multifractional functionh if for allx∈R^d,

lim

λ→0⁺

X(x+λu)−X(x) λ^h(x)

u∈R^d (d)=

T (u)

u∈R^d,

where the non-degenerate field(T (u))_u∈Rd is called the tangent field at pointx.Furthermore the Hölder exponent of the MBM is allowed to vary along the trajectory. Then the MBM is used as a toy-model for modeling mountains because it allows to take into account erosion phenomena. Ayache and Lévy Véhel (see [2]) have generalized the MBM in order to have more irregularity. Their fields remain lass Gaussian fields, like those studied in [4,5].

Nevertheless in practice, one often observes non-Gaussian phenomena, see for instance [11,16] for image modeling, and thus can not use these models. Real harmonizable fractional Lévy motions (in short RHFLMs), introduced by Benassi, Cohen and Istas in [6], make up a class of lass fields which includes non-Gaussian fields and the FBM.

However their increments are stationary and their Hölder exponent is almost surely equal to a constant.

The main aim of this paper is to introduce some non-Gaussian lass fields whose Hölder exponent varies along the trajectory. These fields will be called real harmonizable multifractional Lévy motions (in short RHMLMs).

Let us recall that a RHFLMX_H of indexH (0< H <1) is defined as the stochastic integral:

XH(x)=

R^d

e^−ix·ξ−1

ξ^d/2+H L(dξ ), (1)

whereξis the Euclidean norm ofξ andL(dξ )is a Lévy random measure that has moments of every order. Then RHMLMs are defined by substituting in (1) to the constant parameterH a locally Hölder functionh. WhenL(dξ ) is a Brownian measureW (dξ ), one obtains the harmonizable representation of the MBM.

Moreover, in order to identify the multifractional functionh, one can build some estimators based on generalized quadratic variations, method used in [10,7,5].

The next section is devoted to the construction of RHMLMs. In Section 3, the regularity of the sample paths of RHMLMs and the asymptotic self-similarity property are studied. The last section deals with the identification of the multifractional function, using localized generalized quadratic variations.

2. Construction of non-Gaussian multifractional fields

In this part M(dξ ) is a non vanishing Lévy random measure represented by a Poisson random measure N (dξ, dz)in the sense of Section 3.12 of [14] but with a control measure that has moments of every order greater than two.M(dξ )is a Lévy random measure without Brownian component.

LetN (dξ, dz)be a Poisson random measure onR^d×C. Here the mean measuren(dξ, dz)=EN (dξ, dz)= dξ ν(dz)satisfies:

∀p2,

C

|z|^pν(dz) <+∞. (2)

Moreoverνis a nonvanishing measure such thatν({0})=0.

IfAis a Borel set ofR^d×CthenN (A)is a Poisson random variable of intensityn(A). Moreover if the setI is finite and if the setsAi,i∈I, are pairwise disjoint, then the random variablesN (Ai),i∈I, are independent.

Let us note N=N −n the compensated Poisson measure. It is classical to define the stochastic integral

R^d×Cϕ(ξ, z)N(dξ, dz) for every function ϕ:R^d×C→C such that ϕ is in L²(R^d×C) for the measure n(dξ, dz). By construction the mapϕ→

ϕ dNis an isometry fromL²(R^d×C)onto a subset ofL²(Ω):

E

R^d×C

ϕ(ξ, z)N (dξ, dz) ²

=

R^d×C

ϕ(ξ, z) ²n(dξ, dz), for everyϕ∈L²(R^d×C).

Notice that ifϕis real so is

ϕ dN. Let us denote by(z)the real part of a complexzand by(z)its imaginary part.

The law of the random variable

ϕ dNis given by its characteristic function E

exp

i

u

(ϕ) dN+v

(ϕ) dN

which is equal to exp

R^d×C

exp i

u(ϕ)+v(ϕ)

−1−i

u(ϕ)+v(ϕ)

dξ ν(dz)

, where(u, v)∈R². Let us remark that

ϕ dNis a centered random variable.

Following [6],

R^d

f (ξ ) M(dξ )^def=

R^d×C

f (ξ )z+f (−ξ )z ¯ N (dξ, dz), wheref ∈L²(R^d).

Then if

∀ξ∈R^d, f (−ξ )=f (ξ ) (3)

R^df (ξ ) M(dξ )is a real centered random variable.

As in [6], the control measureν(dz) is assumed to be rotationally invariant. LetP be the mapP (ρe^iθ)= (θ , ρ)∈ [0,2π )×R⁺_∗.The measureν(dz)satisfies the following property:

P ν(dz)

=dθ ν_ρ(dρ), (4)

wheredθ is the uniform measure on[0,2π ).

Then, whenf satisfies (3), E

R^d

f (ξ ) M(dξ ) ²

=4πf²_L2(R^d) +∞

0

ρ²νρ(dρ). (5)

A real harmonizable multifractional Lévy motionX_h will be characterized by a locallyβ-Hölder functionh.

On the other hand, we are interested in the pointwise Hölder exponent ofXh at pointx. Therefore let us precise the definition of these two notions.

Definition 2.1. Letβ >0. Letf:R^d→Rbe a function onR^d. 1. f is aβ-Hölder function onU⊂R^dif

∃C∈R⁺, ∀(x, y)∈U², f (x)−f (y) Cx−y^β.

2. f is a locallyβ-Hölder function onR^dif for every compact setK⊂R^d,f is aβ-Hölder function onK.

3. Letx∈R^d. Then α_f(x)=sup

α >0,lim

y→0

f (x+y)−f (x) y^α =0

is called the pointwise Hölder exponent of the functionf at pointx.

Let us remark that iff is a locallyβ-Hölder function onR^dthen for everyx∈R^d,αf(x)β. For everyU⊂R^d and everyβ >0, let

fβ,U= sup

(u,v)∈K² u=v

|f (u)−f (v)| u−v^β .

In particular,f is a locallyβ-Hölder function if and only if for every compact setK⊂R^d,fβ,K<+∞. Let us now introduce the class of real harmonizable multifractional Lévy motions.

Definition 2.2. Letβ >0 and let h:R^d→]0,1[ be a locally β-Hölder function on R^d. We suppose that the Lévy random measureM(dξ ) satisfies the finite moment assumption (2) and the rotational invariance (4). Let (a, b)∈R². A real harmonizable multifractional Lévy motion (in short RHMLM) is a real valued field which admits a harmonizable representation

Xh(x)=

R^d

e^−ix·ξ−1

ξ^d/2+h(x)L(dξ ),

whereL(dξ )=a M(dξ )+b W (dξ ) is the sum of the Lévy random measure a M(dξ )and of an independent Wiener measureb W (dξ ).

Consequently,X_h is the sum of two independent fields, one of which is a multifractional Brownian motion.

In particular the MBM is the RHMLM obtained forL(dξ )=W (dξ ). Assuming (4), RHFLMs have stationary increments. It does not remain true for RHMLMs.

In all the sequel,h:R^d→]0,1[is a locallyβ-Hölder function onR^dandX_his the RHMLM associated withh.

3. Some properties of real harmonizable multifractional Lévy motions

This section deals with two properties that RHMLMs share with the MBM. The RHMLMXh has locallyH- Hölder sample paths at pointx for everyH <min(h(x), β). Moreover, if for everyx, 0< h(x) < β, thenXhis lass with tangent FBM at pointx. From this last property, one deduces that the Hölder exponent ofXh at pointx is almost surely equal toh(x).

As these properties have already be shown in [8] in the case of the MBM, i.e. in the case whereL(dξ )=W (dξ ), we suppose for sake of simplicity thatL(dξ )=M(dξ ).

3.1. Preliminary lemmas

Usually, to get the regularity of the trajectories and the lass property, one estimates E X_h(x)−Xh(y) ^q

, q∈N^∗,

whereq ∈N^∗=N\{0}. Nevertheless whenh(x) >1−d/2 andq 3, Xh(x)may have an infinite moment of orderq. Thus the fieldXhis split into two fieldsXh=X⁺_h +X_h⁻whereX_h⁺has moments of every order andX⁻_h has almost surely locallyβ-Hölder sample paths. Then one can estimate

E X⁺_h(x)−X⁺_h(y) ^q

, q∈N^∗. Letn∈Nand

Pn(t)= n

k=1

t^k

k!, with conventionP0(t)=0.

Then

g_n⁺(x, ξ )=e^−ix·ξ−1−P_n(−ix·ξ )1_ξ1

ξ^d/2+h(x) (6)

and

g_n⁻(x, ξ )=P_n(−ix·ξ )1_ξ1

ξ^d/2+h(x) (7)

are inL²(R^d)for everyx∈R^d. ThereforeX_h=X_h,n⁺ +X⁻_h,nwith

X⁺_h,n(x)=

R^d

g⁺_n(x, ξ ) L(dξ ) (8)

and X⁻_h,n(x)=

R^d

g_n⁻(x, ξ ) L(dξ ). (9)

Notice that X_h=X_h,0⁺ .

Moreoverg⁺_n(x,·)∈L^q(R^d)for everyq2 such that n+1−d/2−h(x)

q >−d.

Consequently whennd/2,g_n⁺(x,·)∈L^q(R^d)for everyq2 and everyx∈R^d. In this caseX⁺_h,nhas moments of every order (see next proposition given in [6]).

Proposition 3.1. Letp∈N^∗,f ∈L²(R^d)∩L^2p(R^d)and suppose thatf satisfies (3) then

R^df (ξ ) M(dξ )is in L^2p(Ω)and

E

R^d

f (ξ ) M(dξ ) 2p

= p

m=1

(2π )^m

Lm

m

q=1

(2l_q)!f^2l_2l^q_q+∞

0 ρ^2lqν_ρ(dρ) (lq!)² , where

Lmstands for the sum over the set of partitionsLmof{1, . . . ,2p}inmsubsetsKqsuch that the cardinality ofK_qis 2l_qwithl_q1 and wheref2lq is theL^2lq(R^d)norm off.

Using this proposition, one can compute E

X⁺_h,n(x)−X⁺_h,n(y)2p

with the help of someL^2q-norms of the deterministic mapξ→gn(x, y, ξ ), where g_n(x, y, ξ )=g⁺_n(x, ξ )−g⁺_n(y, ξ ).

To estimate these norms,g_nis split intog_n=g_n,1+g_n,2with

gn,1(x, y, ξ )=e^−ix·ξ−e^−iy·ξ+ [P_n(−iy·ξ )−P_n(−ix·ξ )]1_ξ1

ξ^d/2+h(y) (10)

and

gn,2(x, y, ξ )=e^−ix·ξ−1−P_n(−ix·ξ )1_ξ1 ξ^d/2

1

ξ^h(x)− 1 ξ^h(y)

. (11)

IfX_his a RHFLM of indexH, i.e. ifhis equal to a constantH, notice thatg_n,2=0. One of the main difference between the studies of RHFLMs and RHMLMs lies in the study of the properties ofgn,2.

Lemma 3.2. LetK⊂R^dbe a compact set. Suppose thatq2 is such thatq(n−d/2)−d. Then there exists a nonnegative constantC=C(K, q)such that

∀(x, y)∈K², g_n,1(x, y,·)^q

qCx−y^qh(y). Proof. Let(x, y)∈K²and let us noteI₁= g_n,1(x, y,·)^qq.

I₁(x, y)=I₁₁(x, y)+I₁₂(x, y) where

I11(x, y)=

ξ1

|e^−ix·ξ−e^−iy·ξ+P_n(−iy·ξ )−P_n(−ix·ξ )|^q

ξ^qd/2+qh(y) dξ and

I₁₂(x, y)=

ξ1

|e^−ix·ξ−e^−iy·ξ|^q ξ^qd/2+qh(y) dξ.

By Taylor expansion,

e^−ix·ξ−e^−iy·ξ+P_n(−iy·ξ )−P_n(−ix·ξ ) Cx−yξⁿ⁺¹. Let us defineM_K=max_Kh. If 0<ξ1, then

1

ξ^h(y) 1 ξ^MK. Consequently,

I₁₁(x, y)Cx−y^q

ξ1

1

ξ^{qd/2+q(MK−n−1)}dξ.

This last integral is defined sinceq(n−d/2)−dand so I₁₁(x, y)Cx−y^q.

It remains to studyI₁₂. Unfortunately Taylor expansion gives an infinite bound. Let us suppose thatx=y. One splitsI12(x, y)into the integrals

J₁(x, y)=

x−yξ1 ξ1

|e^{−i(x−y)·ξ}−1|^q

ξ^qd/2+qh(y) dξ and J₂(x, y)=

1ξ_x−¹_y

|e^{−i(x−y)·ξ}−1|^q ξ^qd/2+qh(y) dξ.

Then one can easily see that

J₁(x, y)Cx−y^{d(q/2−1)+qh(y)}. By Taylor expansion,

J₂(x, y)Cx−y^q

x−y1

1

ρ^{d−1−qd/2−qh(y)+q}dρ.

Then

J₂(x, y)Cx−y^{q(1+h(y)−MK)}

x−y1

1

ρ^{d−1−qd/2−qMK+q}dρ.

Consequently, by evaluating the last integral, one obtains that J₂(x, y)Cx−y^qh(y). 2

Let us now studyg_n,2.

Lemma 3.3. LetK⊂R^dbe a compact set. Let us notem_K=minKhandM_K=maxKh.

Suppose thatq2 is such thatq(n−d/2)−d. Then

∀(x, y)∈K², g_n,2(x, y,·)^q

qCx−y^qβ, where

C= h^q_β,Ksup

u∈K

J (u)∈R⁺ with

J (u)=

R^d

|e^−iu·ξ−1−Pn(−iu·ξ )1_ξ1|^q|lnξ|^q ξ^dq/2

1

ξ^mK + 1 ξ^MK

_q dξ.

Proof. Let(x, y)∈K²and let us noteI2= gn,2(x, y,·)^qq. By the Mean Value Theorem, 1

ξ^h(x) − 1

ξ^h(y) =−(h(x)−h(y))lnξ ξ^cξ,x,y ,

wherec_ξ,x,yis betweenh(x)andh(y). Thereforem_Kc_ξ,x,yM_K. Furthermore

1

ξ^cξ,x,y 1

ξ^mK1_ξ1+ 1

ξ^MK1ξ<1. Then

1

ξ^cξ,x,y 1

ξ^mK + 1 ξ^MK. Consequently,

1

ξ^h(x) − 1 ξ^h(y)

h(x)−h(y) lnξ 1

ξ^mK + 1 ξ^MK

.

ThereforeI₂(x, y)|h(x)−h(y)|^qJ (x),whereJ (x)is equal to

R^d

|e^−ix·ξ−1−P_n(−ix·ξ )1_ξ1|^q|lnξ|^q ξ^dq/2

1

ξ^mK + 1 ξ^MK

q

dξ.

It is straightforward to prove that sup_u∈KJ (u) <+∞. Moreover sincehis a locallyβ-Hölder function onR^d, h(x)−h(y) hβ,Kx−y^β

and then

I2(x, y)h^q_β,Ksup

u∈K

J (u)x−y^βq. 2

Sincegn=gn,1+gn,2,by applying Minkowski inequality, Proposition 3.1 and Lemmas 3.2 and 3.3, one can prove:

Lemma 3.4. LetK⊂R^dbe a compact set andnd/2, then for everyp∈N^∗, there exists a nonnegative constant C=C(K, p)such that

∀(x, y)∈K², E

X⁺_h,n(x)−X_h,n⁺ (y)2p

Cy−x^2pm, wherem=min(h(y), β).

Let us notice that by symmetry,mcan be replaced by m=min

max

h(x), h(y) , β

.

It remains now to studyX⁻_h,n. Please note that as usual the regularity of the trajectories is given for a modification X⁻ofX⁻_h,n, i.e. for a fieldX⁻such that

∀x∈R^d, PX⁻(x)=X_h,n⁻ (x)

=1.

Lemma 3.5. There exists a modification of the fieldX⁻_h,nthat has, with probability one, locallyβ-Hölder sample paths.

Remark. WhenhisC¹, there exists a modification of the fieldX⁻_h,nsuch that with probability one,X_h,n⁻ ∈C¹. Proof. Notice that for everyx∈R^d

X⁻_h,n(x)=Z_n x, h(x)

,

where the field(Z_n(x, y))_(x,y)∈Rd×]0,1[is defined as follows:

Z_n(x, y)=

ξ1

Pn(−ix·ξ )

ξ^d/2+y L(dξ ). (12)

Then since h is a locally β-Hölder map with values in ]0,1[, it is sufficient to prove that there exists a modification of the fieldZ_nsuch thatP(Z_n∈C¹)=1.

Let us define Yα(y)=

ξ1

i^|α|_d

j=1ξ_j^αj ξ^d/2+y L(dξ ),

whereα∈N^dis such that 1|α| =_n

j=1αj nandy∈ ]0,1[. One shows that for everyα, the fieldYα admits a modification which has almost surelyC¹-sample paths on]0,1[. Then sinceP_nis a polynomial, the same holds forZ_n.

Letα∈N^dsuch that 1_n

j=1α_jnandη∈ ]0,1[. One can prove with Taylor expansion the existence of a constantC >0 such that

1. E[|Y_α(y+δ)−Y_α(y)|²]C|δ|²,for everyy∈ [η,1−η]andδsuch thaty+δ∈ [η,1−η],

2. E[|Yα(y+δ)+Yα(y−δ)−2Yα(y)|²]C|δ|⁴,for everyy ∈ [η,1−η]andδ such that(y+δ, y−δ)∈ [η,1−η]².

According to [9], see p. 69, these statements imply the existence of a modification ofYαwhich has almost surely C¹-sample paths. And so the same holds forZn. 2

3.2. Trajectories regularity

There exist modifications of the MBM whose sample paths are locally Hölder. Here an analogous result in the case of RHMLMs is shown.

Proposition 3.6. LetK⊂R^dbe a compact set.

Then for everyH <min(minKh, β), there exists a modification of the RHMLMXhwhich hasH-Hölder sample paths onK.

Proof. LetH <min(min_Kh, β)andnd/2.

As a consequence of Lemma 3.4 and of Kolmogorov Theorem, there exists a modification of the fieldX_h,n⁺ whose sample paths areH-Hölder onK. Then sinceX_h,n⁻ has almost surely locallyβ-Hölder sample paths (see Lemma 3.5) andX_h=X⁺_h,n+X_h,n⁻ , the proof is done. 2

Then Proposition 3.6 give us a lower bound for the pointwise Hölder exponentα_Xh(x)ofX_hat pointx:

αX_h(x)min h(x), β

. (13)

3.3. Asymptotic self-similarity

Proposition 3.7. Suppose that for everyx∈R^d,h(x) < β. Then the real harmonizable multifractional Lévy motion X_his locally self-similar with multifractional functionhin the sense that for every fixedx∈R^d:

lim

ε→0⁺

X_h(x+εu)−X_h(x) ε^h(x)

u∈R^d (d)=C_x

B_h(x)(u)

u∈R^d, (14)

where the convergence is a convergence in distribution on the space of continuous functions endowed with the topology of the uniform convergence on compact sets,B_h(x)is a standard FBM of indexh(x)and

Cx= 4π

+∞

0

ρ²νρ(dρ)

R^d

|e^ie1·ξ−1|² ξ^d+2h(x) dξ

1/2

,

withe1=(1,0, . . . ,0)∈R^d.

Remark. This proposition is stated in the case whereL(dξ )=M(dξ ). However it remains true whenL(dξ )= a M(dξ )+b W (dξ )is as in Definition 2.2, but with

C_x=

4a²π

+∞

0

ρ²ν_ρ(dρ)+b² 1/2

R^d

|e^ie1·ξ−1|² ξ^d+2h(x) dξ

_1/2 .

Proof. We first prove the convergence of the finite dimensional margins. Next a tightness property is shown. It is a direct consequence of Lemmas 3.4 and 3.5.

Let us recall thatX_h=X_h,0⁺ and that

g0(u, v, ξ )=g⁺₀(u, ξ )−g⁺₀(v, ξ )= e^−iu·ξ−1

ξ^d/2+h(u)− e^−iv·ξ−1 ξ^d/2+h(v). Let us fixedx∈R^d. Then for everyε >0 andu∈R^d,

Y_ε(u)=Xh(x+εu)−Xh(x)

ε^h(x) = 1

ε^h(x)

R^d

g₀(x+εu, x, ξ ) L(dξ ).

In the sequel we shall need to use the decomposition ofg0intog0=g1+g2whereg1=g0,1is defined by (10) andg2=g0,2by (11).

Convergence of the finite dimensional margins. Letp∈N^∗,u=(u1, . . . , up)∈(R^d)^p,v=(v1, . . . , vp)∈R^p and

g_j^ε(ξ, z)= 2 ε^h(x)

z

p

k=1

v_kg_j(x+εu_k, x, ξ )

, j=0,1,2.

Then sincep

k=1vkYε(uk)=

R^dg₀^ε(ξ, z)N (dξ, dz), E

exp

i

p

k=1

v_kY_ε(u_k)

=exp

ψ_ε(u, v) withψ_ε(u, v)=

R^d×C[exp(ig^ε₀(ξ, z))−1−ig^ε₀(ξ, z)]dξ ν(dz).

Thenψ_ε(u, v)=I₁^ε+I₂^ε+I₃^ε, where I₁^ε=

R^d×C

exp

ig^ε₁(ξ, z)

−1−ig^ε₁(ξ, z)

dξ ν(dz),

I₂^ε=

R^d×C

exp

ig^ε₂(ξ, z)

−1−ig^ε₂(ξ, z)

dξ ν(dz),

I₃^ε=

R^d×C

exp

ig^ε₀(ξ, z)

+1−exp

ig^ε₁(ξ, z)

−exp

ig₂^ε(ξ, z)

dξ ν(dz).

Sinceg_j^ε∈L²(R^d×C), these three integrals are defined.

1. Study of I₁^ε. By rotational invariance of the measureν(dz)and by applying the change of variableλ=ε ξ, one obtains

I₁^ε=

R^d×C

l^ε(λ, z) dλ ν(dz),

wherel^ε(λ, z)=_ε¹d[exp(ε^d/2g˜₁(λ, z))−1−ε^d/2g˜₁(λ, z)]with

˜

g₁(λ, z)=2i

z p

k=1

v_k e^−iuk·λ−1 λ^d/2+h(x)

.

Then asε→0₊a dominated convergence argument yields that lim

ε→0+I₁^ε= −2π

+∞

0

ρ²ν_ρ(dρ)

R^d

_p

k=1

v_k|e^−iuk.λ−1|

|λ|^d/2+h(x) 2

dλ.

2. Study ofI₂^ε. Since for everyt∈R,|e^it−1−it||t|²/2, I₂^ε 1

2

R^d×C

g₂^ε(ξ, z) ²dξ ν(dz), i.e. I₂^ε C ε^2h(x)

p

k=1

v_kg₂(x+εu_k, x,·)

2

2

.

One applies Lemma 3.3 and obtains that forε1 and for everyk, g2(x+εuk, x,·)²

2Cε^2β.

Then asβ > h(x), limε→0₊g2(x+εuk, x,·)2/ε^h(x)=0. And so by applying Minkowski inequality, lim

ε→0₊I₂^ε=0.

3. Study ofI₃^ε. One can notice that I₃^ε

R^d×C

g₁^ε(ξ, z) g₂^ε(ξ, z) dξ ν(dz).

Because of Lemma 3.2, forε1

R^d

g₁^ε(ξ, z) ²dξ ν(dz)C.

Then Cauchy–Schwarz inequality implies that lim

ε→0₊I₃^ε=0.

Consequently, lim_ε→0+E[exp(ip

k=1v_kY_ε(u_k))]is equal to exp

−2π

+∞

0

ρ²ν_ρ(dρ)

R^d

_p

k=1

v_k|e^−iuk.λ−1|

|λ|^d/2+h(x) 2

dλ

,

and then lim

ε→0⁺E

exp

i p

k=1

v_kY_ε(u_k)

=exp

−C_x² 2 Var

p

k=1

v_kB_h(x)(u_k)

,

whereB_h(x) is a standard fractional Brownian motion of index h(x)andCx is defined in the statement of the proposition, which concludes the proof of the convergence of the finite margins.

Tightness. Letnd/2,

Y_ε⁺(u)=X⁺_h,n(x+εu)−X_h,n⁺ (x)

ε^h(x) and Y_ε⁻(u)=X⁻_h,n(x+εu)−X⁻_h,n(x)

ε^h(x) .

ThenYε=Y_ε⁺+Y_ε⁻.

SinceX⁻_h,nhas locallyβ-Hölder sample paths (see Lemma 3.5) and sinceh(x) < β, it is clear that lim

ε→0₊

X_h,n⁻ (x+εu)−X_h,n⁻ (x) ε^h(x)

u∈R^d (d)=0.

Consequently,(Y_ε⁻)_ε>0is a tight family. Let us now prove the tightness of(Y_ε⁺)_ε. Notice thatY_ε⁺(0)=0 and so that(Y_ε⁺(0))εis tight.

LetK⊂R^dbe a compact set andr0>0 such that whenε1,x+εK⊂K0whereK0=B(x, r0). Lemma 3.4 is applied to the compact setK₀. Then for everyp∈N^∗, there exists a constantC=C(p, K₀, x) >0 such that for everyε∈ ]0,1]and every(u, v)∈K²,

E

Y_ε⁺(u)−Y_ε⁺(v)2p

Cε^{2p(h(x+εv)−h(x))}u−v^2ph(x+εv). Moreover ashis a locallyβ-Hölder map,

∀ε∈ ]0,1],∀v∈K, ε^{2p(h(x+εv)−h(x))}C.

Therefore for everyp∈N^∗, there exists a constantC=C(p, K, x) >0 such that for everyε∈ ]0,1]and every (u, v)∈K²,

E

Y_ε⁺(u)−Y_ε⁺(v)2p

Cu−v^2pminK0h.

One can choosep∈N^∗such that 2pminK₀h > d, which concludes the proof. 2

Let us fixedx∈R^d. Actually, convergence (14) holds as soon ash(x) < αh(x). Then, one can wonder what happens whenh(x) > αh(x). In this case, suppose that

l(x)=lim

y→0

h(x+y)−h(x) y^αh(x) exists and is not equal to 0. Then

lim

ε→0⁺

X_h(x+εu)−X_h(x) ε^αh(x)

u∈R^d

(d)=l(x)X^∗(x)

u^αh(x)

u∈R^d, (15)

where the limit is in distribution for all finite margins of the fields and where X^∗(x)=

R^d

(e^−ix·ξ−1)lnξ ξ^d/2+h(x) L(dξ ).

Let us remark that in (15), the field(u^αh(x))_u∈Rd is deterministic. Then the randomness of the tangent field is only due to the real-valued random variableX^∗(x)which does not depend onu. Let us precise that one proceeds as in the proof of (14), replacingε^h(x)byε^αh(x)ing_j^ε. The preponderant term is thenI₂^ε.

Corollary 3.8. Suppose that for everyx∈R^d,h(x) < β. Then for everyx∈R^d, the pointwise Hölder exponent of the RHMLMX_hat pointxis almost surely equal toh(x).

Proof. It is classical, see Proposition 3.3, p. 109 in [6], to deduce from the lass property that the pointwise Hölder exponentαX_h(x)of the RHMLMXhat pointxsatisfiesαX_h(x)h(x).

Then by (13),αX_h(x)=h(x). 2

4. Identification

In this section, L(dξ )=M(dξ )+σ W (dξ ) is a Lévy random measure which satisfies the assumptions of Definition 2.2. In particularWandMare two independent measures.

Leth:R^d→]0,1[be a locallyβ-Hölder function andXhbe the RHMLM associated withL(dξ )andh. Then Xh(x)=

R^d

e^−ix·ξ−1

ξ^d/2+h(x)L(dξ ).

Our aim is to identify the multifractional functionhfrom discrete observations of the fieldXhon[0,1]^d. The varianceσ and the control measureν(dz)are unknown. The field is observed at sampling points (^k_N¹, . . . ,^k_N^d), 0k_jN, 1j d.

One uses localized generalized quadratic variations, method introduced in [5]. For anyx∈ ]0,1[^d, we define a (ε, N )-neighborhood ofxby

Vε,N(x)=

p∈Z^d, max

j=1...d

p_j N −xj

< ε

. Let(a)=0...K be a real valued sequence such that:

K

=0

a= K

=0

a=0. (16)

As an example, one can takeK=2, a₀=1, a₁= −2, a₂=1.

For everyk=(k₁, . . . , k_d)∈N^d,a_k=_d

j=1a_kj.Let us note K=

k∈N^d,0k_jK, j=0. . . K , X_p(x)=

k∈K

a_kX_h k+p

N

= K

k₁,...,k_d=0

d

j=1

a_kjX_h k+p

N

, wherep∈Vε,N(x), are the increments ofX_hassociated with the sequencea.

Then the localized generalized quadratic variation at pointx is equal to V_ε,N(x)=

p∈Vε,N(x)

Xp(x)2

. Our aim is to show that

hˆ_N(x)=1 2

Log₂

Vε,N/2(x) V_ε,N(x)

+d

is a consistent estimator ofh(x).

Notations. Let (v_m)_m be a deterministic real valued sequence, (Z_m)_m and (R_m)_m be two sequences of real random variables.

• v_m=O(1)means that the sequence(v_m)_mis bounded.

• Z_m=O_P(1)if and only if

∀ε >0, ∃M >0, sup

m P

|Z_m|> M

< ε.

• Zm=O_P(Rm)meansZm=RmUmwithUm=O_P(1).