Estimation in models driven by fractional brownian motion

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www.imstat.org/aihp 2008, Vol. 44, No. 2, 191–213

DOI: 10.1214/07-AIHP105

Estimation in models driven by fractional Brownian motion

Corinne Berzin

^a

and José R. León

^b

aLabSAD – Université Pierre Mendès-France 1251 Avenue centrale, BP 47, 38040 Grenoble cedex 9, France.

E-mail: Corinne.Berzin@upmf-grenoble.fr

bEscuela de Matemáticas, Facultad de Ciencias, Universidad Central de Venezuela, Paseo Los Ilustres, Los Chaguaramos, A.P. 47197, Caracas 1041-A, Venezuela. E-mail: jleon@euler.ciens.ucv.ve

Received 6 March 2006; accepted 25 July 2006

Abstract. Let{bH(t ), t∈R}be the fractional Brownian motion with parameter 0< H <1. When 1/2< H, we consider diffusion equations of the type

X(t )=c+ _t

0 σ

X(u)

db_H(u)+ _t

0 μ

X(u) du.

In different particular models whereσ (x)=σ orσ (x)=σ xandμ(x)=μorμ(x)=μ x, we propose a central limit theorem for estimators ofH and ofσ based on regression methods. Then we give tests of the hypothesis onσ for these models. We also consider functional estimation onσ (·)in the above more general models based in the asymptotic behavior of functionals of the 2nd-order increments of the fBm.

Résumé. Soit{b_H(t ), t∈R}le mouvement Brownien fractionnaire de paramètre 0< H <1. Lorsque 1/2< H, nous considérons des équations de diffusion de la forme

X(t )=c+ _t

0 σ

X(u)

db_H(u)+ _t

0 μ

X(u) du.

Nous proposons dans des modèles particuliers où,σ (x)=σouσ (x)=σ xetμ(x)=μouμ(x)=μ x, un théorème central limite pour des estimateurs deH et deσ, obtenus par une méthode de régression. Ensuite, pour ces modèles, nous proposons des tests d’hypothèses surσ. Enfin, dans les modèles plus généraux ci-dessus nous proposons des estimateurs fonctionnels pour la fonction σ (·)dont les propriétés sont obtenues via la convergence de fonctionnelles des accroissements doubles du mBf.

AMS 2000 subject classifications:60F05; 60G15; 60G18; 60H10; 62F03; 62F12; 33C45

Keywords:Central limit theorem; Estimation; Fractional Brownian motion; Gaussian processes; Hermite polynomials

1. Introduction

Let{b_H(t ), t∈R}be the fractional Brownian motion with Hurst coefficientH, 0< H <1. ForH >1/2, the integral with respect to fBm can be defined pathwise as the limit of Riemann sums (see [8] and [9]). This allows us to consider, under certain restrictions overσ (·)andμ(·), the “pseudo-diffusion” equations with respect to fBm, that is,

X(t )=c+ _t

0

σ X(u)

dbH(u)+ _t

0

μ X(u)

du. (1)

Our main interest in this work is to provide estimators for the functionσ (·).

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In Section 3.1 we propose simultaneous estimators ofH and ofσ in models such thatσ (x)=σ orσ (x)=σ x andμ(x)=μor μ(x)=μx. Following [4,7] and [8], we can give an explicit expression for the unique solution to each equation that is a function of the trajectory ofb_H(·). Instead of the original process, we use a mollified ver- sion and we assume we observe a smoothed by convolution process, defined asX_ε(t )=ϕ_ε∗X(t ). Hereε, which tends to zero, is the smoothing parameter andϕ_ε(·)is a convolution kernel such thatϕ_ε(·)=¹_εϕ(_ε^·), withϕ(·)aC² positive kernel withL¹norm equal to one. Then we observed functionals of the type1

0h(Xε(u))| ¨Xε(u)|^kdu, with h(x)=1/|x|^k in the case of linearσ (·)andh(x)=1 in the case of constantσ (·). Such an observation could seem unusual but note that in case whereϕ(·)=1_[−1,0]∗1_[−1,0](·), then ε²X¨_ε(u)=X(u+2ε)−2X(u+ε)+X(u).

Although ϕ(·) is not a C² function, all results obtained in this paper can be rewritten with this particular ap- proximation. So our method can be related with variation methods and consists in obtaining some least squares estimators for H and σ in certain regression models. This method can be compared to that of [3], where the model was the fBm i.e. σ (·)≡1 and μ(·)≡0 and the purpose was to estimate H. Indeed, we prove that the asymptotic behavior of such estimators, that is, (Hk −H )/√

ε and (σk −σ )/(√

εlog(ε)) are both equiva- lent to those of certain non-linear functionals of the Gaussian process b^ε_H(·)=ϕε ∗bH(·). As in [3], we show that they satisfy a central limit theorem using the method of moments, via the diagram formula. It is interest- ing to note that the rates of convergence of such estimators are not the same. Furthermore the asymptotic law of (Hk,σk)is a degenerated Gaussian. Hence, we could not provide simultaneous confidence intervals for H and σ. Finally, we proved that the best estimators for H and σ in the sense of minimal variance are obtained for k=2.

In Section 3.2, we get back to more general models of the form (1) and our goal is to provide functional estimation ofσ (·)as in [2]. Indeed, in [2] we considered the case whereμ(·)≡0 and we proved that, ifN_ε^X(x)denotes the number of times the regularized processXε(·)crosses levelx, before time 1, then for 1/2< H <1 and any continuous functionh,

π 2

ε⁽¹⁻^{H )}

˜ σ_2H

_∞

−∞h(x)N_ε^X(x)dx−→^a.s.

₁

0

h X(u)

σ X(u)

du. (2)

Furthermore we got the following result about the rates of convergence proving that there exists a Brownian mo- tionW ( ·)independent ofb_H(·)and a constantσ_g₁ such that for 1/2< H <3/4,

√1 ε

π 2

ε⁽¹⁻^{H )}

˜ σ_2H

_∞

−∞h(x)N_ε^X(x)dx− ₁

0

h X(u)

σ X(u)

du

⇒ε→0σ_g₁ 1

0

h X(u)

σ X(u)

dW (u), (3)

under some assumptions on the functionh.

The proofs of these two last convergences are based on the fact that, on the one hand, when μ(·) ≡0 and because fBm has quadratic variation when H > ¹₂ and σ (·)∈ C¹(R), the solution to Eq. (1) is given by X(t )=K(b_H(t )) where K is the solution to the ordinary differential equation K(t )˙ =σ (K(t )) with K(0)=c (see [8]). On the other hand, by the Banach formula, we have _∞

−∞h(x)N_ε^X(x)dx =1

0h(X_ε(u))| ˙X_ε(u)|du and since X˙ε(u) ˙K(b^ε_H(u))b˙_H^ε (u)=σ (K(b^ε_H(u)))b˙^ε_H(u), we needed to look for the asymptotic behavior of a particular non-linear functional of the regularized fBm b^ε_H(·) and Theorems 1.1 and 3.4 of [2] gave the result.

We need to recall these two convergence results because they can serve as a motivation to the statement of Theo- rems 3.8 and 3.9. Indeed in these two last theorems we will use the same type of approximations and convergence.

Also convergences in (2) and (3) can serve as a motivation to the main interest in the present paper, that is to work with the second derivative of the smoothed process instead of the first one. Indeed in the second convergence result (3), we obtained the restrictive conditionH <3/4 due to the fact that we used the first derivative ofXε(·).

Thus in this work, having in mind to reach all the range ofH <1, we considered the case whereμ(·)is not necessarily null and instead of considering functionals of| ˙Xε(·)|, we worked with functionals of| ¨Xε(·)|^k. This approach allowed us to provide functional estimation ofσ (·)as in (2) and to exhibit the rate of convergence as in (3) for any

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value ofHin]1/2,1[, using a generalization of the two last convergence results whenμ(·)≡0 and then applying the Girsanov theorem (see [5] and [6]).

We observe that the limit convergence in (2) is1

0h(X(u))σ (X(u))du and will become1

0 σ^k(X(u))duin that work in cases whereh(·)≡1. Thus if we get back to the last four models of Section 3.1 and if we take into account the form ofσ (·), that is σ (x)=σ orσ (x)=σ x, the limit integral is now a function ofσ. Thus in the second part of Section 3.1 we propose estimators forσ^kwhenHis known. This supplementary information aboutH leads us to estimators ofσ performing more than those of the first part of Section 3.1 because the rate of convergence will be 1/√

εinstead of 1/(√

εlog(ε))as before.

Finally in Section 3.3, a result similar to the one of part two in Section 3.1 can be obtained under contiguous alternatives forσand provides a test of the hypothesis for such a coefficient.

We work with the techniques described in the last sections but without using the Girsanov theorem that is more tricky to use since under the alternative hypothesisσ depends onε.

2. Hypothesis and notation

Let{b_H(t ), t∈R}be a fractional Brownian motion (fBm) with parameter 0< H <1 (see for instance [10]), i.e.b_H(·) is a centered Gaussian process with the covariance function:

E

b_H(t )b_H(s)

=1 2v²_2H

|t|^2H+ |s|^2H− |t−s|^2H ,

withv_2H² := [(2H+1)sin(πH )]⁻¹. We define, for aC²densityϕ with compact support included in[−1,1], for eacht≥0 andε >0 the regularized processes:

b^ε_H(t ):=1 ε

_∞

−∞ϕ t−x

ε

b_H(x)dx and Z_ε(t ):=ε⁽²⁻^{H )}b¨^ε_H(t ) σ2H

,

with

σ_2H² :=V

ε⁽²⁻^{H )}b¨_H^ε(t )

= 1 2π

_+∞

−∞ |x|⁽³⁻^{2H )}ϕ(ˆ −x)²dx.

We shall use Hermite polynomials, defined by e^{(t x}⁻^t²^/2)=^∞

n=0

H_n(x)tⁿ n! .

They form an orthogonal system for the standard Gaussian measureφ (x)dx and, if h∈L²(φ (x)dx),thenh(x)= _∞

n=0hˆ_nH_n(x)andh²_2,φ:=_∞

n=0hˆ²_nn!.

Letgbe a function inL²(φ (x)dx)such thatg(x)=_∞

n=1gˆ_nH_n(x), withg²_2,φ=_∞

n=1gˆ_n²n!<+∞.

The symbol “⇒” will mean weak convergence of measures.

At this step of the paper it will be helpful to state several theorems obtained in [3] in the aim to enlighten the notations and to make this paper more independent of it.

We proved the two following theorems.

Theorem 2.1. For all0< H <1andk∈N^∗, ₁

0

Zε(u)k

du−→^a.s.

ε→0E[N]^k,

whereN will denote a standard Gaussian random variable.

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Remark. This theorem implies thatZ_ε(·)⇒N whenεgoes to zero,the random variableZ_ε(·)is considered as a variable on([0,1], λ)whereλis the Lebesgue measure.This last convergence implies that for all0< H <1and real k >0,almost surely,₁

0 |Zε(u)|^kdu→E[|N|^k].

Forε >0, define Sg(ε):=ε⁻^1/2

₁

0

g Zε(u)

du.

Theorem 2.2. For all0< H <1, S_g(ε·) ⇒

ε→0 X(·),

where the above convergence is in the sense of finite-dimensional distributions andX(·)is a cylindrical centered Gaussian process with covarianceρ_g(b, c):=E[X(b)X(c)],where forb, c >0,

ρg(b, c):= 1

√bc ∞ n=1

ˆ g²_nn!

_+∞

−∞ ρ_Hⁿ(x, b, c)dx, and forx∈R,

ρ_H(x, b, c):=E

Z_εb(εx+u)Z_εc(u)

=(bc)⁽²⁻^{H )} 2πσ_2H²

_∞

−∞|y|⁽³⁻^{2H )}e^ixyϕ(−by)ϕ(cy)dy.

Note that for fixedc >0,ρ_H(x, c, c)=ρ_H(x/c), where forx∈R, ρ_H(x):=E

Z_ε(εx+u)Z_ε(u)

= 1 2πσ_2H²

_∞

−∞|y|⁽³⁻^{2H )}e^ixyϕ(−y)²dy, so that,ρg(c, c)=σ_g²:=_∞

n=1gˆ_n²n!_+∞

−∞ ρ_Hⁿ(x)dx. Therefore we furthermore got the following remark that was also shown in Corollary 3.2(i) in [2].

Remark. Ifc >0is fixed,Sg(εc)⇒σgN.

For allm∈N^∗, for allc1>0,c2>0, . . . ,c_m>0 and for alld1,d2, . . . , d_m∈R, we will denote

σ_g,m² (c,d):=

m i=1

m j=1

didjρg(ci, cj)=E _m

i=1

diX(ci) 2

.

We also proved in [3] the following lemma.

Lemma 2.1. limε→0E[_m

i=1d_iS_g(εc_i)]²=σ_g,m² (c,d).

Throughout the paper,C(resp.C(ω)) shall stand for a generic constant (resp. for a generic constant that depends onωliving in the space of the trajectories), whose value may change during a proof, and log(·)for the Naperian logarithm.

For k≥1, we shall note N^k_k :=E[|N|^k], and if C(·) is a measurable function we shall note C(·)^k_k :=

1

0|C(u)|^kdu, byC(·)^k_k,ε:=_ε

0|C(u)|^kduand byC(·)^k_k,ε^c:=1

ε |C(u)|^kdu, forε >0.

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3. Results

3.1. Simultaneous estimators ofHand ofσ

As mentioned in the Introduction, we are interested in providing simultaneous estimators of H andσ in the four following models. ForH >1/2 andt≥0

dX(t )=σdbH(t )+μdt, (4)

dX(t )=σdbH(t )+μX(t )dt, (5)

dX(t )=σ X(t )dbH(t )+μX(t )dt, (6)

dX(t )=σ X(t )dbH(t )+μdt, (7)

withX(0)=c.

The solution to these equations are, respectively:

(4) X(t )=σ b_H(t )+μt+c(see [8]), (5) X(t )=σ b_H(t )+exp(μt )[σ μ(_t

0b_H(s)exp(−μs)ds)+c], (6) X(t )=cexp(μt+σ b_H(t ))(see [4] and [7]),

(7) X(t )=exp(σ bH(t ))(c+μt

0exp(−σ bH(s))ds).

We consider the problem of estimating simultaneously H and σ >0. Suppose we observe instead of X(t ) a smoothed by convolution process X_ε(t )=ϕ_ε ∗X(t ), where ϕ_ε(·)=1/εϕ(·/ε), with ϕ(·) as before and where we have extendedX(·)by means ofX(t )=c, ift <0.

For model (7) we will make the additional hypothesis thatμandchave the same sign withμpossibly null.

From now on, we shall note for eacht≥0 andε >0, Z^X_ε(t ):=

⎧⎨

⎩

ε^{(2−H )}X¨_ε(t )

σ_2H for the first two models,

ε⁽²^{−H )}X¨ε(t )

σ2HXε(t ) for the other two.

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Fork≥1, let us denote A_k(ε):= 1

σ^kN^k_k 1

0

Z_ε^X(u)^kdu

−1. (9)

The remark following Theorem 2.1 allows us to state the following theorem.

Theorem 3.1. Fork≥1, (1) A_k(ε)−→^a.s.

ε→00.

(2) Furthermore

√1

εA_k(ε)=S_g_k(ε)+oa.s.(1), where gk(x):= |x|^k

N^k_k −1. (10)

At this step, we can propose estimators ofH andσ, by observingX_ε(u)at several scales of the parameterε, i.e.

h_i=εc_i,c_i>0,i=1, . . . , . In this aim, let us define M_k(ε):=

1

0X¨ε(u)^kdu for the first two models, ₁

0^X^¨^ε^(u)

Xε(u)^kdu for the other two. (11)

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Using assertion (1) of Theorem 3.1, we get ε^k(2^{−H )}Mk(ε)

σ_2H^k σ^kN^k_k

−→a.s.

ε→01, from which we obtain

log M_k(ε)

=k(H−2)log(ε)+log

σ_2H^k σ^kN^k_k

+oa.s.(1). (12)

The following regression model can be written, for each scaleh_i: Y_i=a_kX_i+b_k+ξ_i, i=1, . . . , ,

wherea_k:=k(H −2),b_k:=log(σ_2H^k σ^kN^k_k)and fori=1, . . . , ,Y_i:=log(Mk(h_i)),X_i:=log(hi). Hence, the least squares estimatorsH_k ofHandB_kofb_k are defined as

k(Hk−2):=

i=1

zilog

Mk(hi)

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and

B_k:=1

i=1

log M_k(h_i)

−k(H_k−2)1

i=1

log(hi), (14)

where

zi:= y_i

i=1y²_i and yi:=log(ci)−1

i=1

log(ci).

Note the following property

i=1

z_i =0 and

i=1

z_iy_i=1. (15)

Then we propose σ_2H^k :=σ^k

2H_k, as estimator ofσ_2H^k and

σ^k:= exp(Bk)

σ_2H^k N^k_k, (16)

as estimator ofσ^k. Finally, we proposeσkas estimator ofσ defined by σ_k:=σ^k1/ k

. (17)

Theorem 3.1 and Theorem 2.2 imply the following theorem for all the range ofH belonging to]1/2,1[. Theorem 3.2. Fork≥1,

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(1) H_kis a strongly consistent estimator ofH and

√1

ε(H_k−H )⇒

ε→0N

0, σ_g²_k_,

c,√ c

z k

, wheregk(·)is defined by(10)and

g_k(x)= ∞ n=1

ˆ

g2n,kH2n(x), withgˆ2n,k= 1 (2n)!

n−1 i=0

(k−2i).

(2) σ_k is a weakly consistent estimator ofσ and

√ 1

εlog(ε)(σ_k−σ )⇒

ε→0N

0, σ²σ_g²

k,

c,√

c z

k

.

Remark. As in[3],the varianceσ_g²

k,(c,√

c(z/ k))is minimal fork=2and then the best estimators forHandσ in the sense of minimal variance are obtained fork=2.

Now let us suppose thatH, ¹₂< H <1, is known. Theorem 3.1 also provides estimators forσ. Indeed, if fork≥1 we set,

˜

σ_k:=Z_ε^X(·)_k Nk

see (8) for definition ofZ_ε^X(·) ,

then Theorem 3.1 and the remark following Theorem 2.2 imply the following theorem.

Theorem 3.3. Fork≥1and ifH is known with¹₂< H <1then (1) σ˜k is a strongly consistent estimator ofσ and

(2)

√1

ε(σ˜k−σ )⇒

ε→0N

0,σ² k²σ_g²

k

, whereg_k(·)is defined by(10).

Remark 1. Note that the rate of convergence in assertion(2)is1/√

εinstead of1/(√

εlog(ε))as in assertion(2)in Theorem3.2.This is due to the fact that hereH is known.

Remark 2. The varianceσ_g²

k/ k² is minimal fork=2 and then the best estimator for σ in the sense of minimal variance is obtained fork=2.

Proof of Theorem 3.1.

(1) We need the following lemma for which proof is given in the Appendix.

Lemma 3.1. For0≤t≤1, X¨_ε(t )=

σb¨^ε_H(t )+aε(t ) for the first two models, σ X_ε(t )b¨^ε_H(t )+X_ε(t )a_ε(t ) for the other two, with

aε(t )≤C(ω)

ε^(H⁻²⁻^δ)1_{0≤t≤ε}+ε^(2H⁻²⁻^δ)1_{ε≤t≤1}

, for anyδ >0.

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Remark. Indeed,for the first model one has, a_ε(t )≤C(ω)ε^(H⁻²^−δ)1_{0≤t≤ε}

and for the second one, a_ε(t )≤C(ω)

ε^(H⁻²⁻^δ)1_{0≤t≤ε}+ε^(H⁻¹⁻^δ)1_{ε≤t≤1} .

We have to prove that almost surely, Z_ε^X(·)_k converges to σNk when ε goes to zero. For this, we write Z^X_ε(·)_kas

Z_ε^X(·)

k=σ Zε(·)

k+Z^X_ε(·)

k−σ Zε(·)

k.

By the remark following Theorem 2.1, we know thatZε(·)k converges almost surely toNk when εgoes to zero. Thus, we have just to prove that|Z_ε^X(·)k− σ Zε(·)k|converges almost surely to zero withε. By Lemma 3.1 and using Minkowski’s inequality, one has

Z^X_ε(·)

k−σ Z_ε(·)

k≤Z^X_ε(·)−σ Z_ε(·)

k

=ε⁽²^{−H )} σ2H

a_ε(·)

k≤C(ω)

ε^{(1/ k}⁻^δ)+ε^(H⁻^δ) .

Choosingδ small enough, i.e. 0< δ <inf(¹_k, H ), we proved that the last term in the above inequality tends almost surely to zero withεand assertion (1) follows.

(2) We write

√1

εA_k(ε)=S_g_k(ε)+ 1

√εN^k_kσ^kZ^X_ε(·)^k

k−σ Z_ε(·)^k

k

.

Let us prove now that(Z_ε^X(·)^k_k− σ Zε(·)^k_k)=oa.s.(√

ε). Using the bound |x+y|^k− |x|^k≤2^(k⁻¹⁾k|y|

|x|^(k⁻¹⁾+ |y|^(k⁻¹⁾

, fork≥1, and Hölder’s inequality, we obtain

f^k_k− g^k_k≤f (·)^k−g(·)^k

1

≤2^(k−¹⁾kf−gk

g^(k_k⁻¹⁾+ f−g^(k_k⁻¹⁾

. (18)

Let us apply this inequality tof (·):=Z^X_ε(·)and tog(·):=σ Z_ε(·), successively with the norm · k,εand the norm · k,ε^c.

On the one hand, applying Lemma 3.1, one obtains thatZ^X_ε(·)−σ Z_ε(·)k,ε≤C(ω)ε^{(1/ k}⁻^δ)and thatZ^X_ε(·)− σ Z_ε(·)k,ε^c≤C(ω)ε^(H⁻^δ).

On the other hand, the trajectories ofb_H(·)are(H−δ)-Hölder continuous, in other words for anyδ >0

b_H(u+ε)−b_H(u)≤C(ω)ε^(H⁻^δ). (19)

Using this fact, we get b¨^ε_H(u)=

1 ε²

_∞

−∞ϕ(v)¨

bH(u−εv)−bH(u) dv

≤C(ω)ε^(H⁻²⁻^δ). (20)

We deduceZε(·)k,ε≤C(ω)ε^{(1/ k}⁻^δ)andZε(·)k,ε^c≤C(ω)ε⁻^δ.

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Finally, takingδsmall enough, i.e. 0< δ <¹_k(H−¹₂), we proved that Z_ε^X(·)^k

k−σ Z_ε(·)^k

k≤C(ω)

ε⁽¹⁻^δk)+ε^(H⁻^δk)+ε^k(H⁻^δ)

≤C(ω)ε^(H⁻^δk)=oa.s.

√ε ,

and assertion (2) follows.

(1) By using (13) and (12) we obtain k(H_k−2)=

i=1

z_i

k(H−2)log(εci)+b_k

+oa.s.(1),

and property (15) gives

k(H_k−2)=k(H−2)+oa.s.(1).

We proved thatH_k is a strongly consistent estimator ofH.

Now by using definitions ofA_k(ε)and ofM_k(ε)(see (9) and (11)), one obtains A_k(ε)=ε^k(2⁻^{H )}exp(−b_k)M_k(ε)−1.

With this definition and using a Taylor expansion for the logarithm function one has log

M_k(ε)

=log

ε^k(H⁻²⁾exp(bk) +log

1+A_k(ε)

=k(H−2)log(ε)+b_k+A_k(ε)+A²_k(ε)

−1 2 +ε₁

A_k(ε) . (21)

Let us see that A²_k(ε)

−1 2 +ε₁

A_k(ε) =oP

√ε

. (22)

By assertion (2) of Theorem 3.1 we know that

√1

εAk(ε)=Sg_k(ε)+oa.s.(1), (23)

whereg_k(·)is defined by (10), and by Lemma 2.1, E

S_g²

k(ε)

=O(1), (24)

so √¹

εA²_k(ε)=oP(1)and then (22) is proved.

By using (21)–(23) we obtain log

M_k(ε)

=k(H−2)log(ε)+b_k+√

εS_g_k(ε)+oP

√ε

. (25)

Thus (13), (25) and property (15) entail that

k(H_k−2)=k(H−2)+

i=1

z_i√

εc_iS_g_k(εc_i)+oP

√ε .

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Then

(H_k−H )

√ε =1 k

i=1

z_i√

c_iS_g_k(εc_i)+oP(1).

Theorem 2.2 gives the required result (the computation of the coefficients in the Hermite expansion of functiong_k(·) is explicitly made in the proof of Corollary 3.2 of [3]).

(2) Let us see thatBk is a weakly consistent estimator ofbk. By using (14) and (25), one has

Bk−bk=k

(H−Hk) i=1

log(hi)+√ ε1

i=1

√ciSg_k(εci)+√ εoP(1).

Thus

B_k−b_k=klog(ε)(H−H_k)+k

(H−H_k) i=1

log(ci)+√ ε1

i=1

√c_iS_g_k(εc_i)+√ εoP(1).

Using assertion (1) of Theorem 3.2 and (24), we obtain [Bk−bk]

√εlog(ε) =k

H−Hk

√ε

+oP(1), (26)

and then using again assertion (1) of Theorem 3.2 we proved thatB_k is a weakly consistent estimator ofb_k.

Now using a Taylor expansion of order one for the exponential function, equality (26), the fact thatB_kis a weakly consistent estimator ofb_k and assertion (1) of Theorem 3.2, we finally get

[exp(B_k)−exp(bk)]

√εlog(ε) =kexp(bk)

H−H_k

√ε

+oP(1). (27)

Thus if we get back to the definition ofσ^k(see (16)) and if we use the last equality (27) we get (σ^k−σ^k)

√εlog(ε) =σ^kexp(−bk)

√εlog(ε)

exp(B_k)−exp(bk)

+exp(B_k) 1

σ^k

2Hk

− 1 σ_2H^k σ_2H^k

=kσ^k

H−Hk

√ε

+ σ

σ₂_H

k

exp(−b_k)exp(B_k)[σ_2H^k −σ^k

2H_k]

√εlog(ε) +oP(1).

At this step of the proof we are going to show that (σ^k−σ^k)

√εlog(ε) =kσ^k

H−H_k

√ε

+oP(1). (28)

Using the fact thatBkis a weakly consistent estimator ofbkit is enough to prove the following convergence (σ_2H^k −σ^k

2Hk)

√εlog(ε)

−→P ε→00, which is the same as showing

(σ_2H² −σ²

2H_k)

√εlog(ε)

−→P

ε→00. (29)

(11)

We write (σ_2H² −σ²

2Hk)

√εlog(ε) = 1 2π√

εlog(ε) _∞

−∞|x|⁽³⁻^{2H )}ϕ(−x)²

1−exp

2(H−Hk)log

|x|

dx.

Making a Taylor expansion for the exponential function we obtain (σ_2H² −σ²

2Hk)

√εlog(ε) = (Hk−H ) π√

εlog(ε) _∞

−∞|x|⁽³⁻^{2H )}ϕ(−x)²log

|x| exp

θ_ε(x) dx,

whereθ_ε(x)is a point between 0 and 2(H−H_k)log(|x|). By using assertion (1) of Theorem 3.2, and inequality exp

θ_ε(x)

≤exp

2|H−H_k|log

|x|,

we will get the convergence in (29) by showing the following convergence result _∞

−∞|x|⁽³⁻^{2H )}ϕ(−x)²|log

|x|

|exp

2|H−H_k|log

|x|dx

−→a.s.

ε→0

_∞

−∞|x|⁽³⁻^{2H )}ϕ(−x)²log

|x|dx <+∞. (30)

To prove the convergence in (30) we use the fact that(H−H_k)=oa.s.(1)and then forx=0,

|x|⁽³⁻^{2H )}ϕ(−x)²log

|x|exp

2|H−H_k|log

|x|

−→a.s.

ε→0|x|⁽³⁻^{2H )}ϕ(−x)²log

|x|.

Now, let 0< δ <inf(2H, (4−2H )), then almost surely for allω, there existsε(ω)such that 2|(H−Hk(ω))| ≤δ, whenε≤ε(ω). Furthermore, using the fact thatϕ is a density one has|ϕ(−x)|²≤1. Since forx=0,|ϕ(−x)|²≤ C|x|⁻⁴, then almost surely for allω, for allε≤ε(ω)andx=0, one obtains

|x|⁽³⁻^{2H )}ϕ(−x)²log

|x|exp

2|H−Hk|log

|x|

≤ |x|⁽³⁻^2(H⁺^δ/2))log

|x|1_|x|≤1+ |x|⁽⁻¹⁻^2(H⁻^δ/2))log

|x|1_|x|>1.

Since(H−δ/2) >0 and(4−2(H+δ/2)) >0, we can apply Lebesgue’s dominated convergence theorem to obtain the convergence in (30), thus we proved the convergence in (29) and equality (28) follows.

Now if we remark that the asymptotic behavior of(σ_k/σ−1)(see (17) for definition ofσ_k) is the same as that of (σ^k/σ^k−1)/ k, then by (28) and assertion (1) of Theorem 3.2 the proof of assertion (2) is complete.

Remark. The last step of the proof shows that the asymptotic behavior of((Hˆ_k−H )/√

ε, (σ_k−σ )/(√

εlog(ε)))is a

degenerated Gaussian law.

(1) Assertion (1) follows from assertion (1) of Theorem 3.1.

(2) Assertion (2) of Theorem 3.1 and the remark following Theorem 2.2 imply that √¹

ε([^σ^˜_σ^k]^k−1)converges weakly toσ_g_kN(0,1)that yields assertion (2).

Remark 2 follows from the fact that sincegˆ2,k=^k₂, one has σ_g²

k

k² = 1 k²

∞ n=1

ˆ

g_2n,k² (2n)! _+∞

−∞ ρ_H²ⁿ(x)dx≥ 2 k²gˆ_2,k²

_+∞

−∞ ρ_H²(x)dx=1 2

_+∞

−∞ ρ_H²(x)dx=σ_g²

2

4 .

(12)

3.2. Functional estimation ofσ (·)

Under certain regularity conditions forμ(·)andσ (·), we consider the “pseudo-diffusion” equations (1) with respect tobH(·), that is

X(t )=c+ _t

0

σ X(u)

dbH(u)+ _t

0

μ X(u)

du,

fort≥0,H >1/2 and positiveσ (·). We consider the problem of estimatingσ (·). Suppose we observe, as before, instead ofX(t )the smoothed by convolution processX_ε(t )=¹_ε_∞

−∞ϕ(^t⁻_ε^x)X(x)dx, withϕ(·)as in Section 2, where we have extendedX(·)by means ofX(t )=c, ift <0.

In a previous paper [2], in the case whereμ(·)≡0, estimation ofσ (·)is done, using the first increments ofX(·)or more generally the first derivative ofX_ε(·). Namely we proved the two following theorems.

Theorem 3.4. Let1/2< H <1.Ifh(·)∈C⁰andσ (·)∈C¹then π

2 ε⁽¹⁻^{H )}

σ_2H ₁

0

h

Xε(u)X˙ε(u)du−→^a.s.

₁

0

h X(u)

σ X(u)

du, whereσ˜_2H is defined by

˜

σ_2H² :=V

ε⁽¹^{−H )}b˙^ε_H(t )

= 1 2π

_+∞

−∞ |x|⁽¹⁻^{2H )}ϕ(ˆ −x)²dx.

The rate of convergence is given by Theorem 3.5.

Theorem 3.5. Let us suppose that 1/2 < H <3/4, h(·)∈C⁴, σ (·)∈C⁴, σ (·) is bounded and sup{|σ⁽⁴⁾(x)|,

|h⁽⁴⁾(x)|} ≤P (|x|),whereP (·)is a polynomial,then

√1 ε

π 2

ε⁽¹⁻^{H )}

˜ σ_2H

₁

0

h

Xε(u)X˙ε(u)du− ₁

0

h X(u)

σ X(u)

du , converges stably towards

σg₁

₁

0

h X(u)

σ X(u)

dW (u).

Here,W ( ·)is a standard Brownian motion independent ofb_H(·),g₁(x)= ^π₂|x| −1.

We give an outline of the proof of the last two above theorems in order to generalize these results to our setting, considering the case whereμ(·) is not necessarily null. Indeed, becausebH(·) has zero quadratic variation when H >1/2, Lin (see ([8]) proved that whenσ (·)∈C¹andμ(·)≡0, the solution to the stochastic differential equation (1) can be expressed asX(t )=K(b_H(t )), fort≥0, whereK(t )is the solution to the ordinary differential equation (ODE)

K(t )˙ =σ K(t )

, K(0)=c (31)

(fort <0,X(t )=c).

Heuristic proof of Theorem 3.4. We have shown in [2] that π

2 ε⁽¹⁻^{H )}

˜ σ2H

1 0

h

X_ε(u)X˙_ε(u)du

π 2

ε⁽¹⁻^{H )}

˜ σ_2H

₁

0

h K

b_H^ε(u) σ

K

b^ε_H(u)b˙^ε_H(u)du,