PARAMETER ESTIMATION FOR FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES: NON-ERGODIC CASE

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PARAMETER ESTIMATION FOR FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES:

NON-ERGODIC CASE

Rachid Belfadli, Khalifa Es-Sebaiy, Youssef Ouknine

To cite this version:

Rachid Belfadli, Khalifa Es-Sebaiy, Youssef Ouknine. PARAMETER ESTIMATION FOR FRAC-

TIONAL ORNSTEIN-UHLENBECK PROCESSES: NON-ERGODIC CASE. 2011. �hal-00569387�

PARAMETER ESTIMATION FOR FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES: NON-ERGODIC CASE

Rachid Belfadli¹ Khalifa Es-Sebaiy² Youssef Ouknine³

1 Polydisciplinary Faculty of Taroudant, University Ibn Zohr, Taroudant, Morocco.

[email protected]

2 Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon, France.

[email protected]

3 Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 2390 Marrakesh, Morocco.

[email protected]

February 25, 2011

Abstract

We consider the parameter estimation problem for the non-ergodic fractional Ornstein- Uhlenbeck process defined asdXt =θXtdt+dBt, t≥0, with a parameterθ >0, where B is a fractional Brownian motion of Hurst index H ∈ (¹₂,1). We study the consistency and the asymptotic distributions of the least squares estimator θbt ofθ based on the observation {Xs, s∈[0, t]}ast→ ∞.

Key words and phrases: Parameter estimation, Non-ergodic fractional Ornstein-Uhlenbeck process, Young integral.

2000 Mathematics Subject Classification: 62F12, 60G18, 60G15.

1 Introduction

We consider the Ornstein-Uhlenbeck processX ={X_t, t≥0}given by the following linear stochas- tic differential equation

X0= 0; dXt=θXtdt+dBt, t≥0, (1) whereB is a fractional Brownian motion of Hurst indexH > ¹₂ andθ∈(−∞,∞)is an unknown parameter. An interesting problem is to estimate the parameterθ when one observes the whole trajectory ofX. First, let us recall some results in the case whenBis a standard Brownian motion.

In this special case, the parameter estimation for θ has been well studied by using the classical maximum likelihood method or by using the trajectory fitting method. Ifθ <0(ergodic case), the maximum likelihood estimator (MLE) ofθis asymptotically normal (see Liptser and Shiryaev [9], Kutoyants [8]). Ifθ >0 (non-ergodic case), the MLE of θis asymptotically Cauchy (see Basawa and Scott [3], Dietz and Kutoyants [4]). Recently, in a more general context, several authors extended this study to some generalizations of Ornstein-Uhlenbeck process driven by Brownian motion (for instance, Barczy and Pap [2]). Similar properties of the asymptotic behaviour of MLE has also been obtained with respect to the trajectory fitting estimators (see Dietz and Kutoyants [4]).

WhenBis replaced by anα-stable Lévy motion in the equation (1), Hu and Long [6] discussed the parameter estimation of θ in both the ergodic and the non-ergodic cases. They used the trajectory fitting method combined with the weighted least squares technique.

Now, let us consider a parameter estimation problem of the parameter θ for the fractional Ornstein-Uhlenbeck processX of (1).

In the caseθ <0(corresponding to the ergodic case), Hu and Nualart [7] studied the parameter estimation forθby using the least squares estimator (LSE) defined as

bθt= Rt

0X_sdX_s Rt

0X_s²ds , t≥0. (2)

This LSE is obtained by the least squares technique, that is,bθt(formally) minimizes θ7−→

Z t 0

¯¯¯X˙_s+θX_s¯¯¯²ds.

To obtain the consistency of the LSE θbt, the authors of [7] are forced to consider Rt

0XsdXs as a Skorohod integral rather than Young integral in the definition (2). Assuming Rt

0X_sdX_s is a Skorohod integral andθ <0, they proved the strong consistence ofθbtifH≥ ¹₂, and that the LSE θb_t ofθ is asymptotically normal if H ∈[¹₂,³₄). Their proof of the central limit theorem is based on the fourth moment theorem of Nualart and Peccati [12].

In this paper, our purpose is to study the non-ergodic case corresponding to θ > 0. More precisely, we shall estimateθby the LSEθbtdefined in (2), where in our case, the integralRt

0XsdXs

is interpreted as a Young integral. Indeed in that case, we have θbt = ₂R_t^Xt2

0X_s²ds which converges almost surely toθ, asttends to infinity (see Theorem 1). Moreover, it turned out that the path- wise approach is the preferred way to simulate numerically an estimatorθbt. Our technics used in this work are inspired from the recent paper by Es-Sebaiy and Nourdin [5].

The organization of our paper is as follows. Section 2 contains the presentation of the basic tools that we will need throughout the paper: fractional Brownian motion, Malliavin derivative, Skorohod integral, Young integral and the link between Young and Skorohod integrals. The aim of Section 3 is twofold. Firstly, we prove when H > ¹₂ the strong consistence of the LSEθb_t, that is, θbt converges almost surely toθ, as tgoes to infinity. Secondly, we investigate the asymptotic distribution of our estimatorbθtin the caseH > ¹₂. We obtain that (see Theorem 5)

e^θt

³bθt−θ

´−→law 2θC(1) as t−→ ∞,

withC(1)the standard Cauchy distribution with the probability density function _π(1+x¹ ₂₎; x∈R.

2 Preliminaries

In this section we describe some basic facts on the stochastic calculus with respect to a fractional Brownian motion. For more complete presentation on the subject, see [11], [1] and [10].

The fractional Brownian motion (B_t, t ≥ 0) with Hurst parameter H ∈ (0,1), is defined as a centered Gaussian process starting from zero with covariance

RH(t, s) =E(BtBs) = 1 2

¡t^2H+s^2H− |t−s|^2H¢ .

We assume thatB is defined on a complete probability space(Ω,F, P)such thatF is the sigma- field generated byB. By Kolmogorov’s continuity criterion and the fact

E(B_t−B_s)²=|s−t|^2H; s, t≥ 0,

we deduce thatB has Hölder continuous paths of orderH−ε, for allε∈(0, H).

Fix a time interval [0, T]. We denote by H the canonical Hilbert space associated to the fractional Brownian motion B. That is, H is the closure of the linear span E generated by the indicator functions1[0,t], t∈[0, T]with respect to the scalar product

h1_[0,t],1_[0,s]i=R_H(t, s).

The applicationϕ∈ E −→B(ϕ)is an isometry fromE to the Gaussian space generated byBand it can be extended toH.

If H > ¹₂ the elements of Hmay be not functions but distributions of negative order (see [13]).

Therefore, it is of interest to know significant subspaces of functions contained in it.

Let|H|be the set of measurable functionsϕon[0, T]such that kϕk²_|H| :=H(2H−1)

Z T 0

Z T 0

|ϕ(u)||ϕ(v)||u−v|^2H−2dudv <∞.

Note that, ifϕ, ψ∈ |H|,

E(B(ϕ)B(ψ)) =H(2H−1) Z T

0

Z T 0

ϕ(u)ψ(v)|u−v|^2H−2dudv.

It follows actually from [13] that the space |H| is a Banach space for the norm k.k_|H| and it is included inH. In fact,

L²([0, T])⊂L^H1([0, T])⊂ |H| ⊂ H.

LetC^∞_b (Rⁿ,R)be the class of infinitely differentiable functionsf :Rⁿ−→Rsuch thatf and all its partial derivatives are bounded. We denote by S the class of smooth cylindrical random variables F of the form

F =f(B(ϕ1), ..., B(ϕn)), (3)

wheren≥1,f ∈C^∞_b (Rⁿ,R)andϕ₁, ..., ϕ_n ∈ H.

The derivative operatorDof a smooth and cylindrical random variableFof the form (3) is defined as theH-valued random variable

DtF = XN

i=1

∂f

∂xi

(B(ϕ1), ..., B(ϕn))ϕi(t)

In this way the derivativeDF is an element ofL²(Ω;H). We denote byD^1,2the closure ofSwith respect to the norm defined by

kFk²1,2=E(kFk²) +E(kDFk²_H).

The divergence operator δ is the adjoint of the derivative operator D. Concretely, a random variableu∈L²(Ω;H)belongs to the domain of the divergence operatorDomδ if

E|hDF, uiH| ≤ckFkL²(Ω)

for everyF ∈ S. In this case δ(u)is given by the duality relationship E(F δ(u)) =EhDF, ui_H

for anyF ∈D^1,2. We will make use of the notation δ(u) =

Z T 0

usδBs, u∈Dom(δ).

In particular, forh∈ H,B(h) =δ(h) =RT 0 h_sδB_s.

For every n ≥1, let Hn be the nth Wiener chaos of B , that is, the closed linear subspace of L²(Ω) generated by the random variables {Hn(B(h)), h∈ H,khkH = 1} whereHn is the nth Hermite polynomial. The mappingIn(h^⊗n) =n!Hn(B(h))provides a linear isometry between the symmetric tensor productH^¯n (equipped with the modified normk.k_H^¯n= √¹

n!k.k_H^⊗n) and Hn. For everyf, g∈ H^¯n the following multiplication formula holds

E(In(f)In(g)) =n!hf, gi_H^⊗n.

Finally, It is well-known that L²(Ω) can be decomposed into the infinite orthogonal sum of the spacesHn. That is, any square integrable random variableF∈L²(Ω)admits the following chaotic expansion

F =E(F) + X∞ n=1

In(fn), where thef_n ∈ H^¯n are uniquely determined byF.

Fix T > 0. Let f, g : [0, T] −→ Rare Hölder continuous functions of orders α∈ (0,1) and β∈(0,1)withα+β >1. Young [14] proved that the Riemann-Stieltjes integral (so-called Young integral) RT

0 fsdgs exists. Moreover, if α = β ∈ (¹₂,1) and φ : R² −→ R is a function of class C¹, the integrals R.

0

∂φ

∂f(fu, gu)dfu andR. 0

∂φ

∂g(fu, gu)dguexist in the Young sense and the following change of variables formula holds:

φ(ft, gt) =φ(f0, g0) + Z t

0

∂φ

∂f(fu, gu)dfu+ Z t

0

∂φ

∂g(fu, gu)dgu, 0≤t≤T. (4) As a consequence, ifH > ¹₂ and(ut, t∈[0, T])be a process with Hölder paths of orderα >1−H, the integralRT

0 usdBsis well-defined as Young integral. Suppose moreover that for anyt∈[0, T], ut∈D^1,2, and

P ÃZ T

0

Z T 0

|D_su_t||t−s|^2H−2dsdt <∞

!

= 1.

Then, by [1],u∈Domδand for everyt∈[0, T], Z t

0

usdBs= Z t

0

usδBs+H(2H−1) Z t

0

Z t 0

Dsur|s−r|^2H−2drds. (5) In particular, whenϕis a non-random Hölder continuous function of order α >1−H, we obtain

Z T 0

ϕ_sdB_s= Z T

0

ϕ_sδB_s=B(ϕ). (6)

In addition, for allϕ, ψ∈ |H|, E

ÃZ T 0

ϕ_sdB_s Z T

0

ψ_sdB_s

!

=H(2H−1) Z T

0

Z T 0

ϕ(u)ψ(v)|u−v|^2H−2dudv. (7)

3 Asymptotic behavior of the least squares estimator

Throughout this paper we assume H ∈(¹₂,1) andθ >0. Let us consider the equation (1) driven by a fractional Brownian motionB with Hurst parameterH andθ is the unknown parameter to be estimated from the observationX. The linear equation (1) has the following explicit solution:

Xt=e^θt Z t

0

e^−θsdBs, t≥0, (8)

where the integralRt

0e^−θsdBsis a Young integral.

Let us introduce the following process ξ_t:=

Z t 0

e^−θsdB_s, t≥0.

By using the equation (1) and (8) we can write the LSEbθtdefined in (2) as follows θbt=θ+

Rt 0XsdBs

Rt

0X_s²ds =θ+ Rt

0e^θsξsdBs

Rt

0e^2θsξ_s²ds. (9)

3.1 Consistency of the estimator LSE

The following theorem proves the strong consistency of the LSEθbt. Theorem 1 Assume H ∈(¹₂,1), then

θbt−→θ almost surely

ast−→ ∞.

For the proof of Theorem 1 we need the following two lemmas.

Lemma 2 Suppose thatH > ¹₂. Then

i) For allε∈(0, H), the processξadmits a modification with(H−ε)-Hölder continuous paths, still denotedξ in the sequel.

ii) ξ_t−→ξ_∞:=R_∞

0 e^−θrdB_r almost surely and inL²(Ω) ast−→ ∞. Lemma 3 Let H > ¹₂. Then, ast→ ∞,

e^−2θt Z t

0

X_s²ds=e^−2θt Z t

0

e^2θsξ²_sds−→ ξ²_∞

2θ almost surely.

Proof of Lemma 2. We prove the pointi). We have, for every 0≤s < t, E(ξt−ξs)² = E

µZ t s

e^−θrdBr

¶2

= H(2H−1) Z t

s

Z t s

e^−θue^−θv|u−v|^2H−2dudv

≤ H(2H−1) Z t

s

Z t s

|u−v|^2H−2dudv

= E(Bt−Bs)²=|t−s|^2H.

Thus, by applying the Kolmogorov-Centsov theorem to the centered gaussian processξwe deduce i).

Concerning the second pointii), we first notice that the integralξ_∞=R_∞

0 e^−θrdB_ris well defined.

In fact,

H(2H−1) Z _∞

0

Z _∞

0

e^−θre^−θs|r−s|^2H−2drds

= 2H(2H−1) Z _∞

0

dse^−θs Z s

0

dre^−θr(s−r)^2H−2

= 2H(2H−1) Z _∞

0

dse^−2θs Z s

0

due^θuu^2H−2

= 2H(2H−1) Z _∞

0

due^θuu^2H−2 Z _∞

u

dse^−2θs

= H(2H−1) θ

Z _∞

0

e^−θuu^2H−2du

= H(2H−1)

θ^2H Γ(2H−1) = HΓ(2H)

θ^2H <∞, (10)

withΓdenotes the classical Gamma function. Moreover,ξ_tconverges toξ_∞in L²(Ω). Indeed, E£

(ξt−ξ_∞)²¤

= H(2H−1) Z _∞

t

Z _∞

t

e^−θre^−θs|r−s|^2H−2drds

= 2H(2H−1) Z _∞

t

dse^−θs Z s

t

dre^−θr(s−r)^2H−2

= 2H(2H−1) Z _∞

t

dve^−2θs Z s−t

0

due^θuu^2H−2

= 2H(2H−1)e^−2θt Z _∞

0

dve^−2θv Z v

0

due^θuu^2H−2

= 2H(2H−1)e^−2θt Z _∞

0

due^θuu^2H−2 Z _∞

u

dve^−2θv

= H(2H−1) θ e^−2θt

Z _∞

0

e^−θuu^2H−2du

= HΓ(2H) θ^2H e^−2θt

→ 0 ast→ ∞.

Now, let us show that ξt−→ ξ_∞ almost surely ast → ∞. By using Borel-Cantelli lemma, it is sufficient to prove that, for anyε >0

X

n≥0

P µ

sup

n≤t≤n+1

¯¯¯¯Z _∞

t

e^−θsdBs

¯¯¯¯> ε

¶

<∞. (11)

For this purpose, let ¹₂ < α < H. As in the proof of [Theorem 4, [1]], we can write for everyt >0 Z _∞

t

e^−θsdBs=c⁻_α¹ Z _∞

t

dBse^−θs µZ s

t

dr(s−r)^−α(r−t)^α−1

¶ ,

withc_α=Rs

t(s−r)^−α(r−t)^α−1dr=β(α,1−α), whereβ is the Beta function.

By Fubini’s stochastic theorem (see for example [11]), we have Z _∞

t

e^−θsdBs=c⁻_α¹ Z _∞

t

dr(r−t)^α−1 µZ _∞

r

dBse^−θs(s−r)^−α

¶ .

Cauchy-Schwarz’s inequality implies that

¯¯¯¯Z _∞

t

e^−θsdBs

¯¯¯¯²

≤ c⁻_α² µZ _∞

t

(r−t)^2(α−1)e^−θ(r−t)dr

¶ ÃZ _∞

t

dre^−θ(r−t)¯¯

¯¯Z _∞

r

dB_se^−θs(s−r)^−αe^θ(r−t)¯¯

¯¯²

!

= c⁻_α²Γ(2α−1) θ^2α−1 e^−2θt

Z _∞

t

dre^−θ(r−t)¯¯

¯¯Z _∞

r

dBs(s−r)^−αe^−θ(s−r)¯¯

¯¯² Thus,

sup

n≤t≤n+1

¯¯¯¯Z _∞

t

e^−θsdBs

¯¯¯¯²

≤ c⁻_α²Γ(2α−1)

θ^2α−1 e^−2θne^θ Z _∞

n

dre^−θ(r−n)¯¯

¯¯Z _∞

r

dB_s(s−r)^−αe^−θ(s−r)¯¯

¯¯² On the other hand,

E ﯯ¯

Z _∞

r

(s−r)^−αe^−θ(s−r)dBs

¯¯¯¯²

!

= H(2H−1) Z _∞

r

dv(v−r)^−αe^−θ(v−r) Z _∞

r

du(u−r)^−αe^−θ(u−r)|u−v|^2H−2

= H(2H−1) Z _∞

0

dvv^−αe^−θv Z _∞

0

duu^−αe^−θu|u−v|^2H−2

= 2H(2H−1) Z _∞

0

dvv^−αe^−θv Z v

0

duu^−αe^−θu(v−u)^2H−2

= 2H(2H−1) Z _∞

0

dvv^−αe^−θv Z v

0

du(v−u)^−αe^−θ(v−u)u^2H−2

≤ 2H(2H−1) Z _∞

0

dvv^−αe^−θv Z v

0

du(v−u)^−αu^2H−2

= 2H(2H−1) Z _∞

0

dvv^2H−2α−1e^−θv Z 1

0

duu^2H−2(1−u)^−α

= 2H(2H−1)Γ(2H−2α)β(2H−1,1−α)

θ^2H−2α :=C₁(α, H, θ)<∞. Combining this with the fact thatR_∞

n e^−θ(r−n)dr=¹_θ, we obtain E

à sup

n≤t≤n+1

¯¯¯¯Z _∞

t

e^−θsdB_s¯¯

¯¯²

!

≤ C₂(α, H, θ)e^−2θn, with

C2(α, H, θ) =c⁻_α²Γ(2α−1)e^θ

θ^2α C1(α, H, θ).

Consequently, X

n≥0

P µ

sup

n≤t≤n+1

¯¯¯¯Z _∞

t

e^−θsdB_s¯¯

¯¯> ε

¶

≤ ε⁻²X

n≥0

E Ã

sup

n≤t≤n+1

¯¯¯¯Z _∞

t

e^−θsdB_s¯¯

¯¯²

!

≤ ε⁻²C₂(α, H, θ)X

n≥0

e^−2θn <∞.

This finishes the proof of the claim (11), and thus the proof of Lemma 2.

Proof of Lemma 3. Using (10), we have

E[ξ_∞² ] = HΓ(2H) θ^2H <∞. Henceξ_∞vN(0,^HΓ(2H)_θ2H )and this implies that

P(ξ_∞= 0) = 0. (12)

The continuity ofξentails that, for everyt >0 Z t

0

e^2θsξ²_sds≥ Z t

t 2

e^2θsξ_s²ds≥ t 2e^θt

à inf

t 2≤s≤t

ξ²_s

!

almost surely. (13)

Furthermore, the continuity ofξand the point ii) of Lemma 2 yield

tlim→∞

Ã

tinf

2≤s≤t

ξ_s²

!

=ξ_∞ almost surely.

Combining this last convergence with (13) and (12), we deduce that

tlim→∞

Z t 0

e^2θsξ_s²ds=∞ almost surely.

Hence, we can use L’Hôspital’s rule and we obtain

tlim→∞

Rt

0e^2θsξ²_sds e^2θt = lim

t→∞

ξ_t² 2θ =ξ_∞²

2θ almost surely.

This completes the proof of Lemma 3.

Proof of Theorem 1. Using the change of variable formula (4), we conclude that 1

2e^2θtξ_t²=θ Z t

0

e^2θsξ²_sds+ Z t

0

e^θsξsdBs

Hence

bθ_t−θ= Rt

0e^θsξsdBs

Rt

0e^2θsξ²_sds = ξ_t² 2e^−2θtRt

0e^2θsξ_s²ds−θ.

Combining this with Lemma 2 and Lemma 3, we deduce thatθbt→θalmost surely ast−→ ∞.

3.2 Asymptotic distribution of the estimator LSE

This paragraph is devoted to the investigation of asymptotic distribution of the LSE bθtofθ. We start with the following lemma.

Lemma 4 Suppose thatH > ¹₂. Then, for everyt≥0, we have Z t

0

dBse^θs Z s

0

dBre^−θr = Z t

0

dBse^θs Z t

0

dBre^−θr− Z t

0

δBse^−θs Z s

0

δBre^θr

−H(2H−1) Z t

0

dse^−θs Z s

0

dre^θr|s−r|^2H−2.

Proof. Lett≥0. By the change of variable formula (4) Z t

0

dBse^θs Z s

0

dBre^−θr = Z t

0

dBse^θs Z t

0

dBre^−θr− Z t

0

dBse^−θs Z s

0

dBre^θr. On the other hand, according to (5) and (6),

Z t 0

dBse^−θs Z s

0

dBre^θr

= Z t

0

δB_se^−θs Z s

0

δB_re^θr+H(2H−1) Z t

0

dse^−θs Z s

0

dre^θr|s−r|^2H−2,

which completes the proof.

Theorem 5 Let H > ¹₂ be fixed. Then, as t−→ ∞, e^θt

³θb_t−θ

´−→law 2θC(1),

withC(1) the standard Cauchy distribution.

In order to prove Theorem 5 we need the following two lemmas.

Lemma 6 FixH > ¹₂. LetF be anyσ{B}-measurable random variable such thatP(F <∞) = 1.

Then, as t−→ ∞, µ

F, e^−θt Z t

0

e^θsdBs

¶

−→law

à F,

pHΓ(2H)

θ^H N

! ,

whereN ∼ N(0,1) is independent ofB.

Lemma 7 Let H > ¹₂. Then, ast→ ∞,

e^−θt2 Z t

0

δBse^−θs Z s

0

δBre^θr−→0 inL²(Ω), (14) and

e^−θt2 Z t

0

dse^−θs Z s

0

dre^θr|s−r|^2H−2−→0. (15) Proof of Lemma 6. For anyd≥1,s1. . . sd∈[0,∞), we shall prove that, as t−→ ∞,

µ

B_s1, . . . , B_sd, e^−θt Z t

0

e^θsdB_s

¶

−→law

Ã

B_s1, . . . , B_sd,

pHΓ(2H) θ^H N

!

(16) which is enough to lead to the desired conclusion. Because the left-hand side in the previous convergence is a Gaussian vector (see proof of [Lemma 7, [5]]), to get (16) it is sufficient to check the convergence of its covariance matrix. Let us first compute the limiting variance of

e^−θtRt

0e^θsdBs ast−→ ∞. We have E

"µ e^−θt

Z t 0

e^θsdBs

¶2#

= H(2H−1)e^−2θt Z t

0

Z t 0

e^θse^θr|s−r|^2H−2drds

= 2H(2H−1)e^−2θt Z t

0

dse^θs Z s

0

dre^θr|s−r|^2H−2

= 2H(2H−1)e^−2θt Z t

0

dse^2θs Z s

0

dre^−θrr^2H−2

= 2H(2H−1)e^−2θt Z t

0

dre^−θrr^2H−2 Z t

r

dse^2θs

= H(2H−1) θ

µZ t 0

r^2H−2e^−θrdr−e^−2θt Z t

0

r^2H−2e^θrdr

¶

→ HΓ(2H)

θ^2H ast→ ∞, becausee^−2θtRt

0r^2H−2e^θrdr≤e^−θtRt

0r^2H−2dr=_(2H^t2H−1_−1)eθt →0ast→ ∞. Thus,

lim

t→∞E

"µ e^−θt

Z t 0

e^θsdB_s

¶²#

= HΓ(2H) θ^2H .

Hence, to finish the proof it remains to check that, for all fixeds≥0,

tlim→∞E µ

B_s×e^−θt Z t

0

e^θvdB_v

¶

= 0.

Indeed, fors < t, E

µ

Bs×e^−θt Z t

0

e^θvdBv

¶

= H(2H−1)e^−θt Z t

0

dve^θv Z s

0

du|u−v|^2H−2

= H(2H−1)e^−θt Z s

0

dve^θv Z s

0

du|u−v|^2H−2+H(2H−1)e^−θt Z t

s

dve^θv Z s

0

du(v−u)^2H−2

= H(2H−1)e^−θt Z s

0

dve^θv Z s

0

du|u−v|^2H−2+He^−θt Z t

s

e^θv(v^2H−1−(v−s)^2H−1)dv := It+Jt.

It’s clear thatIt→0 ast→ ∞.

Using integration by parts, the thermJtcan be written as Jt

= He^−θt Z t

s

e^θv(v^2H−1−(v−s)^2H−1)dv

= He^−θt µe^θt

θ [t^2H−1−(t−s)^2H−1]−e^θs

θ s^2H−1+2H−1 θ

Z t s

e^θv[(v−s)^2H−2−v^2H−2]dv

¶

≤ H

θ[t^2H−1−(t−s)^2H−1] +H(2H−1) θ e^−θt

Z t s

e^θv(v−s)^2H−2dv := J_t¹+J_t².

SinceH <1,J_t¹=^H_θ[t^2H−1−(t−s)^2H−1]→0 ast→ ∞. On the other hand,

J_t² = H(2H−1) θ e^−θt

Z t s

e^θv(v−s)^2H−2dv

= H(2H−1)e^θs θ e^−θt

Z t−s 0

e^θuu^2H−2du

≤ H(2H−1)e^θs θ e^−θt

Z t 0

e^θuu^2H−2du

= H(2H−1)e^θs θ

Z t 0

e^−θv(t−v)^2H−2dv

= H(2H−1)e^θs θ t^2H−1

Z 1 0

e^−θtu(1−u)^2H−2du

Fixu∈(0,1). The functiont∈[0,∞)7→t^2H−1e^−θtuattains its maximum att=^2H_θu⁻¹. Then sup

t≥0

(t^2H−1e^−θtu) =ce^−2H−1u u^1−2H ≤cu^1−2H,

withc=¡_2H−1

θ

¢2H−1

. In addition,R1

0 u^1−2H(1−u)^2H−2du <∞, and for anyu∈(0,1), t^2H−1e^−θtu(1−u)^2H−2→0 ast→ ∞.

Therefore, using the dominated convergence theorem, we obtain thatJ_t²converges to0ast→ ∞. Thus, we deduce the desired conclusion.

Proof of Lemma 7. Let us prove the convergence (14). We have e^−θtE

µZ t 0

δBse^−θs Z s

0

δBre^θr

¶2

= e^−θtE µZ t

0

Z s 0

e^−θ|s−r|δB_rδB_s

¶2

= e^−θtE µ1

2I₂(e^−θ|s−r|1_[0,t]2)

¶2

= H²(2H−1)² 2 e^−θt

Z

[0,t]⁴

e^−θ|v−s|e^−θ|u−r||v−u|^2H−2|s−r|^2H−2dudvdrds

≤ H²(2H−1)² 2 e^−θt

Z

[0,t]⁴

|v−u|^2H−2|s−r|^2H−2dudvdrds

= 1

2[E(B_t²)]²e^−θt=1

2t^4He^−θt

→ 0 as t→ ∞. For the convergence (15), we have

H(2H−1)e^−θt2 Z t

0

dse^−θs Z s

0

dre^θr|s−r|^2H−2

≤ H(2H−1)e^−θt2 Z t

0

ds Z s

0

dr|s−r|^2H−2

= t^2H 2 e^−θt2

→ 0 ast→ ∞.