LEAST SQUARES ESTIMATION FOR ORNSTEIN-UHLENBECK PROCESSES DRIVEN BY THE WEIGHTED

Guangjun SHEN（申广君）†Xiuwei YIN（尹修伟）

Department of Mathematics，Anhui Normal University，Wuhu 241000，China

E-mail：gjshen@163.com；xweiyin@163.com

Litan YAN（闫理坦）

Department of Mathematics，Donghua University，Shanghai 201620，China

E-mail：litanyan@dhu.edu.cn



LEAST SQUARES ESTIMATION FOR ORNSTEIN-UHLENBECK PROCESSES DRIVEN BY THE WEIGHTED FRACTIONAL BROWNIAN MOTION∗

Guangjun SHEN（申广君）†Xiuwei YIN（尹修伟）

Department of Mathematics，Anhui Normal University，Wuhu 241000，China

E-mail：gjshen@163.com；xweiyin@163.com

Litan YAN（闫理坦）

Department of Mathematics，Donghua University，Shanghai 201620，China

E-mail：litanyan@dhu.edu.cn

In this article，we study a least squares estimator（LSE）of θ for the Ornstein-Uhlenbeck process X0=0，dXt=θXtdt+，t≥0 driven by weighted fractional Brownian motion Ba，bwith parameters a，b.We obtain the consistency and the asymptotic distribution of the LSE based on the observation｛Xs，s∈［0，t］｝as t tends to infinity.

Weighted fractional Brownian motion；least squares estimator；Ornstein-Uhlenbeck process

2010 MR Subject Classification60G15；60G18；60H05

1　Introduction

As an extension of Brownian motion，the fractional Brownian motion（fBm for short）exhibits long-range dependence and self similarity，having stationary increments.It is the usual candidate to model phenomena in which the self-similarity property can be observed from the empirical data.Recall that the fBm BHwith Hurst index H∈（0，1）is a central Gaussian process with=0 and the covariance function for all t，s≥0.This process was first introduced by Kolmogorov and studied by Mandelbrot and Van Ness［22］，where a stochastic integral representation in terms of a standard Brownian motion was established.For，BHcoincides with the standard Brownian motion B，but BHis neither a semimartingale nor a Markov process unlessSome surveys and literaturecould be found in Biagini et al［5］，Hu［18］，Mishura［24］，and Nualart［26］.On the other hand，many authors proposed to use more general self-similar Gaussian processes and random fields as stochastic models.Such applications raised many interesting theoretical questions about self-similar Gaussian processes and fields in general.Therefore，some generalizations of the fBm were introduced such as bi-fractional Brownian motion，sub-fractional Brownian motion，and weighted fractional Brownian motion.However，in contrast to the extensive studies on the fBm，there was little systematic investigation on other self-similar Gaussian processes.The main reason for this is the complexity of dependence structures for self-similar Gaussian processes that do not have stationary increments.

The weighted fractional Brownian motion（wfBm for short）Ba，bwith parameters a＞−1，|b|＜1，|b|＜a+1 is a centered and self-similar Gaussian process with long/short-range dependence.It admits the relatively simple covariance function

Clearly，for a=0，b=0，Ba，bcoincides with the standard Brownian motion B.For a=0，（1.1）reduces to

which corresponds to the covariance of the fBm with Hurst indexif−1＜b＜1.Hence，the wfBm is a family of processes which extend the fBm，perhaps it may be useful in some applications.This process Ba，b，appeared in Bojdecki et al［7］in a limit of occupation time fluctuations of a system of independent particles moving in Rdaccording a symmetric α-stable L´evy process（0＜α≤2），started from an inhomogeneous Poisson configuration with intensity measureand 0＜γ≤d=1＜α，a=−γ/α，b=1−1/α，the ranges of values of a and b being−1＜a＜0 and 0＜b≤1+a.The process Ba，balso appeared in Bojdecki et al［8］in a high density limit of occupation time fluctuations of the above mentioned particles system，where the initial Poisson configuration has finite intensity measure，with d=1＜α，a=−1/α，b=1−1/α.Moreover，the wfBm was first studied by Bojdecki et al［6］，and it is neither a semimartingale nor a Markov process unless a=0，b=0，so many of the powerful techniques from stochastic analysis are not available when dealing with Ba，b.Recently，Garz´on［13］shown that for certain values of the parameters，the weighted fractional Brownian sheets were obtained as limits in law of occupation time fluctuations of a stochastic particle model.Shen et al［29］studied Berry-Ess´een bounds and almost sure CLT for quadratic variation of the wfBm.The wfBm has properties analogous to those of the fBm（self-similarity，long-range dependence，Hüolder paths）.However，in comparison with the fBm，the wfBm has non-stationary increments and satisfies the following estimates（see Bojdecki et al［6］）：

for s，t≥0.Thus，Kolmogorov'scontinuity criterion implies that the wfBm is Hüolder continuous of order δ for any δ＜（1+b）.For simplicity throughout this article，let ca，b，Ca，b，Ca，b，θstand for positive constants depending only on the subscripts and their value may be different indifferent appearances.We can rewrite its covariance as

which gives

for b＞0.

The Ornstein-Uhlenbeck process X=｛Xt，t≥0｝driven by a certain type of noise Ztis described by the following linear stochastic differential equation（SDE）：

If the parameter θ∈（−∞，+∞）is unknown and if the process｛Xt，t≥0｝can be observed continuously，then an important problem is to estimate the parameter θ based on the（single path）observation｛Xt，t≥0｝.When Ztis the standard Brownian motion，this problem has been extensively studied.The most popular approaches are either the maximum likelihood estimators（MLE）or the least squares estimators（LSE），and in this case they coincide.For θ＜0（ergodic case），the MLE of θ is asymptotically normal（see Liptser and Shiryaev［21］，Kutoyants［20］）.For θ＞0（non-ergodic case），the MLE of θ is asymptotically Cauchy（see Basawa and Scott［3］，Dietz and Kutoyants［10］）.Other types of noise processes were also studied.For example，when Ztis an α−stable process，the MLE does not exist.Hu and Long［14，15］used the trajectory fitting method combining with the weighted least squares technique to study the parameter estimation of θ in both the ergodic and the non-ergodic cases.Hu and Long［16］study the problem of parameter estimation for mean-reverting α-stable motion observed at discrete time instants.When Ztis a fBm，the parameter estimation for θ was extensively studied using the MLE method（see，for example，Kleptsyna and Le Breton［19］，Prakasa Rao［28］）or using the LSE technique（see Hu and Nualart［17］，Es-Sebaiy and Nourdin［11］，Belfadi et al［4］，Es-Sebaiy［12］）.When Ztis a sub-fractional Brownian motion，the MLEs were studied in Diedhiou et al［9］in a more general case，and the LSEs were considered by Mendy［23］.

Motivated by all these results，in this article，we will consider the situation where Ztis a wfBm.Our first goal is to study the parameter estimation problem for the Ornstein-Uhlenbeck process driven by the wfBm

where θ＞0（the non-ergodic case），using the LSE（see，for example，Hu and Nualart［17］）defined by

whose minimum is achieved when

We study the consistency and the asymptotic distribution of the LSEbθtof θ based on the observation｛Xs，s∈［0，t］｝as t→∞.

Our second goal is to study some properties of the Ornstein-Uhlenbeck process driven by the wfBm

where θ＞0（the ergodic case）.

This article is organized as follows.In Section 2，we present the main results.Some preliminaries for the wfBm，some Lemmas，and the proof of main results are given in Section 3.The proof of Lemmas are included in the Appendix.

2　Main Results

In this section，we will study the SDE（1.2）and（1.5）driven by a wfBm Ba，bwith parameter a＞−1，0＜b＜1，b＜a+1 and where θ＞0 is the unknown parameter to be estimated from the observation X.

For the SDE（1.2），the explicit solution is given by

where the stochastic integralRt0e−θsdBa，bsis a Young integral.Let us introduce the following process

Using equations（1.2）and（2.1），we can rewrite the LSEbθtdefined in（1.3）as follows

The following theorems give the strong consistency and the asymptotic distribution of the LSE（the non-ergodic case）.

Theorem 2.1Assume that a＞−1，0＜b＜1，b＜a+1.Then，→θ almost surely as t tends to infinity.

Theorem 2.2Assume that−1＜a＜0，0＜b＜1+a and a+b＞0.Then，

as t tends to infinity，where C（1）denotes the standard Cauchy distribution.

For the SDE（1.5），the explicit solution is given by

where the stochastic integralis a Young integral.We obtain some properties about the solution Xt.

Proposition 2.3Let a＞−1，0＜b＜1，b＜1+a，and Xtbe given by（2.3）.We have

Moreover，when−1＜a＜0，0＜b＜1+a，

where Xtis given by（2.3）.We have

Proposition 2.4Let−1＜a＜0，0＜b＜1+a，and

3　Preliminaries，Main Lemmas and Proof of Results

3.1Preliminaries

Let Ba，bbe a wfBm with parameters a，b（a＞−1，0＜b＜1，b＜a+1），defined on the complete probability space（Ω，F，P）.As wfBm Ba，bis a Gaussian process，it is possible to construct a stochastic calculus of variations with respect to it，which will be related to the Malliavin calculus.Some surveys and references could be found in Al`os et al［1］，and Nualart［26］.We recall here the basic definitions and results of this calculus.

The crucial ingredient is the canonical Hilbert space H（is also said to be reproducing kernel Hilbert space）associated to the wfBm Ba，bwhich is defined as the closure of the linear space ε generated by the indicator functions｛1［0，t］，t∈［0，T］｝，with respect to the scalar product〈1［0，t］，1［0，s］〉H=Ra，b（t，s）.The application ε∋ϕBa，b（ϕ）（Ba，b（ϕ）is a Gaussian process on H such that E［Ba，b（ϕ）Ba，b（ψ）］=〈ϕ，ψ〉Hfor all ϕ，ψ∈H）is an isometry from ε to the Gaussian space generated by Ba，band it can be extended to H.The Hilbert space H can be written as

where

with φ（t，s）=b（t∧s）a（t∨s−t∧s）b−1.We can use the subspace|H|of H that is defined as the set of measurable function ϕ on［0，T］such that

It was shown that|H|is a Banach space with the norm‖ϕ‖|H|and ε is dense in|H|.

For b＞0，we denote by S the set of smooth functionals of the form

The derivative operator Da，bis then a closable operator from L2（Ω）into L2（Ω；H）.We denote by D1，2the closure of S with respect to the norm

The divergence integral δa，bis the adjoint operator of Da，b.That is，we say that a random variable u in L2（Ω；H）belongs to the domain of the divergence operator δa，b，denoted by Domis defined by the duality relationship

for any u∈D1，2.We have D1，2⊂Dom（δa，b）and for any u∈D1，2，

We will use the notation

to express the Skorohod integral of a process u，and the indefinite Skorohod integral is definedFor every n≥1，let Hnbe the n-th Wiener chaos of Ba，b，that is，the closed linear subspace of L2（Ω）generated by the random variables｛Hn（Ba，b（h）），h∈H，‖h‖H=1｝，where Hnis the n-th Hermite polynomial.The mapping In（h⊗n）=n！Hn（Ba，b（h））provides a linear isometry between the symmetric tensor product H⊙n（equipped with the modified normwhere H⊗ndenotes the tensor product，h⊗n∈H⊗n） and Hn.For every f，g∈H⊙n，the following multiplication formula holds，

where In（f）denotes the multiple stochastic integral of function f.

Let f，g：［0，T］→R be Hüolder continuous functions of orders α∈（0，1）and β∈（0，1）with＞1.Young［30］proved that the Riemann-Stieltjes integral（so-called Young integral）Rfsdgsexists.Moreover，if α=β∈）and F：R2→R is a function of class C1， the integralsexist in the Young sense and the following change of variables formula holds：

As a consequence，if is a process with Hüolder paths of orderthe integralis well-defined as Young integral.

Suppose that for any t∈［0，T］，ut∈D1，2（|H|），and

Then，by the same argument as in Al`os and Nualart［2］，we have

In particular，when ϕ is a non-random Hüolder continuous function of order α＞1−1+b2，we have

In addition，for all ϕ，ψ∈|H|，

3.2Main lemmas and proof of results

In order to prove Theorem 2.1 and Theorem 2.2，we need the following lemmas 3.1-3.5.

Lemma 3.1Assume that a＞−1，|b|＜1 and|b|＜1+a.Then

Moreover，when bR ＞0，one can obtain

Lemma 3.2Assume that b＞0，then，

almost surely as t tends to infinity.

Lemma 3.3Assume that b＞0，then，for all t≥0，we have

Lemma 3.4Assume that−1＜a＜0，b＞0，and a+b＞0.Let F be any σ（Ba，b）-measurable random variable such that P（F＜∞）=1.Then，

when t tends to infinity，where constant Ca，b，θ＞0 depends on a，b，θ and N～N（0，1）is independent of Ba，b.

Lemma 3.5Assume that b＞0，then，

Proof of Theorem 2.1 By（2.2），we have

where we use the fact

from（3.3）.Hence，the result follows from Lemmas 3.1 and 3.2. Proof of Theorem 2.2By（2.2）and Lemma 3.3，we have

By Lemma 3.1（ii）and Lemma 3.2，we have almost surely as t tends to infinity.By Lemma 3.4，we obtain as t tends to infinity.Moreover，we have

In contrast，it follows from Lemma 3.2 and Lemma 3.5 that

Proof of Proposition 2.3We first prove that（2.4）holds.Let s＜t，then，

where and for the term B2，we have

This implies（2.4）holds.

For inequality（2.5），we have

If t＜s，then，

If s＜t，then，

Next，we estimate inequality（2.6）.Let s＜t，then，

This completes the proof.

Proof of Proposition 2.4In fact，we have

we can write

In the same way，we can prove that2，3，hold.By triangular inequality，we obtain

This completes the proof.

4　Appendix：Proof of Main Lemmas

Proof of Lemma 3.1Firstly，we prove（i）holds.For any 0≤s＜t，observe that

From the Hüolder continuity of Ba，b，（i）holds.

Secondly，we prove（ii）holds.Note that the integral η∝：=R∝0e−θrdBa，bris well defined. In fact，we have

with Γ denoting the Gamma function.Moreover，ηtconverges to η∝in L2（Ω）.In fact，

In the following，we will prove that ηt→η∝almost surely as t→∞.Using Borel-Cantelli lemma，it is sufficient to prove that for any ε＞0，

Observe that

So，we have

and

From the self-similarity of Ba，b，we have

Combine with（4.3）and（4.4），（4.2）holds.This completes the proof.

Proof of Lemma 3.2By（4.1），we have

The continuity of η implies that for every t＞0，

almost surely.Moreover，it follows from the continuity of η and the point（ii）of Lemma 3.1 that

almost surely.Combining this convergence with（4.5）and（4.6），we have

almost surely.Therefore，by L'Hˆospital's rule，we have

almost surely.This completes the proof.

Proof of Lemma 3.3Let t≥0.By（3.3）

In contrast，by（3.4）and（3.5）

This completes the proof.

To get（4.7），it is sufficient to check the convergence of its covariance matrix，because the lefthand side in the previous convergence is a Gaussian vector（see Es-Sebaiy and Nourdin［11］）. Let us first prove that the limiting variance of e−θtRt0eθsdBa，bsexists as t→∞.We have

Proof of Lemma 3.4For any d≥1，s1···sd∈［0，∞），it is enough to prove that

where

because

and

Because ψ1（t）is increasing with respect to t，the monotone convergence theorem implies that there exists a constant Ca，b，θ＞0 depending on a，b，θ such that

To finish the proof，it remains to check that for all fixed s＞0，

For s＜t，

It is obvious that

and

This completes the proof.

Proof of Lemma 3.5Firstly，we prove the convergence（3.7）.In fact，

Secondly，we prove the convergence（3.8）.In fact，

This completes the proof.

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February 18，2014；revised April 28，2015.Guangjun Shen was supported by the National Natural Science Foundation of China（11271020），the Distinguished Young Scholars Foundation of Anhui Province（1608085J06）.Litan Yan was supported by the National Natural Science Foundation of China（11171062）.

†Corresponding author.