• No results found

Minimum distance estimation for fractional Ornstein Uhlenbeck type process

N/A
N/A
Protected

Academic year: 2020

Share "Minimum distance estimation for fractional Ornstein Uhlenbeck type process"

Copied!
8
0
0

Loading.... (view fulltext now)

Full text

(1)

R E S E A R C H

Open Access

Minimum distance estimation for fractional

Ornstein-Uhlenbeck type process

Zaiming Liu and Na Song

*

*Correspondence:

[email protected] School of Mathematics and Statistics, Central South University, Changsha, China

Abstract

We consider a one-dimensional linear stochastic differential equation defined as

dXt=

θ

Xtdt+

ε

dBH

t,X0=x0, with

θ

the unknown drift parameter, where

{BHt, 0≤tT}is a fractional Brownian motion with

ε

> 0. The consistency and the asymptotic distribution of the minimum Skorohod distance estimator

θ

ε∗of

θ

based on the observation{Xt, 0≤tT}is studied asT→+∞.

Keywords: long-range dependence; minimum distance estimation; consistency; asymptotic distribution

Introduction

Stochastic models having long-range dependence phenomena have been paid much atten-tion to in view of their applicaatten-tions in signal processing, computer networks, and math-ematical finance (see [, ]). The long-range dependence phenomenon is said to occur in a stationary time series{Xn,n≥}if the autocovariance functionsρ(n) :=cov(Xk,Xk+)

satisfy

lim

n→∞

ρ(n)

cnα = 

for some constantcandα∈(, ). In this case, the dependence betweenXkandXk+ndecays

slowly asn→ ∞and∞n=ρ(n) =∞.

Fractional Brownian motions are a special class of long memory processes when the Hurst parameterH>. When one implements the fractional Ornstein-Uhlenbeck model, it is important to estimate the parameters in the model.

In case of diffusion type processes driven by fractional Brownian motions, the most im-portant methods are either maximum likelihood estimation (MLE) or least square estima-tion (LSE). Substantial progress has been made in this direcestima-tion. The problem of parameter estimation in a simple linear model driven by a fractional Brownian motion was studied in [] in the continuous case. For the case of discrete data, the problem of parameter es-timation was studied in [, ]. Hu and Nualart [] studied a least squares estimator for the Ornstein-Uhlenbeck process driven by fractional Brownian motion and derived the asymptotic normality of by using Malliavin calculus. The MLE of the drift parameter has also been extensively studied. Kleptsyna and Le Breton [] considered one-dimensional homogeneous linear stochastic differential equation driven by a fractional Brownian mo-tion in place of the usual Brownian momo-tion. The asymptotic behavior of the maximum likelihood estimator of the drift parameter was analyzed. Tudor and Viens [] applied the

(2)

techniques of stochastic integration with respect to fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the MLE for the drift parameter of stochastic processes satisfying stochastic equations driven by a fractional Brownian motion with any level of Hölder-regularity (any Hurst parameter). Moreover, in recent years, there has been increased interest in studying the asymptotic properties of the MLE for the drift parameter in some fractional diffusion systems (see [–]). However, MLE has some shortcomings; its expressions of likelihood function are not explicitly computable. Moreover, MLE are not robust, which means that the properties of MLE will be changed by a slight perturbation.

In order to overcome this difficulty, the minimum distance estimation approach is proposed. For a more comprehensive discussion of the properties of the minimum dis-tance estimation, we refer to Millar []. In this direction, the parameter estimation for Ornstein-Uhlenbeck process driven by Brownian motions is well developed. Kutoyants and Pilibossian [] and Kutoyants [] proved thatε–(θε∗–θ) converge in probability

to the random variableζT with L,L or supremum norm and he also proved thatζT

is asymptotically normal whenθ>  asT→+∞. Hénaff [] established the same

re-sults in the general case of a norm in some Banach space of functions on [,T]. Diop and Yode [] studied the minimum Skorohod distance estimation for a stochastic differential equation driven by a centered Lévy process. However, there have been very few studies on the minimum distance estimate for the fractional Ornstein-Uhlenbeck process. Prakasa Rao [] studied the minimumL-norm estimatorθε∗of the drift parameter of a fractional

Ornstein-Uhlenbeck type process and proved thatε–(θε∗–θ) converges in probability un-derPθto a random variableζ. Our main motivation is to obtain the minimum Skorohod distance estimator of Ornstein-Uhlenbeck process driven by fractional Brownian motions and study the asymptotic properties of this estimator.

Preliminaries

Let (,F,{Ft}t≥,P) be a stochastic basis satisfying the usual conditions,i.e., a filtered

probability space with a filtration.{Ft}t≥is right continuous andFcontains everyP-null

set. Suppose that the processes discussed in the following are{Ft}t≥-adapted. Further the

natural filtration of a process is understood as theP-completion of the filtration generated by this process.

Consider the parameter estimation problem for a special fractional process,i.e., frac-tional Ornstein-Uhlenbeck type processX={Xt, ≤tT}, which satisfies the following

stochastic integral equation:

Xt=x+θ

t

Xsds+εBHt , ≤tT, ()

where the drift parameterθ= (θ,θ)⊆Ris unknown,ε> , andBH={BHt (t), ≤

tT} is a scalar fractional Brownian motion defined on the probability space (,F,

{Ft}t≥,P). For a fractional Brownian motionBHwith Hurst parameterH∈(, ), we mean

that it is a continuous and centered Gaussian process with the covariance function

EBHsBHt = 

sH+tH–|st|H, t≥,s≥. ()

(3)

Definition  We say that anRd-valued random processX= (X

t)t≥is self-similar or

sat-isfies the property of self-similarity if for everya>  there existsb>  such that

Law(Xat,t≥) =Law(bXt,t≥), ()

whereLaw(·) denotes the law of random variable·.

Remark  Note that () means that the two processesXatandXbthave the same

finite-dimensional distribution functions,i.e., for every choicet, . . . ,tninR,

P(Xat≤x, . . . ,Xatnxn) =P(Xbt≤x, . . . ,Xbtnxn)

for everyx, . . . ,xninR.

Definition  Ifb=aHin the above definition, then we say thatX= (X

t)t≥is a self-similar

process with Hurst indexHor that it satisfies the property of (statistical) self-similar with Hurst indexH. The quantityD= /His called the statistical fractal dimension ofX.

Remark  Note that the law of a Gaussian random variance is determined by its

expecta-tion value and variaexpecta-tion. By (), it is easy to see thatBHis a self-similar process with Hurst indexH. Let

BHT∗:= sup ≤tT

BHt . ()

Then we conclude from the fact thatBHis a self-similar process with Hurst indexHthat

LawBHat∗=LawaHBHt ∗, a> ,t≥. ()

Letxt(θ) be the solution of the above differential equation withε= . It is obvious that

xt(θ) =xeθt, ≤tT. ()

Let

KH(t,s) =H(H– )

d ds

t

s

rH–(rs)H–dr, st. ()

Define the minimum Skorohod distance estimator

θε∗:=arg min

θ∈ρ

X,x(θ), ()

the Skorohod distanceρ(·,·)

ρ(x,y) := inf

λ∈([,T])

H(λ) + sup

t∈[,T]

(t)–y(t) ()

(4)

Here

[,T]:=λ|t∈[,T],λ(t)∈[,T], ()

([,T]) is continuous, strictly increasing such thatλ() =  andλ(T) =T. Let

H(λ) := sup

s,t∈[,T],s=t

log

λ(s) –λ(t)

st

<∞. ()

Note that the spaceD([,T],R) consists of functions which are right continuous with left limits on [,T]. The uniform metric coincides with Skorohod distance when relativized toC([,T],R) the space of continuous functions on [,T].

Denoteθthe true parameter ofθ andP(θε) be the probability measure induced by the

process{Xt}.

The following two lemmas due to Novikov and Valkeila [] and Kutoyants and Pili-bossian [] play an important role in the limit analysis below.

Lemma  Let T> ,BHT∗=suptTBH

t and{BHt (t), ≤tT}be a fractional Brownian

motion with Hurst parameter H,then for every p> ,

EBHTp=K(p,H)TpH, ()

where K(p,H) =E(BH

 )p.

Lemma  Let Zε(u),ε> ,u∈Rbe a sequence of continuous functions and Z(u)a convex function which admits a unique minimumξ ∈R.Let Lε,ε> be a sequence of positive

numbers such that Lε→+∞asε→.We suppose that

lim

ε→|usup|<L

ε

(u) –Z(u) = ,

then

lim

ε→arg min|u|<

(u) =ξ,

where if there are several minima of Zε,we choose an arbitrary one.

For anyδ> , define

g(δ) := inf

|θθ|>δx(θ) –x(θ)∞. ()

Note thatg(δ) >  for anyδ> . Introduce the random variable

ζ:=arg min

u∈Rρ

Y(θ),ux(θ)

, ()

(5)

It can be obtained from () that

Xtxt(θ) =εeθt

t

eθsdBH

s. ()

Let

Yt=t

t

eθsdBH

s. ()

Note that{Yt, ≤tT}is a Gaussian process and can be interpreted as the ‘derivative’

of the process{Xt, ≤tT}with respect toε.

Now we investigate the parameter estimation problem of parameterθ based on the ob-servation of a fractional Ornstein-Uhlenbeck type processX={Xt, ≤tT}satisfying

the following stochastic differential equation:

dXt=θXtdt+εdBHt , ≤tT,X=x, ()

where the drift parameterθ= (θ,θ)⊆Ris unknown andT is a fixed time. We will

study its consistency asε→.

Consistency

Theorem  For every p> ,δ> ,we have

P(ε)

θθ

εθ >δ

≤pTpHK(p,H)e|θT|pg(δ)pεp=Og(δ)pεp. ()

Proof Let the set

A=

ω: inf

|θ–θ|<δρ

X,x(θ)> inf

|θ–θ|>δρ

X,x(θ). ()

Fixδ> ,

P(ε)

θ θε∗–θ >δ

=P(θε)(A). ()

In fact, forωA,

inf

|θ–θ|<δρ

X(ω),x(θ)> inf

|θ–θ|>δρ

X(ω),x(θ)

≥inf

θ∈ωρ

X(ω),x(θ)=ρX(ω),xθε

,

thus|θε∗(ω) –θ|>δ.

Conversely, if|θε∗(ω) –θ|>δ, then

ρX(ω),ε∗≤ inf

|θ–θ|>δρ

X(ω),x(θ) < inf

|θ–θ|<δρ

(6)

Since

Since the process{Xt}satisfies the stochastic differential equation (), it follows that

Xtxt(θ) =x+θ

Applying the Gronwall-Bellman lemma, we obtain

sup

Applying Lemma  and Chebyshev’s inequality, for allp> , we get

(7)

Remark  As a consequence of the above theorem, we obtain the result thatθε∗converges in probability toθunder(ε)-measure asε→. Furthermore, the rate of convergence is

of orderO(εp) for everyp> .

Asymptotic distribution

Theorem  Asε→,the random variableε–(θε∗–θ)converges in probability to a ran-dom variable whose probability distribution is the same as that ofζ under Pθ.

Proof Denotext(θ) =xteθtand let

Also we define the random variable

ζε:=arg min |u|<

Z(u). ()

Note that, with probability , we get

sup

has a unique minimumu∗with probability .

Furthermore, we can choose the interval [–L,L] such that

(8)

and

Pu∗∈(–L,L)≥ –βg(L)–p, ()

whereβ> . The processes{(u),u∈[–L,L]}, and{Z(u),u∈[–L,L]}, satisfy the

Lips-chitz conditions and(u) converges uniformly toZ(u) onu∈[–L,L], so the minimizer

of(u) converges to the minimizer ofZ(u). This completes the proof. Remark  It is not clear what the distribution ofζis. It would be interesting to say some-thing about the distribution ofζ through simulation studies even if an explicit computa-tion of the distribucomputa-tion seems to be difficult.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript. All authors read and approved the final manuscript.

Acknowledgements

We are very grateful to the anonymous referees and the associate editor for their careful reading and helpful comments.

Received: 2 December 2013 Accepted: 28 April 2014 Published:08 May 2014

References

1. Willinger, W, Paxson, V, Riedi, RH, Taqqu, M: Long-range dependence and data network traffic. In: Theory and Applications of Long-Range Dependence, pp. 373-407 (2003)

2. Henry, M, Zafforoni, P: The long-range dependence paradigm for macroeconomics and finance. In: Theory and Applications of Long-Range Dependence, pp. 417-438 (2003)

3. Le Breton, A: Filtering and parameter estimation in a simple linear model driven by a fractional Brownian motion. Stat. Probab. Lett.38, 263-274 (1998)

4. Bertin, K, Torres, S, Tudor, CA: Drift parameter estimation in fractional diffusions driven by perturbed random walks. Stat. Probab. Lett.81, 243-249 (2011)

5. Hu, Y-Z, Nualart, D, Xiao, W-L, Zhang, W-G: Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation. Acta Math. Sci.31, 1851-1859 (2011)

6. Hu, Y-Z, Nualart, D: Parameter estimation for fractional Ornstein-Uhlenbeck processes. Stat. Probab. Lett.80, 1030-1038 (2010)

7. Kleptsyna, ML, Le Breton, A: Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Stat. Inference Stoch. Process.5, 229-248 (2002)

8. Tudor, CA, Viens, FG: Statistical aspects of the fractional stochastic calculus. Ann. Stat.35, 1183-1212 (2007) 9. Brouste, A, Kleptsyna, M: Asymptotic properties of MLE for partially observed fractional diffusion system. Stat.

Inference Stoch. Process.13, 1-13 (2010)

10. Brouste, A: Asymptotic properties of MLE for partially observed fractional diffusion system with dependent noises. J. Stat. Plan. Inference140, 551-558 (2010)

11. Brouste, A, Kleptsyna, M, Popier, A: Design for estimation of the drift parameter in fractional diffusion systems. Stat. Inference Stoch. Process.15(2), 133-149 (2012)

12. Millar, PW: A general approach to the optimality of the minimum distance estimators. Trans. Am. Math. Soc.286(1), 377-418 (1984)

13. Kutoyants, Y, Pilibossian, P: On minimumL1-norm estimate of the parameter of the Ornstein-Uhlenbeck process. Stat. Probab. Lett.20(2), 117-123 (1994)

14. Kutoyants, Y: Identification of Dynamical Systems with Small Noise. Springer, London (1994)

15. Hénaff, S: Asymptotics of a minimum distance estimator of the parameter of the Ornstein-Uhlenbeck process. C. R. Acad. Sci., Sér. 1 Math.325, 911-914 (1997)

16. Diop, A, Yode, AF: Minimum distance parameter estimation for Ornstein-Uhlenbeck processes driven by Lévy process. Stat. Probab. Lett.80(2), 122-127 (2010)

17. Prakasa Rao, BLS: MinimumL1-norm estimation for fractional Ornstein-Uhlenbeck type process. Theory Probab. Math. Stat.71, 181-189 (2005)

18. Biagini, F, Hu, Y, Øksendal, B, Zhang, T: Stochastic Calculus for Fractional Brownian Motion and Applications. Probability and Its Applications, Springer, Berlin (2008)

19. Novikov, A, Valkeila, E: On some maximal inequalities for fractional Brownian motions. Stat. Probab. Lett.44(1), 47-54 (1999)

10.1186/1687-1847-2014-137

References

Related documents

Rather than examining the Ashéninka’s current situation in terms of ideas about ‘cultural change’ my thesis seeks to understand the intrinsic diversity and flexibility

A Principal Lecturer in Design and Technology Education at the University of Surrey Roehampton carried out some research for the 2004 Design and Technology Association

In conversation with former Lundy students they unanimously state that attending as a level- two student was excellent preparation for their third year project, even if it was not done

Purpose of research -This research aims to explore the ways in which ‘Adult Third Culture Kids’ from the UK understand and experience their friendships. Research suggests that

1 Weblogs or just &#34;blogs&#34; are a diary-style form of writing where people write blogs about seeing a recent event, their views on a particular subject, or more typically,

Integrating On-line Learning Technologies into Higher Education: A Case Study of Two UK Universities Abstract This dissertation presents an in-depth, qualitative case study

The diff erence between the code for drawing a circle and the code for drawing a Voronoi diagram using the traditional ‘computational geometry’ approach is huge: the two

Abstract;   The  Information  Communication  Technology  (ICT)  tools  such  as