Minimum distance estimation for fractional Ornstein-Uhlenbeck type process
© Liu and Song; licensee Springer. 2014
Received: 2 December 2013
Accepted: 28 April 2014
Published: 8 May 2014
We consider a one-dimensional linear stochastic differential equation defined as , , with θ the unknown drift parameter, where is a fractional Brownian motion with . The consistency and the asymptotic distribution of the minimum Skorohod distance estimator of θ based on the observation is studied as .
for some constant c and . In this case, the dependence between and decays slowly as and .
Fractional Brownian motions are a special class of long memory processes when the Hurst parameter . 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 important methods are either maximum likelihood estimation (MLE) or least square estimation (LSE). Substantial progress has been made in this direction. 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 estimation was studied in [4, 5]. 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 motion in place of the usual Brownian motion. The asymptotic behavior of the maximum likelihood estimator of the drift parameter was analyzed. Tudor and Viens  applied the 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 [9–11]). 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 distance 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 that converge in probability to the random variable with , or supremum norm and he also proved that is asymptotically normal when as . Hénaff  established the same results in the general case of a norm in some Banach space of functions on . 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 minimum -norm estimator of the drift parameter of a fractional Ornstein-Uhlenbeck type process and proved that converges in probability under to a random variable ζ. 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.
Let be a stochastic basis satisfying the usual conditions, i.e., a filtered probability space with a filtration. is right continuous and contains every ℙ-null set. Suppose that the processes discussed in the following are -adapted. Further the natural filtration of a process is understood as the ℙ-completion of the filtration generated by this process.
By  (see Definitions 1.5.1 and 1.5.2, p.11), we introduce the following.
where denotes the law of random variable ⋅ .
for every in ℝ.
Definition 2 If in the above definition, then we say that is a self-similar process with Hurst index H or that it satisfies the property of (statistical) self-similar with Hurst index H. The quantity is called the statistical fractal dimension of X.
on the Skorohod space .
Note that the space consists of functions which are right continuous with left limits on . The uniform metric coincides with Skorohod distance when relativized to the space of continuous functions on .
Denote the true parameter of θ and be the probability measure induced by the process .
where if there are several minima of , we choose an arbitrary one.
Note that for any .
where is the derivative of with respect to .
Note that is a Gaussian process and can be interpreted as the ‘derivative’ of the process with respect to ε.
where the drift parameter is unknown and T is a fixed time. We will study its consistency as .
This completes the proof. □
Remark 3 As a consequence of the above theorem, we obtain the result that converges in probability to under -measure as . Furthermore, the rate of convergence is of order for every .
Theorem 2 As , the random variable converges in probability to a random variable whose probability distribution is the same as that of ζ under .
where , . From Lemma 2, we get has a unique minimum with probability 1.
where . The processes , and , satisfy the Lipschitz conditions and converges uniformly to on , so the minimizer of converges to the minimizer of . This completes the proof. □
Remark 4 It is not clear what the distribution of ζ is. It would be interesting to say something about the distribution of ζ through simulation studies even if an explicit computation of the distribution seems to be difficult.
We are very grateful to the anonymous referees and the associate editor for their careful reading and helpful comments.
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