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Minimum distance estimation for fractional Ornstein-Uhlenbeck type process
Advances in Difference Equations volume 2014, Article number: 137 (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 .
Stochastic models having long-range dependence phenomena have been paid much attention to in view of their applications in signal processing, computer networks, and mathematical finance (see [1, 2]). The long-range dependence phenomenon is said to occur in a stationary time series if the autocovariance functions satisfy
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.
Consider the parameter estimation problem for a special fractional process, i.e., fractional Ornstein-Uhlenbeck type process , which satisfies the following stochastic integral equation:
where the drift parameter is unknown, , and is a scalar fractional Brownian motion defined on the probability space . For a fractional Brownian motion with Hurst parameter , we mean that it is a continuous and centered Gaussian process with the covariance function
By  (see Definitions 1.5.1 and 1.5.2, p.11), we introduce the following.
Definition 1 We say that an -valued random process is self-similar or satisfies the property of self-similarity if for every there exists such that
where denotes the law of random variable ⋅ .
Remark 1 Note that (3) means that the two processes and have the same finite-dimensional distribution functions, i.e., for every choice in ℝ,
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.
Remark 2 Note that the law of a Gaussian random variance is determined by its expectation value and variation. By (2), it is easy to see that is a self-similar process with Hurst index H. Let
Then we conclude from the fact that is a self-similar process with Hurst index H that
Let be the solution of the above differential equation with . It is obvious that
Define the minimum Skorohod distance estimator
the Skorohod distance
on the Skorohod space .
is continuous, strictly increasing such that and . Let
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 .
Lemma 1 Let , and be a fractional Brownian motion with Hurst parameter H, then for every ,
Lemma 2 Let , , be a sequence of continuous functions and a convex function which admits a unique minimum . Let , be a sequence of positive numbers such that as . We suppose that
where if there are several minima of , we choose an arbitrary one.
For any , define
Note that for any .
Introduce the random variable
where is the derivative of with respect to .
It can be obtained from (1) that
Note that is a Gaussian process and can be interpreted as the ‘derivative’ of the process with respect to ε.
Now we investigate the parameter estimation problem of parameter θ based on the observation of a fractional Ornstein-Uhlenbeck type process satisfying the following stochastic differential equation:
where the drift parameter is unknown and T is a fixed time. We will study its consistency as .
Theorem 1 For every , , we have
Proof Let the set
In fact, for ,
Conversely, if , then
then, for all ,
Since the process satisfies the stochastic differential equation (1), it follows that
Applying the Gronwall-Bellman lemma, we obtain
Applying Lemma 1 and Chebyshev’s inequality, for all , we get
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 .
Proof Denote and let
For the random variable , we get
Also we define the random variable
Note that, with probability 1, we get
where , . From Lemma 2, we get has a unique minimum with probability 1.
Furthermore, we can choose the interval such that
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.
Willinger W, Paxson V, Riedi RH, Taqqu M: Long-range dependence and data network traffic. Theory and Applications of Long-Range Dependence 2003, 373–407.
Henry M, Zafforoni P: The long-range dependence paradigm for macroeconomics and finance. Theory and Applications of Long-Range Dependence 2003, 417–438.
Le Breton A: Filtering and parameter estimation in a simple linear model driven by a fractional Brownian motion. Stat. Probab. Lett. 1998, 38: 263–274. 10.1016/S0167-7152(98)00029-7
Bertin K, Torres S, Tudor CA: Drift parameter estimation in fractional diffusions driven by perturbed random walks. Stat. Probab. Lett. 2011, 81: 243–249. 10.1016/j.spl.2010.10.003
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. 2011, 31: 1851–1859. 10.1016/S0252-9602(11)60365-2
Hu Y-Z, Nualart D: Parameter estimation for fractional Ornstein-Uhlenbeck processes. Stat. Probab. Lett. 2010, 80: 1030–1038. 10.1016/j.spl.2010.02.018
Kleptsyna ML, Le Breton A: Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Stat. Inference Stoch. Process. 2002, 5: 229–248. 10.1023/A:1021220818545
Tudor CA, Viens FG: Statistical aspects of the fractional stochastic calculus. Ann. Stat. 2007, 35: 1183–1212. 10.1214/009053606000001541
Brouste A, Kleptsyna M: Asymptotic properties of MLE for partially observed fractional diffusion system. Stat. Inference Stoch. Process. 2010, 13: 1–13. 10.1007/s11203-009-9035-x
Brouste A: Asymptotic properties of MLE for partially observed fractional diffusion system with dependent noises. J. Stat. Plan. Inference 2010, 140: 551–558. 10.1016/j.jspi.2009.08.001
Brouste A, Kleptsyna M, Popier A: Design for estimation of the drift parameter in fractional diffusion systems. Stat. Inference Stoch. Process. 2012, 15(2):133–149. 10.1007/s11203-012-9067-5
Millar PW: A general approach to the optimality of the minimum distance estimators. Trans. Am. Math. Soc. 1984, 286(1):377–418. 10.1090/S0002-9947-1984-0756045-0
Kutoyants Y, Pilibossian P:On minimum -norm estimate of the parameter of the Ornstein-Uhlenbeck process. Stat. Probab. Lett. 1994, 20(2):117–123. 10.1016/0167-7152(94)90026-4
Kutoyants Y: Identification of Dynamical Systems with Small Noise. Springer, London; 1994.
Hénaff S: Asymptotics of a minimum distance estimator of the parameter of the Ornstein-Uhlenbeck process. C. R. Acad. Sci., Sér. 1 Math. 1997, 325: 911–914.
Diop A, Yode AF: Minimum distance parameter estimation for Ornstein-Uhlenbeck processes driven by Lévy process. Stat. Probab. Lett. 2010, 80(2):122–127. 10.1016/j.spl.2009.09.020
Prakasa Rao BLS:Minimum -norm estimation for fractional Ornstein-Uhlenbeck type process. Theory Probab. Math. Stat. 2005, 71: 181–189.
Biagini F, Hu Y, Øksendal B, Zhang T Probability and Its Applications. In Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, Berlin; 2008.
Novikov A, Valkeila E: On some maximal inequalities for fractional Brownian motions. Stat. Probab. Lett. 1999, 44(1):47–54. 10.1016/S0167-7152(98)00290-9
We are very grateful to the anonymous referees and the associate editor for their careful reading and helpful comments.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
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Liu, Z., Song, N. Minimum distance estimation for fractional Ornstein-Uhlenbeck type process. Adv Differ Equ 2014, 137 (2014). https://doi.org/10.1186/1687-1847-2014-137
- long-range dependence
- minimum distance estimation
- asymptotic distribution