Consider an Ornstein-Uhlenbeck process driven by a fractional Brownian motion. It is an interesting problem to find criteria for whether the process is stable or has a unit root, given a finite sample of observations. Recently, various asymptotic distributions for estimators of the drift parameter have been developed. We illustrate through computer simulations and through a Stein's bound that these asymptotic distributions are inadequate approximations of the finite-sample distribution for moderate values of the drift and the sample size. We propose a new model to obtain asymptotic distributions near zero and compute the limiting distribution. We show applications to regression analysis and obtain hypothesis tests and their asymptotic power. 1. Introduction Stability properties of the ordinary differential equation depend on the sign of the parameter : the equation is asymptotically stable if , neutrally stable if , and instable if . These stability results carry over to the stochastic process driven by noise . When the value of is not known and a trajectory of is observed over a finite time interval , a natural problem is to develop the zero-root test, that is, a statistical procedure for testing the hypothesis versus one of the possible alternatives , , or . While the classical solution to this problem is wellknown (to use the maximum likelihood estimator (MLE) of the parameter as the test statistic), further analysis is necessary because the exact distribution of the MLE is usually too complicated to allow an explicit computation of either the critical region or the power of the test. More specifically, an approximate distribution of the MLE must be introduced and investigated, both in the finite-sample asymptotic and in the limit . There are other potential complications, such as when MLE is not available (e.g., if is a stable Lévy process, see [1]) or when the MLE is difficult to implement numerically (e.g., if is a fractional Brownian motion, see [2]). The objective of this work is the analysis and implementation of the zero root test for (1.1) when , the fractional Brownian motion with the Hurst parameter , and . When , the integral transformation of Jost [3, Corollary 5.2] reduces the corresponding model back to (see [2]). Recall that the fractional Brownian motion , , is a Gaussian process with , mean zero, and covariance Direct computations show that, for every continuous process , (1.1) has a closed-form solution that does not involve stochastic integration When , let denote the corresponding fractional Ornstein-Uhlenbeck process: and let
References
[1]
Y. Hu and H. Long, “Parameter estimation for Ornstein-Ohlenbeck processes driven by -stable Lévy motions,” Communications on Stochastic Analysis, vol. 1, no. 2, pp. 175–192, 2007.
[2]
A. Brouste and M. Kleptsyna, “Asymptotic properties of MLE for partially observed fractional diffusion system,” Statistical Inference for Stochastic Processes, vol. 13, no. 1, pp. 1–13, 2010.
[3]
C. Jost, “Transformation formulas for fractional Brownian motion,” Stochastic Processes and their Applications, vol. 116, no. 10, pp. 1341–1357, 2006.
[4]
R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes. II Applications, vol. 6, Translated from the 1974 Russian original by A. B. Aries, vol. 6, Stochastic Modelling and Applied Probability, Springer, New York, NY, USA, 1978.
[5]
D. Nualart, The Malliavin Calculus and Related Topics, Springer, 2nd edition, 2006.
[6]
Y. Hu and D. Nualart, “Parameter estimation for fractional Ornstein-Uhlenbeck processes,” Statistics and Probability Letters, vol. 80, no. 11-12, pp. 1030–1038, 2010.
[7]
R. Belfadli, K. Es-Sebaiy, and Y. Ouknine, “Parameter estimation for fractional ornstein-uhlenbeck processes: non-ergodic case,” Frontiers in Science and Engineering, vol. 1, no. 1, 2011, Edited by Hassan II Academy of Science and Technology, http://arxiv.org/abs/1102.5491.
[8]
M. A. Arcones, “On the law of the iterated logarithm for Gaussian processes,” Journal of Theoretical Probability, vol. 8, no. 4, pp. 877–903, 1995.
[9]
I. Nourdin and G. Peccati, “Stein's method on wiener chaos,” Probability Theory and Related Fields, vol. 145, no. 1-2, pp. 75–118, 2009.
[10]
J. P. N. Bishwal, “Sharp berry-esseen bound for the maximum likelihood estimator in the Ornstein-uhlenbeck process,” Sankhyā: Series A, vol. 62, no. 1, pp. 1–10, 2000.
[11]
H. B. Mann and A. Wald, “On the statistical treatment of linear stochastic difference equations,” Econometrica, vol. 11, pp. 173–220, 1943.
[12]
J. S. White, “The limiting distribution of the serial correlation coefficient in the explosive case,” Annals of Mathematical Statistics, vol. 29, pp. 1188–1197, 1958.
[13]
P. C. B. Phillips, “Towards a unified asymptotic theory for autoregression,” Biometrika, vol. 74, no. 3, pp. 535–547, 1987.
[14]
P. C. B. Phillips, H. R. Moon, and Z. Xiao, “How to estimate autoregressive roots near unity,” Econometric Theory, vol. 17, no. 1, pp. 29–69, 2001.
[15]
P. C. B. Phillips and P. Perron, “Testing for a unit root in time series regression,” Biometrika, vol. 75, no. 2, pp. 335–346, 1988.
[16]
F. Biagini, Y. Hu, B. ？ksendal, and T. Zhang, “Stochastic calculus for fractional Brownian motion and applications,” in Probability and its Applications, Springer, London, UK, 2008.
[17]
T. E. Duncan, Y. Hu, and B. Pasik-Duncan, “Stochastic calculus for fractional brownian motion. I. Theory,” SIAM Journal on Control and Optimization, vol. 38, no. 2, pp. 582–612, 2000.
[18]
V. Pipiras and M. S. Taqqu, “Are classes of deterministic integrands for fractional Brownian motion on an interval complete?” Bernoulli, vol. 7, no. 6, pp. 873–897, 2001.
[19]
L. C. Young, “An inequality of the h？lder type, connected with stieltjes integration,” Acta Mathematica, vol. 67, no. 1, pp. 251–282, 1936.
[20]
M. L. Kleptsyna and A. Le Breton, “Statistical analysis of the fractional Ornstein-Uhlenbeck type process,” Statistical Inference for Stochastic Processes, vol. 5, no. 3, pp. 229–248, 2002.
[21]
I. Norros, E. Valkeila, and J. Virtamo, “An elementary approach to a girsanov formula and other analytical results on fractional Brownian motions,” Bernoulli, vol. 5, no. 4, pp. 571–587, 1999.
[22]
N. Lin and S. V. Lototsky, “Undamped harmonic oscillator driven by additive gaussian white noise: a statistical analysis,” Communications on Stochastic Analysis, vol. 5, no. 1, pp. 233–250, 2011.
[23]
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55, Dover Publications, 1964.
[24]
Y. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer, London, UK, 2004.
