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Statistical Analysis of the Fractional Ornstein–Uhlenbeck Type Process

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Abstract

We consider the fractional analogue of the Ornstein–Uhlenbeck process, that is, the solution of a one-dimensional homogeneous linear stochastic differential equation driven by a fractional Brownian motion in place of the usual Brownian motion. The statistical problem of estimation of the drift and variance parameters is investigated on the basis of a semimartingale which generates the same filtration as the observed process. The asymptotic behaviour of the maximum likelihood estimator of the drift parameter is analyzed. Strong consistency is proved and explicit formulas for the asymptotic bias and mean square error are derived. Preparing for the analysis, a change of probability method is developed to compute the Laplace transform of a quadratic functional of some auxiliary process.

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References

  • Comte, F.: Simulation and estimation of long memory continuous-time models, J. Time Ser. Anal. 17(1) (1996), 19-36.

    MATH  MathSciNet  Google Scholar 

  • Decreusefond, L. and Üstünel, A. S.: Stochastic analysis of the fractional Brownian motion, Pot ent. Anal. 10 (1999), 177-214.

    Article  MATH  Google Scholar 

  • Duncan, T. E., Hu, Y. and Pasik-Duncan, B.: Stochastic calculus for fractional Brownian motion I. Theory, SIAM J. Cont. Optimiz. 38(2) (2000), 582-612.

    Article  MATH  MathSciNet  Google Scholar 

  • Kleptsyna, M. L. and Le Breton, A.: Some explicit statistical results about elementary fractional type models, Nonlinear Anal. Th. Meth. Appl. 47(7) (2001), 4783-4794.

    Article  MATH  MathSciNet  Google Scholar 

  • Kleptsyna, M. L., Le Breton, A. and Roubaud, M.-C.: Parameter estimation and optimal filtering for fractional type stochastic systems, Statist. Inf. Stochast. Proces. 3 (2000), 173-182.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Le Breton, A. and Pham, D. T.: On the bias of the least squares estimator for the first order autoregressive process, Ann. Inst. Statist. Math. 41(3) (1989), 555-563.

    Article  MATH  MathSciNet  Google Scholar 

  • Liptser, R. S. and Shiryaev, A. N.: Statistics of Random Processes, Springer-Verlag, New York, 1978.

    Google Scholar 

  • Norros, I., Valkeila, E. and Virtamo, J.: An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5(4) (1999), 571-587.

    Article  MATH  MathSciNet  Google Scholar 

  • Novikov, A. A.: Sequential estimation of parameters of diffusion processes, Theor. Probab. Appl. 16 (1971), 391-393.

    MATH  Google Scholar 

  • Watson, G. N.: A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1995.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Information Transmission Problems, Bolshoi Karetnii per., 19, Moscow, 101475, Russia

    M.L. Kleptsyna

  2. Laboratoire de Modélisation et Calcul / Université J. Fourier, BP 53, 38041, Grenoble Cedex 9, France

    A. Le Breton

Authors
  1. M.L. Kleptsyna
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  2. A. Le Breton
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Correspondence to A. Le Breton.

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Kleptsyna, M., Le Breton, A. Statistical Analysis of the Fractional Ornstein–Uhlenbeck Type Process. Statistical Inference for Stochastic Processes 5, 229–248 (2002). https://doi.org/10.1023/A:1021220818545

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