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Regularity conditions and the maximum likelihood estimation in dynamical systems with small fractional Brownian noise. (English) Zbl 1115.62080

The author deals with a dynamical system with small fractional Brownian noise, i.e., a process described by the equation \[ X_t=x_0+\int_0^tS(\theta,u,X_u)\,du+\varepsilon B_t,\quad 0\leq t\leq T, \] where \(B_t\) is a fractional Brownian motion (fBm) with the Hurst parameter \(H\in(1/2,1)\) and \(\theta\in\Theta\subset\mathbb R^d\) is an unknown parameter. Under some, not too restrictive, conditions on the function \(S\) there exists a unique solution \(X\) to the equation in a pathwise sense for every \(\theta\in\Theta\subset\mathbb R^d\) and \(\varepsilon>0\). Following the terminology of I. A. Ibragimov and R. Z. Khas’minskij [Asymptotic Theory of Estimation. (Russian) (1979; Zbl 0467.62025)] we may say that the equation generates a set of statistical experiments \[ \mathbb E_{\varepsilon}=\left\{C([0,T]),\,\mathcal B,\,P_{\theta}^{(\varepsilon)},\;\theta\in\Theta \right\}, \] in which we observe the trajectory of a solution \(X=X^{\varepsilon}\). The index \(\varepsilon>0\) refers to the noise intensity in the experiment. The author presented sufficient conditions under which this dynamical system with small fractional Brownian noise generates a set of regular statistical experiments in the sense of I. A. Ibragimov and R. Z. Khas’minskij definition. As a corollary, it is shown that the maximum likelihood estimator of the unknown parameter based on the observation of a trajectory is consistent, uniformly asymptotically normal and its moments converge to moments of the standard normal distribution.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62M09 Non-Markovian processes: estimation
62F12 Asymptotic properties of parametric estimators
60G15 Gaussian processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Citations:

Zbl 0467.62025

Software:

PDEFIT
Full Text: DOI

References:

[1] Ibragimov I.A., Has’minskii R.Z. Asymptotical theory of estimation, ”Nauka”, Moscow (1979).
[2] Liptser R.S., Shiryayev A.N. Statistics of Random Processes, ”Nauka”, Moscow (1974). · Zbl 0364.60004
[3] Kutoyants Yu. A. Identification Of Dynamical Systems With Small Noise, Mathematics and Its Applications, Vol 300. Kluwer Academic Publisher, p. 298 (1994). · Zbl 0831.62058
[4] Kleptsyna M.L., Le Breton A., Roubaud M.C. Parameter estimation and optimal filtering for fractional stochastic systems, Statistical inference for stochastic processes 3, 173-182 (2000). · Zbl 0966.62069
[5] Kleptsyna M.L., Le Breton A. Statistical analysis of the fractional Ornstein-Uhlenbeck type processes, Statistical inference for stochastic processes 5, 229-248 (2002). · Zbl 1021.62061
[6] DOI: 10.1163/156939703771378581 · Zbl 1053.62089 · doi:10.1163/156939703771378581
[7] Kukush A., Mishura Yu., Valkeila E. Statistical Inference with Fractional Brownian Motion, Statistical inference for stochastic processes 8, 71-93 (2005). · Zbl 1107.62355
[8] Zahle M, Probab. Theory Relat. Field 111 pp 33– (1998)
[9] Zahle M. On the link between fractional and stochastic calculus, Stochastic dynamics, Bremen, pp. 305-325, Springer (1999).
[10] Nualart D., Collect. Math. 53 pp 1– (2002)
[11] DOI: 10.2307/3318691 · Zbl 0955.60034 · doi:10.2307/3318691
[12] Androshchuk T.O. Asymptotical normality of family of measures generated by solutions of a stochastic differential equation with small fractional Brownian noise, Theory of probability and mathematical statistics, 71, 1-14 (2004). · Zbl 1097.60045
[13] Kutoyants Yu. A. Parameter Estimation for Stochastic Processes, Heldermann, Berlin (1984). · Zbl 0542.62073
[14] Lifshits M.A. Gaussian Random Functions, TV and MS, Kiev (1995), 245 p.
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