Asymptotic law of limit distribution for fractional Ornstein-Uhlenbeck process
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RESEARCH
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Asymptotic law of limit distribution for fractional Ornstein-Uhlenbeck process Liang Shen1,2 and Qingsong Xu1* * Correspondence: [email protected] 1 School of Mathematics and Statistics, Central South University, Changsha, China Full list of author information is available at the end of the article
Abstract We consider the minimum L1 -norm estimator θε∗ of the parameter θ of a linear stochastic differential equation dXt = θ Xt dt + ε dBHt , X0 = x0 , where {BHt , 0 ≤ t ≤ T} is a fractional Brownian motion. The asymptotic law of its limit distribution is studied for T → +∞, when ε → 0. Keywords: fractional Ornstein-Uhlenbeck process; minimum L1 -norm estimator; fractional Brownian motion; asymptotic law
1 Introduction Stochastic differential equations driven by Brownian motions are used widely in variety of sciences as stochastic modeling to describe some phenomena. There are many applications such as mathematical finance, economic processes as well as signal processing. The Ornstein-Uhlenbeck process, which is also called the Vasicek model in finance, is being extensively used in finance over the last few decades as the one-factor short-term interest rate model. Statistical inference for the process of Ornstein-Uhlenbeck type driven by Brownian motions has been an active research area, and a comprehensive survey of various methods is given in Prakasa Rao []. As fractional Brownian motion plays an important role in the modeling of financial time series, there has been a growing interest in the study of similar problems for stochastic processes driven by fractional Brownian motion (fBm) in view of their applications to long-range dependence of time series. A stationary sequence (Xn )n∈N exhibits long-range dependence if the autocovariance functions ρ(n) := cov(Xk , Xk+ ) satisfy lim
n→∞
ρ(n) = cn–α
for some constant c and α ∈ (, ). In this case, the dependence between Xk and Xk+n decays slowly as n → ∞ and ∞
ρ(n) = ∞
n=
(see, e.g., [, Definition .., p.]). The long-range dependence was first observed by the hydrologist Hurst [] on projects involving the design of reservoirs along the Nile river. It was also observed that a similar phenomenon occurs in problems concerning traffic ©2014 Shen and Xu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Shen and Xu Advances in Difference Equations 2014, 2014:75 http://www.advancesindifferenceequations.com/content/2014/1/75
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patterns of packet flows in high-speed data networks such as the Internet (see [, ]) and in macroeconomics and finance (see []). The problem of parameter estimation and filtering in a simple linear model driven by a fractional Brownian motion was studied by Le Breton [] in the continuous case. Prakasa Rao [, ] studied parametric estimation for more general classes
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