%0 Journal Article
%T Hypothesis Testing in a Fractional Ornstein-Uhlenbeck Model
%A Michael Moers
%J International Journal of Stochastic Analysis
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/268568
%X 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 L¨¦vy 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
%U http://www.hindawi.com/journals/ijsa/2012/268568/