Drift parameter estimation for fractional Ornstein–Uhlenbeck process of the second kind

Abstract

Abstract. Fractional Ornstein-Uhlenbeck process of the second kind (fOU2) is solution of the Langevin equation dXt = −θXt dt+dY , we prove that the least squares estimator θT introduced in [[7], Statist. Probab. Lett. 80, no. 11-12, 1030Lett. 80, no. 11-12, -1038, provides a consistent estimator. Moreover, using central limit theorem for multiple Wiener integrals, we prove asymptotic normality of the estimator valid for the whole range H ∈ ( 1 2 , 1).

Optimal rates for parameter estimation of stationary Gaussian processes

“…Arguably, as long as one can compute λ f 2 ,n (Z (1) ) and relate it to a parameter of interest, this improvement allows one to take advantage of the best rate of convergence, that of Berry-Esséen order. A study of one example of what it means to extract a parameter from λ f 2 ,n (Z (1) ) is given in Section 6.3.3, for the fractional Ornstein-Uhlenbeck process.…”

“…It should be noted that in [14], the full LS estimator relies on an unobservable Skorohod integral, and the authors proposed a modified version of this estimator which can be computed based on in-fill asymptotics; however, this modified estimator bears no immediate relation to an LS one (see [13] for examples of what constitutes a discretization of an LS estimator for fBm models, and for a comparison with MLE methods, which coincide with LS methods if and only if H = 1/2). In the ergodic case, the statistical inference for several fractional Ornstein-Uhlenbeck (fOU) models via LS methods was recently developed in the papers [14], [1], [2], [13], [15], [6], [21]. The case of non-ergodic fOU process of the first kind and of the second kind can be found in [3], [11] and [12] respectively.…”

“…In fact, we show that one can develop estimators valid for any Gaussian sequence, with suitable conditions on the sequence's auto-correlation function, and then apply them to fBm-driven SDEs of interest. In this way, we are able to provide estimators for many other models, while the models studied in [15], [1], [2], [13] become particular cases in our approach.…”

Berry-Esséen bounds for parameter estimation of general Gaussian processes

“…A typical example of such a process is the fractional Brownian motion with Hurst parameter H > 1/2. In that case, any α can be chosen in (1/2, H), and g can be any process with the same regularity as the fBm, and thus with β = α; this enables a stochastic calculus immediately, in which no "Itô"-type correction terms occur, owing to (1). For instance, for any Lipshitz non-random function g, this integration-by-parts formula holds for any a.s. Hölder-continuous Gaussian process f , since then α > 0 and β = 1.…”

Maximum likelihood estimators of a long-memory process from discrete observations

“…Hu and Nualart [15] obtained least squares estimation (hereafter LSE) for the fractional OrnsteinUhlenbeck process and proposed the asymptotic normality of LSE using Malliavin calculus. More recently, there has been increased interest in studying asymptotic properties of LSE for the drift parameter in the univariate case with fractional processes (see, Azmoodeh and Morlanes [16]; Cheng et al [17]). Moreover, Xiao and Yu [18] and Xiao and Yu [19] considered the LSE in fractional Vasicek models in the stationary case, the explosive case, and the null recurrent case.…”