Parameter estimation for the discretely observed fractional Ornstein-Uhlenbeck process and the Yuima R package

Statistical inference for parameters of ergodic diffusion processes observed on discrete increasing grid have been much studied. Local asymptotic normality (LAN) property of the likelihoods have been shown in [10] for elliptic ergodic diffusion, under proper conditions for the drift and the diffusion coefficient, and a mesh satisfying ∆ N -→ 0 and N ∆ N -→ +∞ when the size of the sample N grows to infinity. Estimation procedure have been studied by many authors, mainly in the one-dimensional case (see, for instance, [8,14] and [26] in the multidimensional setting). All estimators in the previous works are based on contrasts (for contrasts framework, see [9]), assuming in the general case, that for some p > 1, as n -→ +∞, N ∆ p N -→ 0. In particular, for Ornstein-Uhlenbeck process, transitions densities are known, and all have been treated, as remarked in [13].

In the fractional case, we consider the fraction Ornstein-Uhlenbeck process (fOU), the solution of dY t = -λY t dt + σdW H t where W H = W H t , t ≥ 0 is a normalized fractional Brownian motion (fBM), i.e. the zero mean Gaussian processes with covariance function

with Hurst exponent H ∈ (0, 1). The fOU process is neither Markovian nor a semimartingale for H = 1 2 but remains Gaussian and ergodic (see [5]). For H > 1 2 , it even presents the long-range dependance property that makes it useful for different applications in biology, physics, ethernet traffic or finance. Statistical large sample properties of Maximum Likelihood Estimator of the drift parameter in the continuous observations case have been treated in [1,4,6,15] for different applications. Moreover, asymptotical properties of the Least Squares Estimator have been studied in [11].

In the discrete case and fractional case, we can cite few works on the topic. On the one hand, very recent works give methods to estimate the drift λ by contrast procedure [17,20] or the drift λ and the diffusion coefficient σ with discretization procedure of integral transform [25]. In these papers, the Hurst exponent is supposed to be known and only consistency is obtained. On the other hand, methods to estimate the Hurst exponent H and the diffusion coefficient are presented in [3] with classical order 2 variations convolution filters.

To the best of our knowledge, nothing have been done, to have a complete estimation procedure that could estimate all Hurst exponent, diffusion coefficient and drift parameter with central limit theorems and this is the gap we fill in this paper. Moreover, estimates of H, σ and λ presented in this paper slightly differ from all those studied previously.

In Section 2 we review the basic facts of stochastic differential equations driven by the fractional Brownian motion and we introduce the basic notations and assumptions. Section 3 presents consistent and asymptotically Gaussian estimators of the parameters of the fractional Ornstein-Uhlenbeck process from discrete observations. In Section 4 we present ready-to-use software for the R statistical environment which allows the user to simulate and estimate the parameters of the fOU process. We further present Monte-Carlo experiments to test the performance of the estimators under different sampling conditions.

Let X = (Y t , t ≥ 0) be a fractional Ornstein-Uhlenbeck process (fOU), i.e. the solution of

where unknown parameter ϑ = (λ, σ, H) belongs to an open subset Θ of (0, Λ) × [σ, σ] × (0, 1), 0 < Λ < +∞, 0 < σ < σ < +∞ and W H = (W H t , t ≥ 0) is a standard fractional Brownian motion [16,18] of Hurst parameter H ∈ (0, 1), i.e. a Gaussian centered process of covariance function

It is worth emphasizing that in the case

is the classical Wiener process The fOU process is neither Markovian nor a semimartingale for H = 1 2 but remains Gaussian and ergodic. For H > 1 2 , it even presents the long-range dependance property (see [5]).

The present work exposes an estimation procedure for estimating all three components of ϑ given the regular discretization of the sample path

where

In the following, convergences

-→ stand respectively for the convergence in law, the convergence in probability and the almost-sure convergence.

Contrary to the previous works on the subject, we consider here the problem of estimation of H, σ and λ when all parameters are unknown, using discrete observations from the fractional Ornstein-Uhlenbeck process. Due to the fact that one can estimate H and σ without the knowledge of λ, our approach consists naturally in a two step procedure.

The key point of this paper is that the Hurst exponent H and the diffusion coefficient σ can be estimated without estimating λ.

Let a = (a 0 , . . . , a K ) be a discrete filter of length K + 1, K ∈ N, and of order

(

Let it be normalized with

In the following, we will also consider dilatated filter a 2 associated to a defined by

k r a k , filter a 2 as the same order than a. We denote by

the generalized quadratic variations associated to the filter a (see for inst

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