Least Squares Estimation for 𝛼-Fractional Bridge with Discrete Observations

Abstract

We consider a fractional bridge defined as , where is a fractional Brownian motion of Hurst parameter and parameter is unknown. We are interested in the problem of estimating the unknown parameter . Assume that the process is observed at discrete time , and denotes the length of the “observation window.” We construct a least squares estimator of which is consistent; namely, converges to in probability as .

1. Introduction

Self-similar stochastic processes with long range dependence are of practical interest in various applications, including econometrics, internet traffic, and hydrology. These are processes whose dependence on the time parameter is self-similar, in the sense that there exists a (self-similarity) parameter such that, for any constant , and have the same distribution. These processes are often endowed with other distinctive properties.

The fractional Brownian motion (fBm) is the usual candidate to model phenomena in which the self-similarity property can be observed from the empirical data. The fBm is a suitable generalization of the standard Brownian motion, which exhibits long-range dependence and self-similarity and has stationary increments. Some surveys and complete literatures could be found in Biagini et al. [1], Hu [2], Mishura [3], and Nualart [4].

Recently, Es-Sebaiy and Nourdin [5] study the asymptotic properties of a least squares estimator for the parameter of a fractional bridge defined as where is a fBm with Hurst parameter and the process was observed continuously. In particular, when , Barczy and Pap [6, 7] study the various problems related to the -Wiener bridge. The parametric estimation problems for fractional diffusion processes based on continuous-time observations have been studied, for example, in Tudor and Viens [8], Hu and Nualart [9], and Belfadli et al. [10].

In applications usually the process cannot be observed continuously. Only discrete-time observations are available. There exists a rich literature on the parameter estimation problem for diffusion processes driven by fBm based on discrete observations (see, e.g., Hu and Song [11], Es-Sebaiy [12]).

Motivated by all these results, in this paper, we will consider the fractional bridge (1). Assume that the process is observed equidistantly in time with the step size , , and denotes the length of the “observation window.” We also assume that and when . Our goal is to study the asymptotic behavior of the least squares estimator (LSE for short) of based on the sampling data , . Our technics used in this work are inspired from Es-Sebaiy [12].

The least squares estimator aims to minimize

This is a quadratic function of . The minimum is achieved when

By (1), we can get the following result where , .

The paper is organized as follows. In Section 2 some known results that we will use are recalled. The consistency of estimator is proved in Section 3.

2. Preliminaries

Recall that fBm with index is a mean zero Gaussian process with and the covariance for all . For , coincides with the standard Brownian motion . is neither a semimartingale nor a Markov process unless , so many of the powerful techniques from stochastic analysis are not available when dealing with . It is possible to construct a stochastic calculus of variations with respect to the Gaussian process , which will be related to the Malliavin calculus. Some surveys and complete literatures could be found in Alòs et al. [13], Nualart [4] and the reference. We recall here the basic definitions and results of this calculus. The crucial ingredient is the canonical Hilbert space (it is also said to be reproducing kernel Hilbert space) associated with the fBm which is defined as the closure of the linear space generated by the indicator functions with respect to the scalar product The mapping can be extended to a linear isometry between and the Gaussian space associated with . We will denote the isometry by . For we denote by the set of smooth functionals of the form where and . The Malliavin derivative of a functional as above is given by and this operator can be extended to the closure () of with respect to the norm where denotes the fold symmetric tensor product and the th derivative is defined by iteration. The divergence integral is the adjoint operator of . Concretely, a random variable belongs to the domain of the divergence operator (in symbol ) if for every . In this case is given by the duality relationship for any , and we have the following integration by parts: for any , such that . It follows that where is the adjoint of in the Hilbert space , and where and, for , we have We denote by the subspace of , which is defined as the set of measurable functions on with

Note that, if , then

It follows actually from Pipiras and Taqqu [14] that the space is a Banach space for the norm . Moreover, If , , then we have (Nualart [4]) and if , then As a consequence, we have

For every , let be the th Wiener chaos of , that is, the closed linear subspace of generated by the random variables , where is the th Hermite polynomial. The mapping provides a linear isometry between the symmetric tensor product (equipped with the modified norm ) and . For every the following multiplication formula holds

Let be Hölder continuous functions of orders and with . Young proved that the Riemann-Stieltjes integral (so-called Young integral) exists. Moreover, if and is a function of class , the integrals and exist in the Young sense and the following change of variables formula holds:

As a consequence, if and (, ) is a process with Hölder paths of order , the integral is well defined as Young integral. Suppose that, for any , , and Then, following from Alòs and Nualart [15], we have In particular, when is a nonrandom Hölder continuous function of order , we have In addition, for all ,

3. Asymptotic Behavior of the Least Squares Estimator

Throughout this paper we assume . We will study (1) driven by a fractional Brownian motion with Hurst parameter and being the unknown parameter to be estimated for discretely observed . It is readily checked that we have the following explicit expression for : where the integral can be understood as Young integral. In order to study the asymptotic behavior of the least squares estimator, let us introduce the following processes: Hence, we have For simplicity, we assume that the notation means that there exists positive constants (depending only on , and its value may differ from line to line) so that We firstly give the following lemmas.

Lemma 1. Let , . Then where

Proof. By (26), we have On the other hand, This completes the proof.

The following Lemma 2 comes from Lemma 3.2 of Es-Sebaiy and Nourdin [5].

Lemma 2. Letting , , one has

Lemma 3. Assume , , and let . Then

Proof. By the isometry property of the double stochastic integral , the variance of is given by where Now, we study , by setting We have . By of Es-Sebaiy and Nourdin [5], we have Similarly Thus, the proof is finished.

The following theorem gives the consistency of the least squares estimator of .

Theorem 4. Let . If , as , and , then, one has where means convergence in probability.

Proof. By (4), we have
Letting , we obtain First, we considering the term , we have For the term , using Lemma 2, we obtain So, we get Hence, For the term , it follows the fact that, for , We have Using inequality (22) and , , we have On the other hand, So, we get Thus, Hence For the term , by setting and by using Lemma 3, we get As a consequence,
Second, we estimate the term : We firstly consider , since By Markov inequality, we obtain Now, we estimate the term . Applying the change of variable formula (24), we get Hence, By Markov inequality and Lemma 2, we obtain Therefore Combining (59) and (66), this completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank anonymous earnest referee whose remarks and suggestions greatly improved the presentation of the paper. Guangjun Shen is partially supported by the National Natural Science Foundation of China (11271020) and the Natural Science Foundation of Anhui Province (1208085MA11). Xiuwei Yin is partially supported by Mathematical Tianyuan Foundation of China (11326171) and the Natural Science Foundation of Anhui Province (1308085QA14).