On comparison of the estimators of the Hurst index and the diffusion coefficient of the fractional Gompertz diffusion process

We study some estimators of the Hurst index and the diffusion coefficient of the fractional Gompertz diffusion process and prove that they are strongly consistent and most of them are asymptotically normal. Moreover, we compare the asymptotic behavior of these estimators with the aid of computer simulations.


Introduction
Many applications make use of processes that are described by stochastic differential equations (SDEs). Recently, much attention has been paid to SDEs driven by the fractional Brownian motion (fBm) and to the problems of statistical estimation of model parameters. Statistical aspects of the models driven by the fBm have been studied in many articles. Especially much attention has been paid to the estimation of the parameters of drift. We focus on estimators of the Hurst index and the diffusion coefficient. Recently some new estimators of the Hurst index and of the diffusion coefficient have been proposed (see [1], [3], [14], [13]). This paper aims to compare them using discrete observations of the sample paths of the solution of the SDE.
As the test process we will consider the fractional Gompertz diffusion process (fGd) where α, β = 0, and σ are real parameters and B H is a fBm with the Hurst index H ∈ (1/2, 1). Almost all sample paths of B H have bounded p-variation for each p > 1/H on [0, T ] for every T > 0. The second integral in (1) is the pathwise Riemann-Stieltjes integral with respect to the process having finite p-variation.
The reasons we have chosen fGd as the test process are as follows. Firstly, it is a non-linear process.
To the equation (1) it is possible to apply a pathwise approach and use a chain rule for the composition of a smooth function and a function of bounded p-variation with 1 < p < 2. This approach allows to easily obtain the unique explicit solution of the equation (1) for H ∈ (1/2, 1) in the class of processes, almost all sample paths of which have bounded p-variation with 1 < p < 2. Secondly, the structure of the increments of fGd allows us to apply a wider class of estimators without imposing additional restrictions on the process. The normalization of quadratic variation by the square of the process value at a fixed point allows us to derive the asymptotic normality of these estimators. The application of this approach allows to consider similar statistics for the equations with time-dependent coefficients. Moreover, in case of the standard Brownian motion, i.e. for H = 1/2, this process plays an important role in the modeling of population growth.
Dung [7] proved that a class of fractional geometric mean reversion processes expressed by a fractional SDE of the form where W H is a fractional Brownian motion of the Liouville form, has a unique solution. It follows from his results that, if the coefficients in the equation above are constant, its solution will be of the form In the Appendix it will be shown that the equation (1) has the solution of the same form even without the assumption required by Dung.
In case of the fractional Ornstein-Uhlenbeck process and the geometric Brownian motion a comparison of various estimators of the Hurst index was presented in [11]. The behavior of the estimators based on quadratic variations was compared with that of some of the other known estimators. It should be noted that these estimators are not asymptotically normal. Moreover, only one of the estimators considered in the aforementioned paper is included in the comparison presented in this article.
A reader interested in the existence of the solution of the Gompertz diffusion process with respect to the standard Brownian motion and the estimation of its parameters is encouraged to read [15], [9], [8] and the references therein.
The structure of the paper is as follows. Section 2 presents the estimators considered in the rest of the paper. Section 3 contains the numerical comparison of the estimators' performance. Sections 4-6 are dedicated to proofs of strong consistency of the considered estimators in case of the fractional Gompertz diffusion process. In the Appendix the existence and uniqueness of the solution of equation (1) is proved.

Estimators
In the rest of the paper we will deal with the problem of estimating the Hurst index and the diffusion coefficient of the fractional Gompertz diffusion process based on discrete observations of its sample paths. The estimation of the trend parameters α and β, although not included in the present paper, can be performed using the least squares method. Using the change of variable Zt = ln Xt the equation (1) can be reduced to the fractional Vasicek model, to which the least squares method is then applied (see, e.g., [17]).

Remark 2
The estimators H (i) n , i = 1, 2, 3, 4, were considered in [13], [12], [3], and [1]. The estimator H (2) n can be used to estimate the Hurst index of the generic form of the SDE with an additional restriction on the diffusion coefficient.

Diffusion coefficient estimators
In this section, we describe four estimators of the diffusion coefficient. The application of the fourth is not explicitly justified, however this can be performed. It was proposed in [3] for the fractional geometric Brownian motion. The aforementioned paper shows it to be a weakly consistent estimator of the diffusion coefficient σ 2 .
For the purposes of comparison we shall also consider σ4,n = exp( B) where ni and H (3) n are defined in Theorem 1.

Remark 4
The estimators σ 2 i,n , i = 1, 2, are similar to the estimators used in the book [3] for the evaluation of the diffusion coefficient σ of the solutions of linear SDE when H is known. The estimator σ 2 3,n is used to estimate the diffusion coefficient of the fractional Ornstein-Uhlenbeck process when H is known (see [17]). We have shown that this restriction can be lifted.

Modeling of the estimators
The goal of this section is to describe the numerical simulations that were performed in order to compare the behavior of the estimators considered in this paper.
The sample paths of the fractional Brownian motion, which were further used to construct the sample paths of the fractional Gompertz diffusion process, were simulated using the Wood-Chan circulant matrix embedding method [16]. The values of the constants involved in these simulations were, unless explicitly stated otherwise, X0 = 3, α = 0.5, β = 2, and σ = 1.5. We considered these sample paths on the unit interval, hence T = 1. The number of replicates was 300 in all of the considered cases. In what follows we present the dependencies of the estimators both on the true parameter values (H, σ) and on the sample size (n). We have also checked for possible dependencies of the estimators of the Hurst index and the diffusion coefficient on the values of the other parameters of the considered equation, namely the drift coefficients α and β and the initial condition X0. No such dependencies of significant impact have been observed.  Figure 1, the same sample sizes n = 2 10 were used for all of the considered estimators, which does suggest that the estimators H (4) n and H (2) n would be a-priori less efficient. However, in practical applications the sample size is usually fixed, hence the motivation was to see what kind of performance the considered estimators would show given the exact same number of observations. In Figure 2, the value of the Hurst index was chosen as H = 0.75. The values of rj were taken to be powers of 2 (more precisely, rj = 2 j−1 , j = 1, . . . , l) and, further, the values of ni were taken as ni = n/ri where n denotes the (fixed) maximum sample size length. The value of l was (arbitrarily) taken to be 4, as simulation results suggested that both considerably smaller (f.e., 2) and considerably larger (f.e., log2n − 1) values yielded inferior performance. It does appear plausible that for much bigger sample sizes it might be beneficial to increase this value further, however in this study sample sizes exceeding 6400 points were not considered. It can be seen that the performance of the estimator H (4) n is slightly lacking compared to that of the other estimators, which, despite imposing rather different requirements on the sample sizes, show similar precision.

Modeling of the diffusion coefficient estimators
In order to calculate the estimators σ 2 1,n , σ 2 2,n and σ 2 3,n we need to supply them with the estimated values of the Hurst index. In the Figures 3 and 4 presented below, the diffusion coefficient estimator σ 2 i,n , using the Hurst index estimator H (j) n , is denoted as 'si hj', i, j = 1, 2, 3. The estimator σ 2 4,n is denoted as 's4'. The graphs present the relative differences, namely ( σi,n − σ)/σ. In Figure 3, the sample size was chosen to be n = 2 10 for all of the considered estimators. In Figure 4, the value of the diffusion coefficient was chosen as σ = 1. It can be seen that the performance of all the considered estimators is roughly similar. The convergence rate of σ 2 4,n appears slower, although it seems to perform better for the values of σ close to zero. For the other estimators, it appears that using H
In the rest of the paper Wp( Below we list several facts used in the sequel. For details we refer the reader to [6]. • Let f ∈ Wq and h ∈ Wp with p, q ∈ (0, ∞) : 1/p + 1/q > 1. Then an integral b a f dh exists as the Riemann-Stieltjes integral provided f and h have no common discontinuities. If the integral exists, the Love-Young inequality Cp,qVq f Vp h holds for all y ∈ [a, b], where Cp,q = ζ(p −1 + q −1 ) and ζ(s) = n 1 n −s . Moreover, The chain rule is based on the Riemann-Stieltjes integrals.

Several results on fBm
Recall that the fBm (B H t ) t∈[0,T ] with the Hurst index H ∈ (0, 1) is a real-valued continuous centered Gaussian process with the covariance given by In order to consider the strong consistency and asymptotic normality of the given estimators we need several facts about B H (see [2], [3], [4], [10]).
Limit results. For consideration of the asymptotic properties of the estimators H (i) n , i = 1, 2, we shall use the following results. Let In order to prove the asymptotic normality of the estimator H n we need the following result obtained in [3]. Let ni = rin, i = 1, . . . , , where ri, n ∈ N, and zi, i = 1, . . . , , are defined in Theorem 1. Then where r = (r1, . . . , r ), z = (z1, . . . , z ), If ki = kj then Then sup where Oω is defined in subsection 5.

Properties of the increments of the Gompertz diffusion process
The fractional Gompertz diffusion process X has the explicit solution given by The proof of this can be found in the Appendix. Now we will consider the structure of increments of the Gompertz diffusion process.
Lemma 8 Suppose that X satisfies (1), ε ∈ (0, H − 1 2 ) and partition πn of the interval [0, T ] is uniform. Then the following relations hold: where dn = τ mn k − τ mn k−1 and dn → 0 as n → ∞. Moreover, EOω(1) < ∞. Proof. For the sake of simplicity we will omit the index mn for the points τ mn k . Let the sample path t → Xt be continuous. We first prove (10). Note that It is clear that From the Chain rule it follows that Provided it follows that since |e x − 1| |x|e |x| for all x ∈ R. Consequently, and Since (see subsection 4.2 and [5]) for all p 1, then EOω(1) < ∞.
Next we prove (11). Taking into account (12) and (13) we get (2) n considered in Theorem 1 follows from Lemma 8. Indeed, the asymptotics of the increments of the solution X of the equation (1) are the same as the asymptotics of the increments of the solution of the equation with polynomial drift in [13]. Thus in order to establish the convergence of the estimator H (1) n it suffices to repeat the proof of Theorem 2 in [13]. Further, note that hypotheses (H) and (H1) in [14] are satisfied for the solution of the equation (1), i.e.
It follows from Lemma 8 and the a.s. continuity of t → Xt. Thus it suffices to apply Theorem 2.2 in [14].
We will notice the following properties: Using those we get zi yi − yi + ln(rin) + Oω(n −1/2 ln 1/2 n) ln ri + Oω(n −1/2 ln 1/2 n) = H + Oω(n −1/2 ln 1/2 n). (15) So the estimator H n is strongly consistent. Now we prove the asymptotic normality of the estimator H n . From (14) and (15) it follows that and we obtain the asymptotic normality of the estimator H n by the application of the limit results from subsection 4.2.
3. It remains to determine the convergence of H (4) n . Denote This statistic was introduced in [1]. Further on, we will require the following lemma which is a simple modification of Lemma 3.1 in [1]. In this lemma we have lifted the requirement for the random variables Z1 and Z2 to be independent. This became possible due to the application of less precise estimators of the partial derivatives.

The convergence rate of H
Proceeding along the lines of the proof of Theorem 2.2 from [14], it can be concluded that for almost all ω belongs to CWH ε ([0, T ]) and is the unique solution of (1).