Quasi-Likelihood Estimation in the Fractional Black–Scholes Model

Abstract

In this paper, we consider the parameter estimation for the fractional Black–Scholes model of the form $S_t^H=S_0^H+\mu {\int }_0^t{S}_s^H\mathrm{d}s+\sigma {\int }_0^t{S}_s^H\mathrm{d}{B}_s^H,$ where $\sigma >0$ and $\mu \in \mathbb{R}$ are the parameters to be estimated. Here, $B^H=\{B_t^H,t\ge 0\}$ denotes a fractional Brownian motion with Hurst index $0<H<1$. Using the quasi-likelihood method, we estimate the parameters $\mu$ and $\sigma$ based on observations taken at discrete time points $\{t_i=ih,i=0,1,2,\dots ,n\}$. Under the conditions $h=h\left(n\right)\to 0$, $nh\to \infty$, and $h^{1+\gamma }n\to 1$ for some $\gamma >0$, as $n\to \infty$, the asymptotic properties of the quasi-likelihood estimators are established. The analysis further reveals how the convergence rate of $n{h}^{1+\gamma }-1$ approaching zero affects the accuracy of estimation. To validate the effectiveness of our method, we conduct numerical simulations using real-world stock market data, demonstrating the practical applicability of the proposed estimation framework.

1. Introduction

In classical financial theory, the market is assumed to be arbitrage free and complete, and so standard Brownian motion is often used as noise to characterize the prices of financial derivatives. However, in real-world financial markets, prices often exhibit long-range dependence and non-stationarity, which are inconsistent with the characteristics of standard Brownian motion. Consequently, many authors have proposed using fractional Brownian motion (fBm) to construct market models (see, for examples, Mandelbrot and Van Ness [1]), with its simple structure and properties in memory noise. Unfortunately, starting with Rogers [2], there has been an ongoing dispute on the proper usage of fractional Brownian motion in option pricing theory. A troublesome problem arises because fBm is not a semimartingale and therefore, “no arbitrage pricing” cannot be used. Although this is a consensus, the consequences are not clear. The orthodox explanation is simple: fBm is not a suitable candidate for the price process. But, as shown by Cheridito [3], assuming that market participants cannot react immediately, any theoretical arbitrage opportunities will disappear. On the other hand, in 2003, Hu and ksendal [4] used the Wick–Itô-type integral to define a fractional market and showed the market was arbitrage free and complete. In that case, the prices of financial derivatives satisfied the following fractional Black–Scholes model:

$$d{S}_t^H=\mu S_t^{H}dt+\sigma S_t^{H}d{B}_t^H$$

(1)

with $S_0^H>0$, where $B^H=\{B_t^H,t\ge 0\}$ is a fractional Brownian motion with Hurst index ${\displaystyle \frac{1}{2}}\le H<1$, $\mu \in \mathbb{R}$ and $\sigma >0$ are two parameters, and the integral ${\int }_0^t{S}_s^{H}d{B}_s^H$ denotes the fractional Itô integral (Skorohod integral). For further studies on fractional Brownian motion in Black–Scholes models, refer to works by Bender and Elliott [5], Biagini [6], Bjork-Hult [7], Cheridito [3], Elliott and Chan [8], Greene-Fielitz [9], Necula [10], Lo [11], Mishura [12], Rogers [2], Izaddine [13], and additional references cited therein.

In this paper, we consider the application of the quasi-likelihood method in continuous stochastic systems. Our goal is to establish the quasi-likelihood estimations for parameters $\mu$ and ${\sigma }^2$ in Equation (1) and to establish their asymptotic behaviors. As is well known, there are many papers on parameter estimation of stochastic differential equations, but the use of the quasi-likelihood method to deal with parameter estimation problems of stochastic differential equations without independent increments has not been seen so far. Clearly, the solution of (1) does not have independent increments unless $H={\displaystyle \frac{1}{2}}$. We briefly describe the quasi-likelihood method as follows.

Let $X=\{X_t,t\ge 0\}$ be a stochastic process such that its distribution contains unknown parameters $\theta \in \mathsf{\Theta }\subset {\mathbb{R}}^k$ with $k\ge 1$. Assume that $X_{t_1},X_{t_2},\dots ,X_{t_n}$ are samples extracted from X, and that $x\mapsto f_j\left(x\right)$ is the probability function (e.g., density function) of the increment $X_{t_j}-X_{t_{j-1}}$ for $j\in \{1,2,\dots ,n\}$. Since the process X generally does not have independent increments, the function

$$L\left(\theta \right)=\prod _{j=1}^n{f}_j\left(X_{t_j}-X_{t_{j-1}}\right)$$

is generally not a likelihood function. However, we can still use the usual method to obtain an estimator, which is called a quasi-likelihood estimator.

Let $B^H=\left\{B_t^H,t\ge 0\right\}$ be fractional Brownian motion with Hurst index $H\in (0,1)$ defined on the probability space $(\mathsf{\Omega },\mathcal{F},P,\left({\mathcal{F}}_t\right))$. Consider the fractional Black–Scholes model as follows:

$$\begin{array}{c}\hfill S_t^H=S_0^H+\mu {\int }_0^t{S}_s^{H}ds+\sigma {\int }_0^t{S}_s^{H}d{B}_s^H\end{array}$$

(2)

with $S_0^H>0$, where $0<H<1$, and the stochastic integral is the fractional Itô integral [14]. By using the Itô formula, we get

$$\begin{array}{c}\hfill S_t^H=S_0^{H}exp\left(\mu t-{\displaystyle \frac{1}{2}}{\sigma }^2{t}^{2H}+\sigma B_t^H\right),\end{array}$$

with $t\ge 0$, which is called the geometric fractional Brownian motion (gmfBm).

In this paper, for simplicity, throughout, we let H be known. Denote $f\left(t\right)=\mu t-{\displaystyle \frac{1}{2}}{\sigma }^2{t}^{2H}$$(t\ge 0)$ and

$$\begin{array}{c}\hfill Y_t^H:=log{S}_t^H-log{S}_0^H=\sigma B_t^H+\mu t-{\displaystyle \frac{1}{2}}{\sigma }^2{t}^{2H}=\sigma B_t^H+f\left(t\right).\end{array}$$

(3)

Now, let $H\in (0,1)$ be known, and let the gmfBm $S^H=\{S_t^H,t\ge 0\}$ be observed at some discrete time instants $\{t_i=ih,i=0,1,2,\dots ,n\}$ satisfying the following conditions:

(C1)

$h=h\left(n\right)\downarrow 0$ and $t_n=nh\to +\infty$ as $n\to \infty$.

(C2)

There exists $\gamma >0$ such that $n{h}^{1+\gamma }⟶1$ as $n\to \infty$.

We get a quasi-likelihood function of parameter $\mu$ and ${\sigma }^2$ as follows

$$\begin{array}{cc}\hfill L_1(\mu ,{\sigma }^2):& =\prod _{i=1}^n{f}_i\left(Y_{t_i}-Y_{t_{i-1}}\right)\hfill \\ \hfill & =\prod _{i=1}^n{\displaystyle \frac{1}{\sigma \sqrt{2\pi h^{2H}}}}exp\left\{-{\displaystyle \frac{1}{2{\sigma }^2{h}^{2H}}}{\left(Y_{t_i}-Y_{t_{i-1}}-\mu h+{\displaystyle \frac{1}{2}}{\sigma }^2\left(t_i^{2H}-t_{i-1}^{2H}\right)\right)}^2\right\},\hfill \end{array}$$

where $f_i(·)$ is the density of the random variable $Y_{t_i}-Y_{t_{i-1}}$. Then, the logarithmic quasi-likelihood function is given by

$$\begin{array}{c}\hfill log{L}_1(\mu ,\theta )=-{\displaystyle \frac{n}{2}}log\theta -{\displaystyle \frac{n}{2}}log2\pi h^{2H}-{\displaystyle \frac{1}{2\theta h^{2H}}}\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-\mu h+{\displaystyle \frac{1}{2}}\theta {\widehat{t}}_i\right)}^2\end{array}$$

with $\theta ={\sigma }^2$, where ${\widehat{t}}_i=t_i^{2H}-t_{i-1}^{2H}$. By using the quasi-likelihood function, we get that the estimators ${\widehat{\mu }}_n$ and ${\widehat{\theta }}_n$ of $\mu$ and $\theta ={\sigma }^2$ satisfy the equations

$$\left\{\begin{array}{c}{\widehat{\mu }}_n={\displaystyle \frac{1}{nh}}{\displaystyle \sum _{i=1}^n}\left(Y_{t_i}-Y_{t_{i-1}}+{\displaystyle \frac{1}{2}}{\widehat{\theta }}_n{\widehat{t}}_i\right)\hfill \\ {\widehat{\theta }}_n={\displaystyle \frac{1}{{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2}}\left(-2n{h}^{2H}+2\sqrt{n^2{h}^{2H}+{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2{\displaystyle \sum _{i=1}^n}{\left(Y_{t_i}-Y_{t_{i-1}}-{\widehat{\mu }}_{n}h\right)}^2}\right).\hfill \end{array}\right.$$

(4)

When $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},1)$, by solving the above equation system, we get the estimators of $\mu$ and $\theta ={\sigma }^2$ as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\widehat{\mu }}_n={\displaystyle \frac{1}{nh}}{Y}_{t_n}^H+{\displaystyle \frac{{\beta }_n}{{\rho }_n}}\left(-2n{h}^{2H}+2\sqrt{n^2{h}^{4H}+{\rho }_n{\displaystyle \sum _{i=1}^n}{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}^H\right)}^2}\right),\hfill \\ {\widehat{\theta }}_n={\displaystyle \frac{1}{{\rho }_n}}\left(-2n{h}^{2H}+2\sqrt{n^2{h}^{4H}+{\rho }_n{\displaystyle \sum _{i=1}^n}{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}^H\right)}^2}\right),\hfill \end{array}\right.\end{array}$$

(5)

where ${\beta }_n={\displaystyle \frac{1}{2}}{\left(nh\right)}^{2H-1}$ and ${\rho }_n={\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2-n^{4H-1}{h}^{4H}$ for every $n\ge 1$. When $H={\displaystyle \frac{1}{2}}$, we have ${\rho }_n=0$ and the random variables

$$Y_{t_i}^H-Y_{t_{i-1}}^H\sim N\left((\mu -{\sigma }^2)h,{\sigma }^2\right),i=1,2,\dots n$$

are independent identical distributions, the above logarithm quasi-likelihood function is a classical logarithm likelihood function, and we have

$$\left\{\begin{array}{c}{\widehat{\mu }}_n={\displaystyle \frac{1}{nh}}{Y}_{t_n}^H+{\displaystyle \frac{{\beta }_n}{nh}}{\displaystyle \sum _{i=1}^n}{\left(Y_{t_i}^H-Y_{t_{i-1}}^H\right)}^2-{\displaystyle \frac{{\beta }_n}{n^{2}h}}{\left(Y_{t_n}^H\right)}^2,\hfill \\ {\widehat{\theta }}_n={\displaystyle \frac{1}{nh}}{\displaystyle \sum _{i=1}^n}{\left(Y_{t_i}^H-Y_{t_{i-1}}^H\right)}^2-{\displaystyle \frac{1}{n^{2}h}}{\left(Y_{t_n}^H\right)}^2.\hfill \end{array}\right.$$

(6)

and the asymptotic behavior of the two estimators can be easily established, so in the discussion later in this paper, unless otherwise stated, it is assumed that $H\ne {\displaystyle \frac{1}{2}}$.

Our study focuses on the asymptotic properties of two estimators. Given the Gaussian properties of the sample, we expect these estimators to exhibit quadratic variation, facilitating the derivation and simplification of their asymptotic behavior using fractional Brownian motion. To fully characterize this behavior, we rely on key properties of fractional Brownian motion, which not only underpin the theoretical understanding of complex stochastic processes but also provide a foundation for applying quasi-likelihood methods in parameter estimation.

The structure of this paper is as follows. In Section 2, we briefly describe the basic properties of fractional Brownian motion. In Section 3 and Section 4 we discuss the strong consistency and asymptotic normality of the estimator ${\widehat{\theta }}_n$ and analyze the asymptotic behavior under the cases where the parameter $\mu$ is known and unknown. To prove these two asymptotic behaviors, we rely on two key results related to fractional Brownian motion. Although these results have been proven for a finite observation interval, they also hold when the observation length $nh$ tends to infinity. In Section 5, we consider the asymptotic behavior of the estimator ${\widehat{\mu }}_n$. In Section 6, we provide numerical verification and empirical analysis of the estimators ${\widehat{\mu }}_n$ and ${\widehat{\theta }}_n$. In Section 7, we conclude that the proposed fractional Brownian motion quasi-likelihood method performs well theoretically and empirically, offering a practical framework for financial parameter estimation.

2. Preliminaries

In this section, we briefly recall some basic results on fractional Brownian motion. For more aspects on the material, we refer to Bender [15], Biagini et al. [6], Cheridito-Nualart [16], Gradinaru et al. [17], Hu [4], Mishura [12], Nourdin [18], Nualart [19], Tudor [20], and references therein.

A zero mean Gaussian process $B^H=\{B_t^H,0\le t\le T\}$ defined on a complete probability space $(\mathsf{\Omega },\mathcal{F},P,\left({\mathcal{F}}_t\right))$ is called the fBm with Hurst index $H\in (0,1)$ provided that $B_0^H=0$ and

$$\mathbb{E}\left[B_t^H{B}_s^H\right]={\displaystyle \frac{1}{2}}\left[t^{2H}+s^{2H}-{|t-s|}^{2H}\right]$$

for $t,s\ge 0$. Let $\mathcal{H}$ be the completion of the linear space $\mathcal{E}$ generated by the indicator functions $1_{[0,t]},t\in [0,T]$ with respect to the inner product

$${\langle 1_{[0,s]},1_{[0,t]}\rangle }_{\mathcal{H}}={\displaystyle \frac{1}{2}}\left[t^{2H}+s^{2H}-{|t-s|}^{2H}\right].$$

When $H={\displaystyle \frac{1}{2}}$, we know that $\mathcal{H}=L^2$, and when ${\displaystyle \frac{1}{2}}<H<1$, we have

$${\parallel \phi \parallel }_{\mathcal{H}}^2=H(2H-1){\int }_0^T{\int }_0^T\phi \left(t\right)\phi \left(s\right){|t-s|}^{2H-2}dsdt$$

for all $\phi \in \mathcal{H}$. The application

$$\mathcal{E}\ni \phi \mapsto B^H\left(\phi \right):={\int }_0^T\phi \left(s\right)d{B}_s^H$$

is an isometry from $\mathcal{E}$ to the Gaussian space generated by $B^H$, and it can be extended to $\mathcal{H}$. Denote by $𝒮$ the set of smooth functionals of the form

$$F=f(B^H\left({\phi }_1\right),B^H\left({\phi }_2\right),\dots ,B^H\left({\phi }_n\right)),$$

where $f\in C_b^{\infty }\left({\mathbb{R}}^n\right)$ (f and all its derivatives are bounded) and ${\phi }_i\in \mathcal{H}$. The derivative operator $D^H$ (the Malliavin derivative) of a functional F of the above form is defined as

$$D^{H}F=\sum _{j=1}^n{\displaystyle \frac{\partial f}{\partial x_j}}(B^H\left({\phi }_1\right),B^H\left({\phi }_2\right),\dots ,B^H\left({\phi }_n\right)){\phi }_j.$$

The derivative operator $D^H$ is then a closable operator from $L^2\left(\mathsf{\Omega }\right)$ into $L^2(\mathsf{\Omega };\mathcal{H})$. We denote by ${\mathbb{D}}^{1,2}$ the closure of $𝒮$ with respect to the norm

$${\parallel F\parallel }_{1,2}:=\sqrt{{E\left|F\right|}^2+E{\parallel D^{H}F\parallel }_{\mathcal{H}}^2}.$$

The divergence integral ${\delta }^H$ is the adjoint of derivative operator $D^H$. That is, we say that a random variable u in $L^2(\mathsf{\Omega };\mathcal{H})$ belongs to the domain of the divergence operator ${\delta }^H$, denoted by $\mathrm{Dom}\left({\delta }^H\right)$, if

$$\mathbb{E}\left|{\langle D^{H}F,u\rangle }_{\mathcal{H}}\right|\le c{\parallel F\parallel }_{L^2\left(\mathsf{\Omega }\right)}$$

for every $F\in {\mathbb{D}}^{1,2}$. In this case, ${\delta }^H\left(u\right)$ is defined by the duality relationship

$$\mathbb{E}\left[F{\delta }^H\left(u\right)\right]=E{\langle D^{H}F,u\rangle }_{\mathcal{H}}$$

for any $F\in {\mathbb{D}}^{1,2}$. Generally, the divergence ${\delta }^H\left(u\right)$ is also called the Skorohod integral of a process u and denoted as

$${\delta }^H\left(u\right)={\int }_0^T{u}_{s}d{B}_s^H,$$

and the indefinite Skorohod integral is defined as ${\int }_0^t{u}_{s}d{B}_s^H={\delta }^H\left(u{1}_{[0,t]}\right)$. If the process $u=\{u_t,t\ge 0\}$ is adapted, the Skorohod integral is called the fractional Itô integral, and the Itô formula

$$f(B_t^H,t)=f(0,0)+{\int }_0^t{\displaystyle \frac{\partial }{\partial x}}f(B_s^H,s)d{B}_s^H+{\int }_0^t{\displaystyle \frac{\partial }{\partial t}}f(B_s^H,s)ds+H{\int }_0^t{\displaystyle \frac{{\partial }^2}{\partial x^2}}f(B_s^H,s)s^{2H-1}ds$$

(7)

holds for all $t\in [0,T]$ and $f\in C^{2\times 1}(\mathbb{R}\times {\mathbb{R}}_{+})$.

3. The Strong Consistency of the Estimator ${\widehat{\mathit{\theta }}}_{\mathit{n}}$

In this section, we obtain the consistency of the estimator ${\widehat{\theta }}_n$ in two cases. To establish the consistency of estimator ${\widehat{\theta }}_n$, we need four lemmas, and proving these four lemmas requires several more statements. Therefore, we have placed the proof of these results at the end of this section. For simplicity, we denote by $\stackrel{a.s}{\to }$ the convergence with probability one, as n tends to infinity; moreover, the symbol $\stackrel{a.s}{\sim }$ means that both sides have the same limit with probability one, as n tends to infinity.

 Lemma 1. 

Let $B^H=\{B_t^H,t\ge 0\}$ be a fractional Brownian motion with Hurst index $0<H<1$, and let $t_i=ih,i=0,1,2,\dots ,n$ such that the condition $\left(C1\right)$ holds. Then, with probability one, we have

$$M_n\left(B^H\right):={\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n{\left(B_{t_i}^H-B_{t_{i-1}}^H\right)}^2⟶1(n\to \infty ).$$

 Lemma 2. 

Let the conditions of Lemma 1 and condition $\left(C1\right)$ hold. Denote

$$N_n\left(Y^H\right):={\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H\right)}^2.$$

 (1) 

If $0<H<{\displaystyle \frac{1}{2}}$, we have $N_n\left(Y^H\right)\to {\sigma }^2$ almost surely, as n tends to infinity.

 (2) 

If ${\displaystyle \frac{1}{2}}<H<1$ and the condition $\left(C2\right)$ holds with $0<\gamma <{\displaystyle \frac{1-H}{2H-1}}$, $N_n\left(Y^H\right)$ converges to ${\sigma }^2$ almost surely, as n tends to infinity.

 (3) 

If ${\displaystyle \frac{1}{2}}<H<1$ and the condition $\left(C2\right)$ holds with $\gamma ={\displaystyle \frac{1-H}{2H-1}}$, $N_n\left(Y^H\right)$ converges to ${\sigma }^2+{\displaystyle \frac{H^2{\sigma }^4}{4H-1}}$ almost surely, as n tends to infinity.

3.1. The Strong Consistency of ${\widehat{\theta }}_n$ When $\mu$ Is Known

First, we consider the case where $\mu$ is known. By transforming ${\widehat{\theta }}_n$ based on Equation (4), we obtain the following result:

$$\begin{array}{cc}\hfill {\widehat{\theta }}_n& ={\displaystyle \frac{2n{h}^{2H}}{{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2}}\left(\sqrt{1+{\displaystyle \frac{1}{n^2{h}^{4H}}}{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2·{\displaystyle \sum _{i=1}^n}{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-\mu h\right)}^2}-1\right)\hfill \\ & ={\displaystyle \frac{2n{h}^{2H}}{{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2}}\left(\sqrt{1+{\Delta }_n\left(H\right)}-1\right),\hfill \end{array}$$

(8)

where

$${\Delta }_n\left(H\right)={\displaystyle \frac{1}{n^2{h}^{4H}}}{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2·{\displaystyle \sum _{i=1}^n}{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-\mu h\right)}^2.$$

 Lemma 3. 

Let $\mu \in \mathbb{R}$ and $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},1)$, and let condition $\left(C1\right)$ hold.

 (1) 

For $0<H<{\displaystyle \frac{1}{2}}$, we have $\underset{n\to 0}{lim}{\Delta }_n\left(H\right)\stackrel{a.s}{=}0.$

 (2) 

For ${\displaystyle \frac{1}{2}}<H<1$, if the condition $\left(C2\right)$ hold with $0<\gamma <{\displaystyle \frac{1-H}{2H-1}}$, $\underset{n\to 0}{lim}{\Delta }_n\left(H\right)\stackrel{a.s}{=}0.$

 Theorem 1. 

Let μ be known and let the condition $\left(C1\right)$ hold.

 (1) 

If $0<H<{\displaystyle \frac{1}{2}}$, the estimator ${\widehat{\theta }}_n$ is strongly consistent.

 (2) 

If ${\displaystyle \frac{1}{2}}<H<1$ and the condition $\left(C2\right)$ holds with $0<\gamma <{\displaystyle \frac{1-H}{2H-1}}$, then the estimator ${\widehat{\theta }}_n$ is strongly consistent.

 Proof. 

When $0<H<{\displaystyle \frac{1}{2}}$, by the fact that $\sqrt{1+x}-1\sim {\displaystyle \frac{1}{2}}x(x\to 0)$, Lemma 2, and Lemma 3, we obtain

$$\begin{array}{cc}\hfill {\widehat{\theta }}_n& ={\displaystyle \frac{2n{h}^{2H}}{{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2}}\left(\sqrt{1+{\displaystyle \frac{1}{n^2{h}^{4H}}}\sum _{i=1}^n{\left({\widehat{t}}_i\right)}^2·\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-\mu h\right)}^2}-1\right)\hfill \\ \hfill & \stackrel{a.s}{\sim }{\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-\mu h\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n{(Y_{t_i}^H-Y_{t_{i-1}}^H)}^2-2\mu h^{1-2H}·{\displaystyle \frac{1}{n}}\sum _{i=1}^n(Y_{t_i}^H-Y_{t_{i-1}}^H)+{\mu }^2{h}^{2-2H}⟶{\sigma }^2\hfill \end{array}$$

(9)

almost surely, as $n\to \infty$, and statement (1) follows. Similarly, we may obtain statement (2). □

3.2. The Strong Consistency of ${\widehat{\theta }}_n$ When $\mu$ Is Unknown

In this subsection, we assume that both parameters $\mu$ and $\sigma$ are unknown and establish the strong consistency of estimator ${\widehat{\theta }}_n$ as defined in (5). In this case, for $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},1)$, we obtain

$$\begin{array}{cc}\hfill {\widehat{\theta }}_n& ={\displaystyle \frac{1}{{\rho }_n}}\left(-2n{h}^{2H}+2\sqrt{n^2{h}^{4H}+{\rho }_n\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2}\right)\hfill \\ \hfill & ={\displaystyle \frac{2n{h}^{2H}}{{\rho }_n}}\left(\sqrt{1+{\displaystyle \frac{1}{n^2{h}^{4H}}}{\rho }_n\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2}-1\right)={\displaystyle \frac{2n{h}^{2H}}{{\rho }_n}}\left(\sqrt{1+{\tilde{\Delta }}_n\left(H\right)}-1\right)\hfill \end{array}$$

(10)

for every $n\ge 1$, where ${\rho }_n={\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2-n^{4H-1}{h}^{4H}$ and

$${\tilde{\Delta }}_n\left(H\right)={\displaystyle \frac{1}{n^2{h}^{4H}}}{\rho }_n\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2.$$

 Lemma 4. 

Let $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},1)$, and let condition $\left(C1\right)$ hold.

 (1) 

For $0<H<{\displaystyle \frac{1}{2}}$ we have $\underset{n\to 0}{lim}{\tilde{\Delta }}_n\left(H\right)\stackrel{a.s}{=}0.$

 (2) 

For ${\displaystyle \frac{1}{2}}<H<1$, if the condition $\left(C2\right)$ hold with $0<\gamma <{\displaystyle \frac{1-H}{2H-1}}$, we have $\underset{n\to 0}{lim}{\tilde{\Delta }}_n\left(H\right)\stackrel{a.s}{=}0.$

 Theorem 2. 

Let μ be unknown and let the condition $\left(C1\right)$ hold.

 (1) 

For $0<H<{\displaystyle \frac{1}{2}}$, the estimator ${\widehat{\theta }}_n$ is strongly consistent.

 (2) 

For ${\displaystyle \frac{1}{2}}<H<1$, if the condition $\left(C2\right)$ holds with $0<\gamma <{\displaystyle \frac{1-H}{2H-1}}$, then ${\widehat{\theta }}_n$ is strongly consistent.

 Proof. 

By (10), we first show that

$${\tilde{\Delta }}_n\left(H\right):={\displaystyle \frac{1}{n^2{h}^{4H}}}{\rho }_n\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2⟶0$$

(11)

as n tends to infinity.

We first prove statement (2). When ${\displaystyle \frac{1}{2}}<H<1$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{n{h}^H}}{Y}_{t_n}& ={\displaystyle \frac{\sigma B_{t_n}^H}{n{h}^H}}+\mu h^{1-H}-{\displaystyle \frac{1}{2}}{\sigma }^2{h}^H{n}^{2H-1}\stackrel{a.s}{⟶}0(n\to \infty ),\hfill \end{array}$$

and combined with the fact that $\sqrt{1+x}-1\sim {\displaystyle \frac{1}{2}}x(x\to 0)$, Lemma 2, and Lemma 4, we obtain

$$\begin{array}{cc}\hfill {\widehat{\theta }}_n& ={\displaystyle \frac{2n{h}^{2H}}{{\rho }_n}}\left(\sqrt{1+{\displaystyle \frac{1}{n^2{h}^{4H}}}{\rho }_n\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2}-1\right)\stackrel{a.s}{\sim }{\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2\hfill \\ & ={\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H\right)}^2-{\displaystyle \frac{1}{n^2{h}^{2H}}}{Y}_{t_n}^2\stackrel{a.s}{⟶}{\sigma }^2(n\to +\infty ).\hfill \end{array}$$

By conditions (C1) and (C2), when ${\displaystyle \frac{1}{2}}<H<1$ and $0<\gamma <{\displaystyle \frac{1-H}{2H-1}}$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{n{h}^H}}{Y}_{t_n}& ={\displaystyle \frac{\sigma B_{t_n}^H}{n{h}^H}}+\mu h^{1-H}-{\displaystyle \frac{1}{2}}{\sigma }^2{h}^H{n}^{2H-1}\stackrel{a.s}{⟶}0\left(n\to \infty ,\right).\hfill \end{array}$$

Thus, similar to the case where $0<H<{\displaystyle \frac{1}{2}}$, we obtain statement (1). □

3.3. Proofs of Lemmas in Section 3

In this subsection, we complete the proof of several lemmas that were not proven above. To prove these lemmas and for the convenience of expression, we provide four lemmas.

 Lemma 5 

(Etemadi [21]). Let $\left\{{\xi }_n\right\}$ be a sequence of nonnegative random variables with finite second moments and $S_n={\displaystyle \sum _{i=1}^n}{\xi }_i$ such that:

 (1) 

The sequence $\{{\omega }_n=E{\xi }_n\}$ satisfies ${\left({\displaystyle \sum _{i=1}^n}{\omega }_i\right)}^{-1}{\omega }_n\to 0$ and ${\displaystyle \sum _{i=1}^n}{\omega }_i\to \infty$ as $n\to \infty$.

 (2) 

The following series converges:

$${\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{{\left(E{S}_n\right)}^2}}\sum _{i=1}^n{\mathrm{Cov}}^{+}({\xi }_n,{\xi }_i).$$

Then, as $n\to \infty$, ${\displaystyle \frac{S_n}{E{S}_n}}\to 1$ almost surely.

 Lemma 6. 

For all $0<r^{\prime }<s^{\prime }<r<s$ and $0<H<1$, we have

$$\left|E\left[(B_s^H-B_r^H)(B_{s^{\prime }}^H-B_{r^{\prime }}^H)\right]\right|\le {\displaystyle \frac{(s-r)(s^{\prime }-r^{\prime })}{{(r-s^{\prime })}^{2-2H}}}.$$

(12)

 Proof. 

When $0<H<{\displaystyle \frac{1}{2}}$, the lemma is obtained from the proof of Lemma 3.3 in Yan et al. [22], and we can also prove the case ${\displaystyle \frac{1}{2}}<H<1$ in a completely similar way. □

 Lemma 7. 

Let $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},1)$ and denote $K_n\left(H\right):={\displaystyle \sum _{i=1}^n}{(i^{2H}-{(i-1)}^{2H})}^2$ for $n\ge 1$. Then, we have:

 (1) 

For $0<H<{\displaystyle \frac{1}{4}}$,$\underset{n\to \infty }{lim}{K}_n\left(H\right)$ is finite and nonzero.

 (2) 

For $H={\displaystyle \frac{1}{4}}$, $K_n\left(H\right)=O\left(logn\right)$, as $n\to \infty$.

 (3) 

For ${\displaystyle \frac{1}{4}}<H<1$, as $n\to \infty$, we have

$$K_n\left(H\right)={\displaystyle \frac{4H^2}{4H-1}}{n}^{4H-1}+2H^2{h}^{4H}{n}^{4H-2}+h^{4H}O\left(n^{4H-3}\right).$$

 Proof. 

The lemma is a simple calculus exercise. □

 Lemma 8. 

Let $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},1)$ and $f\left(t\right)=\mu t-{\displaystyle \frac{1}{2}}{\sigma }^2{t}^{2H}$. Denote

$${\mathsf{\Psi }}_H\left(n\right):={\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n{\left[f\left(t_i\right)-f\left(t_{i-1}\right)\right]}^2,$$

where $t_i=ih$ with conditions (C1) and (C2). Then, $\underset{n\to \infty }{lim}{\mathsf{\Psi }}_H\left(n\right)=0$ for $0<H<{\displaystyle \frac{1}{2}}$, and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\mathsf{\Psi }}_H\left(n\right)=\left\{\begin{array}{cc}0,\hfill & 0<\gamma <{\displaystyle \frac{1-H}{2H-1}},\hfill \\ {\displaystyle \frac{H^2{\sigma }^4}{4H-1}},\hfill & \gamma ={\displaystyle \frac{1-H}{2H-1}},\hfill \\ +\infty ,\hfill & \gamma >{\displaystyle \frac{1-H}{2H-1}},\hfill \end{array}\right.\end{array}$$

for all ${\displaystyle \frac{1}{2}}<H<1$.

 Proof. 

By Lemma 7, it follows that

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_H\left(n\right)& ={\mu }^2{h}^{2-2H}-\mu {\sigma }^{2}h{n}^{2H-1}+{\displaystyle \frac{1}{4n{h}^{2H}}}{\sigma }^4\sum _{i=1}^n{\left({\left(t_i\right)}^{2H}-{\left(t_{i-1}\right)}^{2H}\right)}^2\hfill \\ \hfill & ={\mu }^2{h}^{2-2H}-\mu {\sigma }^{2}h{n}^{2H-1}+{\displaystyle \frac{1}{4}}{\sigma }^4{h}^{2H}{\displaystyle \frac{1}{n}}{K}_n\left(H\right)⟶0\hfill \end{array}$$

for all $0<H<{\displaystyle \frac{1}{2}}$, as n tends to infinity. Moreover, when ${\displaystyle \frac{1}{2}}<H<1$, by conditions (C1) and (C2), we have

$$\underset{n\to \infty }{lim}h{n}^{2H-1}=\left\{\begin{array}{cc}0,\hfill & 0<\gamma <{\displaystyle \frac{2-2H}{2H-1}},\hfill \\ 1,\hfill & \gamma ={\displaystyle \frac{2-2H}{2H-1}},\hfill \\ +\infty ,\hfill & \gamma >{\displaystyle \frac{2-2H}{2H-1}},\hfill \end{array}\right.\mathrm{and}\underset{n\to \infty }{lim}{h}^{2H}{n}^{4H-2}=\left\{\begin{array}{cc}0,\hfill & 0<\gamma <{\displaystyle \frac{1-H}{2H-1}},\hfill \\ 1,\hfill & \gamma ={\displaystyle \frac{1-H}{2H-1}},\hfill \\ +\infty ,\hfill & \gamma >{\displaystyle \frac{1-H}{2H-1}},\hfill \end{array}\right.$$

which imply

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_H\left(n\right)& ={\mu }^{2}h-\mu {\sigma }^2{h}^{2H}{n}^{2H-1}+{\sigma }^2{\displaystyle \frac{1}{4n}}{h}^{4H-1}{K}_n\left(H\right)⟶\left\{\begin{array}{cc}0,\hfill & 0<\gamma <{\displaystyle \frac{1-H}{2H-1}},\hfill \\ {\displaystyle \frac{H^2{\sigma }^4}{4H-1}},\hfill & \gamma ={\displaystyle \frac{1-H}{2H-1}},\hfill \\ +\infty ,\hfill & \gamma >{\displaystyle \frac{1-H}{2H-1}}\hfill \end{array}\right.\hfill \end{array}$$

for all ${\displaystyle \frac{1}{2}}<H<1$, as n tends to infinity. □

Now let us prove Lemma 1, Lemma 2, Lemma 3, and Lemma 4 one by one.

 Proof of Lemma 1. 

When $H={\displaystyle \frac{1}{2}}$, the lemma follows from the strong law of large numbers. Now, let $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},1)$. Then, the sequence $\left\{\mathbb{E}{(B_{t_n}^H-B_{t_{n-1}}^H)}^2\right\}$ satisfies condition (1) in Lemma 5. On the other hand, by the fact that

$$\mathbb{E}\left[{(B_{t_n}^H-B_{t_{n-1}}^H)}^2{(B_{t_i}^H-B_{t_{i-1}}^H)}^2\right]=h^{4H}+2{\left(E(B_{t_n}^H-B_{t_{n-1}}^H)(B_{t_i}^H-B_{t_{i-1}}^H)\right)}^2$$

with $n>i$ and Lemma 6, we see that

$$\begin{array}{cc}\hfill \sum _{i=1}^n& \left|\mathrm{Cov}\left({(B_{t_n}^H-B_{t_{n-1}}^H)}^2,{(B_{t_i}^H-B_{t_{i-1}}^H)}^2\right)\right|=\sum _{i=1}^n\left|\mathbb{E}\left[{(B_{t_n}^H-B_{t_{n-1}}^H)}^2{(B_{t_i}^H-B_{t_{i-1}}^H)}^2\right]-h^{4H}\right|\hfill \\ & =2\sum _{i=1}^n{\left(E(B_{t_n}^H-B_{t_{n-1}}^H)(B_{t_i}^H-B_{t_{i-1}}^H)\right)}^2\hfill \\ & =2h^{4H}+2{\left(E(B_{t_n}^H-B_{t_{n-1}}^H)(B_{t_{n-1}}^H-B_{t_{n-2}}^H)\right)}^2+2\sum _{i=1}^{n-2}{\left(E(B_{t_n}^H-B_{t_{n-1}}^H)(B_{t_i}^H-B_{t_{i-1}}^H)\right)}^2\hfill \\ \hfill & \le 2h^{4H}+{(2-2^{2H})}^2{h}^{4H}+2h^{4H}\sum _{i=1}^{n-2}{\displaystyle \frac{1}{{(n-1-i)}^{4-4H}}}=h^{4H}\left(2+{(2-2^{2H})}^2+2\sum _{j=1}^{n-2}{\displaystyle \frac{1}{j^{4-4H}}}\right)\hfill \\ \hfill & \le \left\{\begin{array}{cc}C{h}^{4H},\hfill & 0<H<{\displaystyle \frac{3}{4}},\hfill \\ C{h}^3(1+logn),\hfill & H={\displaystyle \frac{3}{4}},\hfill \\ C{h}^{4H}{n}^{4H-3},\hfill & {\displaystyle \frac{3}{4}}<H<1\hfill \end{array}\right.\hfill \end{array}$$

for all $n\ge 1$. It follows that

$$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }& {\displaystyle \frac{1}{{\left\{{\sum }_{i=1}^n\mathbb{E}{(B_{t_i}^H-B_{t_{i-1}}^H)}^2\right\}}^2}}\sum _{i=1}^n\left|\mathrm{Cov}\left({(B_{t_n}^H-B_{t_{n-1}}^H)}^2,{(B_{t_i}^H-B_{t_{i-1}}^H)}^2\right)\right|<\infty \hfill \end{array}$$

for all $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},1)$. Thus, condition (2) in Lemma 5 holds, and the lemma follows. □

 Proof of Lemma 2. 

Recall that

$$Y_t^H=\sigma B_t^H+f\left(t\right),t\ge 0,$$

where $f\left(t\right)=\mu t-{\displaystyle \frac{1}{2}}{\sigma }^2{t}^{2H}$. It follows that

$$\begin{array}{cc}\hfill N_n\left(Y^H\right)& ={\displaystyle \frac{{\sigma }^2}{n{h}^{2H}}}\sum _{i=1}^n{\left(B_{t_i}^H-B_{t_{i-1}}^H\right)}^2+{\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n{\left[f\left(t_i\right)-f\left(t_{i-1}\right)\right]}^2\hfill \\ \hfill & +{\displaystyle \frac{2\sigma }{n{h}^{2H}}}\sum _{i=1}^n\left[f\left(t_i\right)-f\left(t_{i-1}\right)\right]\left(B_{t_i}^H-B_{t_{i-1}}^H\right)\hfill \end{array}$$

(13)

for all $0<H<1$ and $n\ge 1$.

When $H={\displaystyle \frac{1}{2}}$, statement (1) follows from the strong law of large numbers for independent random variables.

When $0<H<{\displaystyle \frac{1}{2}}$, by Cauchy’s inequality, Lemma 8, and Lemma 1, one may show that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{n{h}^{2H}}}& \sum _{i=1}^n\left|\left[f\left(t_i\right)-f\left(t_{i-1}\right)\right]\left(B_{t_i}^H-B_{t_{i-1}}^H\right)\right|\hfill \\ & \le {\left\{{\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n{\left[f\left(t_i\right)-f\left(t_{i-1}\right)\right]}^2·{\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n{\left(B_{t_i}^H-B_{t_{i-1}}^H\right)}^2\right\}}^{1/2}\stackrel{a.s.}{⟶}0.\hfill \end{array}$$

almost surely, as n tends to infinity. It follows from (13), Lemma 8, and Lemma 1 that

$$N_n\left(Y^H\right)\stackrel{a.s}{⟶}{\sigma }^2$$

(14)

for all $0<H<{\displaystyle \frac{1}{2}}$, as n tends to infinity. This shows that statement (1) holds. Similarly, we can also obtain statements (2) and (3). □

 Proof of Lemma 3. 

When $0<H<{\displaystyle \frac{1}{2}}$, by Lemma 7, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(\widehat{t_i}\right)}^2& ={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left({\left(t_i\right)}^{2H}-{\left(t_{i-1}\right)}^{2H}\right)}^2={\displaystyle \frac{1}{n}}{h}^{4H}{K}_n\left(H\right)=h^{4H}O\left(n^{4H-2}\right)⟶0(n\to \infty ).\hfill \end{array}$$

It follows from Lemma 2 and the fact that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{n}}\sum _{i=1}^n(Y_{t_i}^H-Y_{t_{i-1}}^H)={\displaystyle \frac{1}{n}}{Y}_{t_n}^H={\displaystyle \frac{1}{n}}\sigma B_{t_n}^H+\mu h-{\displaystyle \frac{1}{2}}{\sigma }^2{n}^{2H-1}{h}^{2H}\stackrel{a.s}{⟶}0(n\to \infty )\end{array}$$

that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{n^2{h}^{4H}}}& \sum _{i=1}^n{\left({\widehat{t}}_i\right)}^2·\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-\mu h\right)}^2\hfill \\ \hfill & =C{n}^{4H-2}{h}^{2H}{\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n\left\{{(Y_{t_i}^H-Y_{t_{i-1}}^H)}^2-2\mu h(Y_{t_i}^H-Y_{t_{i-1}}^H)+{\left(\mu h\right)}^2\right\}\stackrel{a.s}{⟶}0\hfill \end{array}$$

for all $H\in (0,{\displaystyle \frac{1}{2}})$, as $n\to \infty$. This gives statement (1).

When ${\displaystyle \frac{1}{2}}<H<1$ and $0<\gamma <{\displaystyle \frac{1}{2H-1}}$, by Lemma 7, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(\widehat{t_i}\right)}^2& ={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left({\left(t_i\right)}^{2H}-{\left(t_{i-1}\right)}^{2H}\right)}^2={\displaystyle \frac{1}{n}}{h}^{4H}{K}_n\left(H\right)=h^{4H}O\left(n^{4H-2}\right)⟶0\hfill \end{array}$$

and

$${\displaystyle \frac{h^{1-2H}}{n}}\sum _{i=1}^n(Y_{t_i}^H-Y_{t_{i-1}}^H)={\displaystyle \frac{h^{1-2H}}{n}}{Y}_{t_n}^H={\displaystyle \frac{h^{1-2H}}{n}}\sigma B_{t_n}^H+\mu h^{2-2H}-{\displaystyle \frac{1}{2}}{\sigma }^2{n}^{2H-1}h\stackrel{a.s}{⟶}0,$$

(15)

as n tends to infinity. It follows from the Lemma 2 that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{n^2{h}^{4H}}}& \sum _{i=1}^n{\left({\widehat{t}}_i\right)}^2·\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-\mu h\right)}^2\hfill \\ \hfill & =C{n}^{4H-2}{h}^{2H}{\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n\left\{{(Y_{t_i}^H-Y_{t_{i-1}}^H)}^2-2\mu h(Y_{t_i}^H-Y_{t_{i-1}}^H)+{\left(\mu h\right)}^2\right\}\hfill \\ \hfill & =C{n}^{4H-2}{h}^{2H}\left\{N_n\left(Y^H\right)-2\mu h^{1-2H}{\displaystyle \frac{Y_{t_n}^H}{n}}+{\mu }^2{h}^{2-2H}\right\}\stackrel{a.s}{⟶}0,\hfill \end{array}$$

as n tends to infinity, if ${\displaystyle \frac{1}{2}}<H<1$ and $0<\gamma <{\displaystyle \frac{1-H}{2H-1}}$. This gives statement (2). □

 Proof of Lemma 4. 

By Lemma 7, we see that

$${\rho }_n=\sum _{i=1}^n{\left({\widehat{t}}_i\right)}^2-n^{4H-1}{h}^{4H}=h^{4H}\left(K_H\left(n\right)-n^{4H-1}\right)=\left\{\begin{array}{cc}O\left(h^{4H}\right),\hfill & \mathrm{if}0<H\le {\displaystyle \frac{1}{4}},\hfill \\ hO\left(logn\right),\hfill & \mathrm{if}H={\displaystyle \frac{1}{4}},\hfill \\ h^{4H}O\left(n^{4H-1}\right),\hfill & \mathrm{if}{\displaystyle \frac{1}{4}}<H<{\displaystyle \frac{1}{2}},\hfill \end{array}\right.$$

as $n\to \infty$. Let $H\in (0,{\displaystyle \frac{1}{4}})$. This yields

$$\begin{array}{cc}\hfill n^{-{\displaystyle \frac{3}{2}}}{Y}_{t_n}& =n^{-{\displaystyle \frac{3}{2}}}\sigma B_{t_n}^H+\mu h{n}^{-{\displaystyle \frac{1}{2}}}-{\displaystyle \frac{1}{2}}{\sigma }^2{h}^{2H}{n}^{2H-{\displaystyle \frac{3}{2}}}\stackrel{a.s}{⟶}0(n\to \infty )\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill n^{2H-2}{Y}_{t_n}& =n^{2H-2}\sigma B_{t_n}^H+\mu h{n}^{2H-1}-{\displaystyle \frac{1}{2}}{\sigma }^2{h}^{2H}{n}^{4H-2}\stackrel{a.s}{⟶}0\left(n\to \infty \right),\hfill \end{array}$$

and we can obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{n^2{h}^{4H}}}{\rho }_n\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2\stackrel{a.s}{\sim }C{n}^{-2}\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2\hfill \\ \hfill & =C{n}^{-1}{h}^{-2H}{N}_n\left(Y^H\right)-C{n}^{-3}{\left(Y_{t_n}\right)}^2\stackrel{a.s}{⟶}0(n\to \infty ).\hfill \end{array}$$

Similarly, when $H={\displaystyle \frac{1}{4}}$,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{n^2{h}^{4H}}}{\rho }_n\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2\stackrel{a.s}{\sim }C{n}^{-2}\left(logn\right)\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2\hfill \\ \hfill & =C\left(logn\right)n^{-1}{h}^{-2H}{N}_n\left(Y^H\right)-C{n}^{-3}\left(logn\right){\left(Y_{t_n}\right)}^2\stackrel{a.s}{⟶}0(n\to \infty ),\hfill \end{array}$$

and when $H\in ({\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}})$,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{n^2{h}^{4H}}}{\rho }_n\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2\stackrel{a.s}{\sim }C{n}^{4H-3}\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2\hfill \\ \hfill & =C{n}^{4H-2}{h}^{2H}{N}_n\left(Y^H\right)-C{n}^{4H-4}{\left(Y_{t_n}\right)}^2\stackrel{a.s}{⟶}0(n\to \infty ).\hfill \end{array}$$

This gives statement (1).

Now, let ${\displaystyle \frac{1}{2}}<H<1$ and $0<\gamma <{\displaystyle \frac{1-H}{2H-1}}$. Lemma 7 implies that

$${\rho }_n=\sum _{i=1}^n{\left({\widehat{t}}_i\right)}^2-n^{4H-1}{h}^{4H}=h^{4H}\left(K_n\left(H\right)-n^{4H-1}\right)=h^{4H}O\left(n^{4H-1}\right)\left(n\to \infty \right).$$

Noting that

$$\begin{array}{cc}\hfill n^{2H-2}{Y}_{t_n}& =n^{2H-2}\sigma B_{t_n}^H+\mu h{n}^{2H-1}-{\displaystyle \frac{1}{2}}{\sigma }^2{h}^{2H}{n}^{4H-2}\stackrel{a.s}{⟶}0\left(n\to \infty \right),\hfill \end{array}$$

we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{n^2{h}^{4H}}}{\rho }_n\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2\stackrel{a.s}{⟶}0\left(n\to \infty \right),\end{array}$$

and statement (2) follows. □

4. Asymptotic Normality of Estimator ${\widehat{\mathsf{\theta }}}_{\mathbf{n}}$

In this section, we examine the asymptotic distribution of ${\widehat{\theta }}_n$. We keep the notations from Section 3, and denote by $\stackrel{d}{\to }$ and $\stackrel{P}{\to }$ the convergence in distribution and probability, as n tends to infinity, respectively. From the structure of estimator ${\widehat{\theta }}_n$, one can find its asymptotic distribution depends on the asymptotic distribution of $\{N_n\left(Y^H\right),n\ge 0\}$. By the definition of $N_n\left(Y^H\right)$, we can check that

$$N_n\left(Y^H\right)-{\sigma }^2={\sigma }^2\left(M_n\left(B^H\right)-1\right)+{\mathsf{\Psi }}_H\left(n\right)+2\sigma {\mathsf{\Phi }}_H\left(n\right)$$

(16)

for $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},1)$, where ${\mathsf{\Psi }}_H\left(n\right)$ is given in Lemma 8 and

$${\mathsf{\Phi }}_H\left(n\right)={\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n\left[f\left(t_i\right)-f\left(t_{i-1}\right)\right]\left(B_{t_i}^H-B_{t_{i-1}}^H\right)$$

with $f\left(t\right)=\mu t-{\displaystyle \frac{1}{2}}{\sigma }^2{t}^{2H}$. From the proof later given, we find that the two terms $\left(M_n\left(B^H\right)-1\right)$ and ${\mathsf{\Phi }}_H\left(n\right)$ admit same asymptotic velocity under some suitable assumptions of $\gamma$. However, when ${\displaystyle \frac{3}{4}}\le H<1$ and conditions (C1) and (C2) hold, we know that (see proof of Lemma 9 in the following)

$$n^{4-4H}E\left[{\mathsf{\Phi }}_H{\left(n\right)}^2\right]=O{\left(nh\right)}^{2H}+O\left(nh\right)+O{\left(nh\right)}^{2-2H}⟶\infty$$

(17)

almost surely, as $n\to \infty$. But $n^{2-2H}\left(M_n\left(B^H\right)-1\right)$ converges in $L^2$ for ${\displaystyle \frac{3}{4}}<H<1$, and $\sqrt{{\displaystyle \frac{n}{logn}}}\left(M_n\left(B^H\right)-1\right)$ converges in distribution for $H={\displaystyle \frac{3}{4}}$. This indicates that ${\mathsf{\Phi }}_H\left(n\right)$ and $M_n\left(B^H\right)-1$ do not have the same asymptotic velocity for all $\gamma >0$, which means that such models have inflection points when $H=3/4$. The reason for this situation is that $nh$ tends to infinity. If we assume that $nh$ tends to infinity logarithmically, the scenario is different. The following lemma provides the asymptotic normality of $N_n\left(Y^H\right)$, and its proof is given at the end of this section.

 Lemma 9. 

Let $N_n\left(Y^H\right)$ be defined in Lemma 2, and let conditions $\left(C1\right)$ and $\left(C2\right)$ hold.

 (1) 

When $0<H<{\displaystyle \frac{1}{2}}$, we have

$$\begin{array}{c}\hfill \sqrt{n}\left(N_n\left(Y^H\right)-{\sigma }^2\right)\stackrel{d}{⟶}\left\{\begin{array}{cc}N\left(0,{\sigma }^4\left(2+{\lambda }_H\right)\right),\hfill & 0<\gamma <3-4H,\hfill \\ N\left({\mu }^2,{\sigma }^4\left(2+{\lambda }_H\right)\right),\hfill & \gamma =3-4H,\hfill \end{array}\right.\end{array}$$

where $N(a,b^2)$ denotes the normal random variable with mean a and variance $b^2$, and

$${\lambda }_H=\sum _{n=1}^{\infty }{\left({(n+1)}^{2H}+{(n-1)}^{2H}-2n^{2H}\right)}^2.$$

 (2) 

When ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$, we obtain

$$\begin{array}{c}\hfill \sqrt{n}\left(N_n\left(Y^H\right)-{\sigma }^2\right)\stackrel{d}{⟶}\left\{\begin{array}{cc}N\left(0,{\sigma }^4\left(2+{\lambda }_H\right)\right),\hfill & 0<\gamma <{\displaystyle \frac{3-4H}{8H-3}},\hfill \\ N\left({\displaystyle \frac{{\sigma }^4{H}^2}{4H-1}},{\sigma }^4\left(2+{\lambda }_H\right)\right),\hfill & \gamma ={\displaystyle \frac{3-4H}{8H-3}}.\hfill \end{array}\right.\end{array}$$

 (3) 

When ${\displaystyle \frac{3}{4}}\le H<1$, we have

$$\begin{array}{cc}\hfill & N_n(Y^H;1):={\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}\left(N_n\left(Y^H\right)-{\sigma }^2\right)\stackrel{a.s}{⟶}{\displaystyle \frac{H^2}{4H-1}}{\sigma }^4\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & N_n(Y^H;2):={\left(nh\right)}^{2H-1}\left(N_n(Y^H;1)-{\displaystyle \frac{H^2{\sigma }^4}{4H-1}}\right)\stackrel{a.s}{⟶}-\mu {\sigma }^2\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & {\left(nh\right)}^{1-H}\left(N_n(Y^H;2)+\mu {\sigma }^2\right)\stackrel{d}{⟶}N\left(0,4\beta \left(H\right){\sigma }^6\right),\hfill \end{array}$$

(20)

where $\beta \left(H\right)={\displaystyle \frac{2H-1}{3H-1}}\mathbb{B}(2H,2H-1)H^3$.

4.1. The Asymptotic Distribution of ${\widehat{\theta }}_n$ When $\mu$ Is Known

In this subsection, we obtain the asymptotic distribution of ${\widehat{\theta }}_n$, provided $\mu$ is known. By (8), Lemma 3, and the fact that $\sqrt{1+x}-1={\displaystyle \frac{1}{2}}x+O\left(x^2\right)$$(x\to 0)$, for all $n\ge 1$, we get

$$\begin{array}{cc}\hfill {\widehat{\theta }}_n-{\sigma }^2& \stackrel{a.s}{=}\left({\displaystyle \frac{1}{n{h}^{2H}}}{\displaystyle \sum _{i=1}^n}{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-\mu h\right)}^2-{\sigma }^2\right)+{\displaystyle \frac{2n{h}^{2H}}{{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2}}O\left({\Delta }_n{\left(H\right)}^2\right)\hfill \\ \hfill & =\left(N_n\left(Y^H\right)-{\sigma }^2\right)-{\displaystyle \frac{2\mu }{n{h}^{2H-1}}}{Y}_{t_n}^H+{\mu }^2{h}^{2-2H}+{\displaystyle \frac{2n{h}^{2H}}{{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2}}O\left({\Delta }_n{\left(H\right)}^2\right)\hfill \end{array}$$

(21)

with $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}})$.

 Lemma 10. 

Let the condition $\left(C1\right)$ hold, $0<H<{\displaystyle \frac{3}{4}}$, and denote

$$T_n\left(H\right):={\displaystyle \frac{n{h}^{2H}}{{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2}}{\Delta }_n{\left(H\right)}^2.$$

 (1) 

For $0<H\le {\displaystyle \frac{3}{8}}$, we have $T_n\left(H\right)\sqrt{n}\stackrel{a.s.}{\to }0$ as $n\to \infty$.

 (2) 

For ${\displaystyle \frac{3}{8}}<H<{\displaystyle \frac{3}{4}}$, we have $T_n\left(H\right)\sqrt{n}\stackrel{a.s.}{\to }0$, as $n\to \infty$, provided that condition $\left(C2\right)$ holds with $0<\gamma <{\displaystyle \frac{3-4H}{8H-3}}$.

 (3) 

For ${\displaystyle \frac{3}{4}}\le H<1$, we have

$$\begin{array}{c}\hfill T_n\left(H\right)·{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}\stackrel{a.s.}{⟶}{\displaystyle \frac{4H^2}{4H-1}}{\sigma }^4\end{array}$$

as n tends to infinity, provided that condition $\left(C2\right)$ holds with $0<\gamma <{\displaystyle \frac{1-H}{2H-1}}$.

 Proof. 

Let $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},1)$. By Lemma 7 we have

$$\begin{array}{cc}\hfill T_n\left(H\right)& ={\displaystyle \frac{K_n\left(H\right)}{n^3{h}^{2H}}}{\left(\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H\right)}^2-2\mu h{Y}_{t_n}^H+{\mu }^2{h}^{2}n\right)}^2\hfill \\ \hfill & \stackrel{a.s}{\sim }{\displaystyle \frac{4H^2{n}^{4H-4}}{(4H-1)h^{2H}}}{\left(\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H\right)}^2-2\mu h{Y}_{t_n}^H+{\mu }^2{h}^{2}n\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{4H^2}{(4H-1)}}{h}^{2H}{n}^{4H-2}{\left(N_n\left(Y^H\right)-2\mu h^{1-2H}{\displaystyle \frac{1}{n}}{Y}_{t_n}^H+{\mu }^2{h}^{2-2H}\right)}^2\hfill \end{array}$$

for all $n\ge 2$. Clearly, $h^{2H}{n}^{4H-{\displaystyle \frac{3}{2}}}\to 0$ for $0<H\le {\displaystyle \frac{3}{8}}$ and $h^{2H}{n}^{4H-{\displaystyle \frac{3}{2}}}=O\left(h^{{\displaystyle \frac{3}{2}}-2H-\gamma (4H-{\displaystyle \frac{3}{2}})}\right)\to 0$ for ${\displaystyle \frac{3}{8}}<H<{\displaystyle \frac{3}{4}}$ if $0<\gamma <{\displaystyle \frac{3-4H}{8H-3}}$. It follows from (15) that

$$\begin{array}{cc}\hfill \sqrt{n}{T}_n\left(H\right)& \stackrel{a.s}{\sim }{\displaystyle \frac{4H^2}{(4H-1)}}{h}^{2H}{n}^{4H-{\displaystyle \frac{3}{2}}}{\left(N_n\left(Y^H\right)-2\mu h^{1-2H}{\displaystyle \frac{1}{n}}{Y}_{t_n}^H+{\mu }^2{h}^{2-2H}\right)}^2⟶0,\hfill \end{array}$$

as n tends to infinity under the conditions of statements (1) and (2).

We now verify statement (3). Let ${\displaystyle \frac{3}{4}}\le H<1$. It follows from Lemma 2 that

$$T_n\left(H\right)·{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}={\displaystyle \frac{4H^2}{(4H-1)}}{\left(N_n\left(Y^H\right)-2\mu h^{1-2H}{\displaystyle \frac{1}{n}}{Y}_{t_n}^H+{\mu }^2{h}^{2-2H}\right)}^2\stackrel{a.s}{⟶}{\displaystyle \frac{4H^2}{4H-1}}{\sigma }^4$$

(22)

as n tends to infinity, provided that $0<\gamma <{\displaystyle \frac{1-H}{2H-1}}$ since

$$\begin{array}{cc}\hfill 2\mu h^{1-2H}{\displaystyle \frac{1}{n}}{Y}_{t_n}^H& =2\mu {\displaystyle \frac{1}{n}}{h}^{1-2H}\left(\sigma B_{t_n}^H+\mu nh-{\displaystyle \frac{1}{2}}{\sigma }^2{\left(nh\right)}^{2H}\right)\hfill \\ \hfill & =2\mu \sigma {\displaystyle \frac{h^{1-H}}{n^{1-H}}}·{\displaystyle \frac{B_{t_n}^H}{{\left(nh\right)}^H}}+2{\mu }^2{h}^{2-2H}-\mu {\sigma }^2{n}^{2H-1}h\stackrel{a.s}{\sim }-\mu {\sigma }^2{n}^{2H-1}h\hfill \end{array}$$

as n tends to infinity. □

 Theorem 3. 

Let μ be known and let conditions (C1) and (C2) hold

 (1) 

Let $0<H<{\displaystyle \frac{1}{2}}$ and $0<\gamma \le 3-4H$, then, as $n\to \infty$, we have

$$\begin{array}{c}\hfill \sqrt{n}\left({\widehat{\theta }}_n-{\sigma }^2\right)\stackrel{d}{⟶}N\left(0,(2+{\lambda }_H){\sigma }^4\right).\end{array}$$

 (2) 

Let ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$, then, as $n\to \infty$, we have

$$\begin{array}{c}\hfill \sqrt{n}\left({\widehat{\theta }}_n-{\sigma }^2\right)\stackrel{d}{⟶}\left\{\begin{array}{cc}N\left(0,(2+{\lambda }_H){\sigma }^4\right),\hfill & 0<\gamma <{\displaystyle \frac{3-4H}{8H-3}},\hfill \\ N\left({\displaystyle \frac{{\sigma }^4{H}^2}{4H-1}},(2+{\lambda }_H){\sigma }^4\right),\hfill & \gamma ={\displaystyle \frac{3-4H}{8H-3}}.\hfill \end{array}\right.\end{array}$$

 Proof. 

Let $0<H<{\displaystyle \frac{3}{4}}$. Then, we have

$$\underset{n\to \infty }{lim}\sqrt{n}{h}^{2-2H}=\left\{\begin{array}{cc}0,\hfill & 0<\gamma <3-4H,\hfill \\ 1,\hfill & \gamma =3-4H,\hfill \\ +\infty ,\hfill & \gamma >3-4H.\hfill \end{array}\right.$$

(23)

Moreover, we have $\underset{n\to \infty }{lim}{n}^{2H-{\displaystyle \frac{1}{2}}}h=0$ for $0<H\le {\displaystyle \frac{1}{4}}$, and

$$\underset{n\to \infty }{lim}{n}^{2H-{\displaystyle \frac{1}{2}}}h=\left\{\begin{array}{cc}0,\hfill & 0<\gamma <{\displaystyle \frac{3-4H}{4H-1}},\hfill \\ 1,\hfill & \gamma ={\displaystyle \frac{3-4H}{4H-1}},\hfill \\ +\infty ,\hfill & \gamma >{\displaystyle \frac{3-4H}{4H-1}}\hfill \end{array}\right.$$

(24)

for all ${\displaystyle \frac{1}{4}}<H<{\displaystyle \frac{3}{4}}$ and $H\ne {\displaystyle \frac{1}{2}}$.

For statement (1), we have

$${\displaystyle \frac{1}{\sqrt{n}}}{h}^{1-2H}{B}_{t_n}^H=n^{H-{\displaystyle \frac{1}{2}}}{h}^{1-H}{\displaystyle \frac{B_{t_n}^H}{{\left(nh\right)}^H}}\stackrel{a.s}{⟶}0(n\to \infty )$$

for all $0<H<{\displaystyle \frac{1}{2}}$, and by (23) and (24), we also have

$$\begin{array}{cc}\hfill {\displaystyle \frac{h^{1-2H}}{\sqrt{n}}}{Y}_{t_n}^H& ={\displaystyle \frac{\sigma }{\sqrt{n}}}{h}^{1-2H}{B}_{t_n}^H+\mu \sqrt{n}{h}^{2-2H}-{\displaystyle \frac{{\sigma }^2}{2}}h{n}^{2H-{\displaystyle \frac{1}{2}}}\stackrel{a.s}{⟶}\left\{\begin{array}{cc}0,\hfill & 0<\gamma <3-4H,\hfill \\ \mu ,\hfill & \gamma =3-4H,\hfill \\ +\infty ,\hfill & \gamma >3-4H\hfill \end{array}\right.(n\to \infty )\hfill \end{array}$$

since $3-4H<{\displaystyle \frac{3-4H}{4H-1}}$ for ${\displaystyle \frac{1}{4}}<H<{\displaystyle \frac{1}{2}}$. Combining these with (21), (23), Lemma 9, Lemma 10, and Slutsky’s theorem, we obtain statement (1).

For statement (2), we have

$${\displaystyle \frac{1}{\sqrt{n}}}{h}^{1-2H}{B}_{t_n}^H=n^{H-{\displaystyle \frac{1}{2}}}{h}^{1-H}{\displaystyle \frac{B_{t_n}^H}{{\left(nh\right)}^H}}\stackrel{a.s}{=}O\left(h^{{\displaystyle \frac{3}{2}}-2H-\gamma (H-{\displaystyle \frac{1}{2}})}\right)\stackrel{a.s}{⟶}0(n\to \infty )$$

for all ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$ if $0<\gamma <{\displaystyle \frac{3-4H}{2H-1}}$. It follows from (23) and (24) that

$$\begin{array}{cc}\hfill {\displaystyle \frac{h^{1-2H}}{\sqrt{n}}}{Y}_{t_n}^H& ={\displaystyle \frac{\sigma }{\sqrt{n}}}{h}^{1-2H}{B}_{t_n}^H+\mu \sqrt{n}{h}^{2-2H}-{\displaystyle \frac{{\sigma }^2}{2}}h{n}^{2H-{\displaystyle \frac{1}{2}}}\stackrel{a.s}{⟶}\left\{\begin{array}{cc}0,\hfill & 0<\gamma <{\displaystyle \frac{3-4H}{4H-1}},\hfill \\ 1,\hfill & \gamma ={\displaystyle \frac{3-4H}{4H-1}},\hfill \\ +\infty ,\hfill & \gamma >{\displaystyle \frac{3-4H}{4H-1}}\hfill \end{array}\right.(n\to \infty )\hfill \end{array}$$

for all ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$ since ${\displaystyle \frac{3-4H}{4H-1}}<3-4H<{\displaystyle \frac{3-4H}{2H-1}}$. Combining these with (21), (23), Lemma 9, Lemma 10, and Slutsky’s theorem, we obtain statement (2) because ${\displaystyle \frac{3-4H}{8H-3}}<{\displaystyle \frac{3-4H}{4H-1}}$. □

 Theorem 4. 

Let μ be known and ${\displaystyle \frac{3}{4}}\le H<1$. If conditions (C1) and (C2) hold with $0<\gamma <{\displaystyle \frac{2-2H}{5H-2}}$. We then have

$${\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left({\widehat{\theta }}_n-{\sigma }^2\right)\stackrel{d}{⟶}N\left(0,4\beta \left(H\right){\sigma }^6\right)$$

(25)

as n tends to infinity, where $\beta \left(H\right)$ is given in Lemma 9.

 Proof. 

By (8), Lemma 3, and the fact $\sqrt{1+x}-1={\displaystyle \frac{1}{2}}x-{\displaystyle \frac{1}{8}}{x}^2+O\left(x^3\right)$$(x\to 0)$, for all $n\ge 1$, we get

$$\begin{array}{cc}\hfill {\widehat{\theta }}_n-{\sigma }^2=& (N_n\left(Y^H\right)-{\sigma }^2)-2\mu h^{1-2H}{n}^{-1}{Y}_{t_n}^H+{\mu }^2{h}^{2-2H}-{\displaystyle \frac{1}{4}}{T}_n\left(H\right)+{\displaystyle \frac{2n{h}^{2H}}{{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2}}·O\left({\Delta }_n{\left(H\right)}^3\right)\hfill \\ \hfill & =(M_n\left(B^H\right)-1){\sigma }^2+2\sigma {\mathsf{\Phi }}_H\left(n\right)+{\displaystyle \frac{H^2}{4H-1}}{\sigma }^4{h}^{2H}{n}^{4H-2}-2\mu \sigma n^{H-1}{h}^{1-H}{\displaystyle \frac{B_{tn}^H}{{\left(nh\right)}^H}}\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}{T}_n\left(H\right)+{\displaystyle \frac{2n{h}^{2H}}{{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2}}·O\left({\Delta }_n{\left(H\right)}^3\right).\hfill \end{array}$$

(26)

On the other hand, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left({\displaystyle \frac{1}{4}}{T}_n\left(H\right)-{\displaystyle \frac{H^2}{4H-1}}{\sigma }^4{h}^{2H}{n}^{4H-2}\right)& ={\displaystyle \frac{H^2}{4H-1}}\{\left(N_n\left(Y^H\right)-{\sigma }^2\right){\left(nh\right)}^H\hfill \\ \hfill & -2\mu n^{H-1}{h}^{1-H}{Y}_{t_n}^H+{\mu }^2{n}^H{h}^{2-H}\}\left\{N_n\left(Y^H\right)-2\mu h^{1-2H}{\displaystyle \frac{1}{n}}{Y}_{t_n}^H+{\mu }^2{h}^{2-2H}+{\sigma }^2\right\}\hfill \\ \hfill & \stackrel{a.s}{⟶}0(n\to \infty )\hfill \end{array}$$

(27)

when $0<\gamma <{\displaystyle \frac{2-2H}{5H-2}}$. Therefore, the asymptotic normality follows from (26), (27), Lemma 9, and Slutsky’s theorem, and we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left({\widehat{\theta }}_n-{\sigma }^2\right)& ={\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left((M_n\left(B^H\right)-1){\sigma }^2\right)+2\sigma {\mathsf{\Phi }}_H\left(n\right){\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}-2\mu \sigma n^{2H-1}h{\displaystyle \frac{B_{tn}^H}{{\left(nh\right)}^H}}\hfill \\ \hfill & +{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left({\displaystyle \frac{1}{4}}{T}_n\left(H\right)-{\displaystyle \frac{H^2}{4H-1}}{\sigma }^4{h}^{2H}{n}^{4H-2}\right)+{\displaystyle \frac{2n{h}^{2H}}{{\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2}}·O\left({\Delta }_n{\left(H\right)}^3\right){\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\hfill \\ & \stackrel{d}{⟶}N\left(0,4\beta \left(H\right){\sigma }^6\right)\hfill \end{array}$$

(28)

when $0<\gamma <{\displaystyle \frac{2-2H}{5H-2}}$ and as n tends to infinity. □

4.2. The Asymptotic Distribution of ${\widehat{\theta }}_n$ When $\mu$ Is Unknown

In this subsection, we consider the asymptotic distribution of estimator ${\widehat{\theta }}_n$ when $\mu$ is unknown. Based on (10), Lemma 4, and the fact that $\sqrt{1+x}-1={\displaystyle \frac{1}{2}}x+O\left(x^2\right),(x\to 0)$, we obtain the following result

$$\begin{array}{cc}\hfill \sqrt{n}\left({\widehat{\theta }}_n-{\sigma }^2\right)& \stackrel{a.s}{=}\sqrt{n}\left({\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}^H\right)}^2-{\sigma }^2\right)+{\displaystyle \frac{2n{h}^{2H}}{{\rho }_n}}O\left({\tilde{\Delta }}_n{\left(H\right)}^2\right)·\sqrt{n}\hfill \\ \hfill & =\sqrt{n}\left(N_n\left(Y^H\right)-{\sigma }^2\right)-{\left(n^{-{\displaystyle \frac{3}{4}}}{h}^{-H}{Y}_{t_n}^H\right)}^2+{\displaystyle \frac{2n{h}^{2H}}{{\rho }_n}}O\left({\tilde{\Delta }}_n{\left(H\right)}^2\right)·\sqrt{n}\hfill \end{array}$$

(29)

with $0<H<{\displaystyle \frac{3}{4}}$, where ${\rho }_n={\displaystyle \sum _{i=1}^n}{\left({\widehat{t}}_i\right)}^2-n^{4H-1}{h}^{4H}$. As a corollary of Lemma 4, the following lemma provides an estimate for the remainder term

$${\tilde{T}}_n\left(H\right):={\displaystyle \frac{n{h}^{2H}}{{\rho }_n}}{\tilde{\Delta }}_n{\left(H\right)}^2.$$

 Lemma 11. 

Let conditions $\left(C1\right)$ and $\left(C2\right)$ hold.

 (1) 

For $0<H\le {\displaystyle \frac{3}{8}}$, we have $\underset{n\to \infty }{lim}{\tilde{T}}_n\left(H\right)·\sqrt{n}\stackrel{a.s.}{=}0.$

 (2) 

For ${\displaystyle \frac{3}{8}}<H<{\displaystyle \frac{3}{4}}$, we have $\underset{n\to \infty }{lim}{\tilde{T}}_n\left(H\right)·\sqrt{n}\stackrel{a.s.}{=}0$, provided $0<\gamma <{\displaystyle \frac{3-4H}{8H-3}}$.

 Proof. 

By Lemma 7 and the proof of Lemma 4, we get

$$\begin{array}{cc}\hfill {\tilde{T}}_n\left(H\right)\sqrt{n}& ={\displaystyle \frac{n{h}^{2H}}{{\rho }_n}}{\tilde{\Delta }}_n{\left(H\right)}^2·\sqrt{n}={\displaystyle \frac{{\rho }_n\sqrt{n}}{n^3{h}^{6H}}}{\left(\sum _{i=1}^n{\left(Y_{t_i}^H-Y_{t_{i-1}}^H-{\displaystyle \frac{1}{n}}{Y}_{t_n}\right)}^2\right)}^2\hfill \\ \hfill & \stackrel{a.s}{\sim }C{\displaystyle \frac{\sqrt{n}}{n^{4-4H}{h}^{2H}}}{\left(n{h}^{2H}{N}_n\left(Y^H\right)-{\displaystyle \frac{1}{n}}{\left(Y_{t_n}\right)}^2\right)}^2\hfill \\ & =C{\left(h^H{n}^{2H-{\displaystyle \frac{3}{4}}}{N}_n\left(Y^H\right)-h^{-H}{n}^{2H-{\displaystyle \frac{11}{4}}}{\left(Y_{t_n}\right)}^2\right)}^2\hfill \end{array}$$

(30)

for all $n\ge 1$. Clearly, $\underset{n\to \infty }{lim}{n}^{2H-{\displaystyle \frac{3}{4}}}{h}^H=0$ and $\underset{n\to \infty }{lim}{n}^{H-{\displaystyle \frac{3}{8}}}{h}^{1-{\displaystyle \frac{1}{2}}H}=0$ for all $0<H\le {\displaystyle \frac{3}{8}}$. Moreover, when ${\displaystyle \frac{3}{8}}<H<{\displaystyle \frac{3}{4}}$, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{n}^{2H-{\displaystyle \frac{3}{4}}}{h}^H=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<\gamma <{\displaystyle \frac{3-4H}{8H-3}},\hfill \\ 1,\hfill & \mathrm{if}\gamma ={\displaystyle \frac{3-4H}{8H-3}},\hfill \end{array}\right.\mathrm{and}\underset{n\to \infty }{lim}{n}^{H-{\displaystyle \frac{3}{8}}}{h}^{1-{\displaystyle \frac{1}{2}}H}=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<\gamma <{\displaystyle \frac{11-12H}{8H-3}},\hfill \\ 1,\hfill & \mathrm{if}\gamma ={\displaystyle \frac{11-12H}{8H-3}}.\hfill \end{array}\right.\end{array}$$

Similarly, $\underset{n\to \infty }{lim}{n}^{3H-{\displaystyle \frac{11}{8}}}{h}^{{\displaystyle \frac{3}{2}}H}=0$ for all $0<H\le {\displaystyle \frac{11}{24}}$ and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{n}^{3H-{\displaystyle \frac{11}{8}}}{h}^{{\displaystyle \frac{3}{2}}H}=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<\gamma <{\displaystyle \frac{11-12H}{24H-11}},\hfill \\ 1,\hfill & \mathrm{if}\gamma ={\displaystyle \frac{11-12H}{24H-11}}\hfill \end{array}\right.\end{array}$$

for all ${\displaystyle \frac{11}{24}}<H<{\displaystyle \frac{11}{12}}$. Noting that ${\displaystyle \frac{3-4H}{8H-3}}<{\displaystyle \frac{11-12H}{24H-11}}$ and ${\displaystyle \frac{3-4H}{8H-3}}<{\displaystyle \frac{11-12H}{8H-3}}$ for all ${\displaystyle \frac{3}{8}}<H<{\displaystyle \frac{3}{4}}$, we obtain that

$$\begin{array}{c}\hfill h^{-{\displaystyle \frac{1}{2}}H}{n}^{H-{\displaystyle \frac{11}{8}}}{Y}_{t_n}=\mu n^{H-{\displaystyle \frac{3}{8}}}{h}^{1-{\displaystyle \frac{1}{2}}H}-{\displaystyle \frac{1}{2}}{\sigma }^2{n}^{3H-{\displaystyle \frac{11}{8}}}{h}^{{\displaystyle \frac{3}{2}}H}+n^{H-{\displaystyle \frac{11}{8}}}{h}^{-{\displaystyle \frac{1}{2}}H}\sigma B_n^H,\end{array}$$

converges almost surely to 0 for $0<H\le {\displaystyle \frac{3}{8}}$ and that it converges almost surely to 0 for ${\displaystyle \frac{3}{8}}<H<{\displaystyle \frac{3}{4}}$ provided $0<\gamma <{\displaystyle \frac{3-4H}{8H-3}}$. Thus, the lemma follows from Lemma 4 and (30). □

 Theorem 5. 

Let μ be unknown and let conditions (C1) and (C2) hold.

 (1) 

For $0<H<{\displaystyle \frac{1}{2}}$, if $0<\gamma \le 3-4H$, we have

$$\sqrt{n}\left({\widehat{\theta }}_n-{\sigma }^2\right)\stackrel{d}{⟶}N\left(0,(2+{\lambda }_H){\sigma }^4\right).$$

 (2) 

For ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$, we have

$$\begin{array}{c}\hfill \sqrt{n}\left({\widehat{\theta }}_n-{\sigma }^2\right)\stackrel{d}{⟶}\left\{\begin{array}{cc}N\left(0,(2+{\lambda }_H){\sigma }^4\right),\hfill & 0<\gamma <{\displaystyle \frac{3-4H}{8H-3}},\hfill \\ N\left({\displaystyle \frac{{\sigma }^4{H}^2}{4H-1}}+{\displaystyle \frac{1}{4}}{\sigma }^4,(2+{\lambda }_H){\sigma }^4\right),\hfill & \gamma ={\displaystyle \frac{3-4H}{8H-3}}.\hfill \end{array}\right.\end{array}$$

 Proof. 

Clearly, we have ${lim}_{n\to \infty }{n}^{2H-{\displaystyle \frac{3}{4}}}{h}^H=0$ for all $0<H\le {\displaystyle \frac{3}{8}}$ and

$$\underset{n\to \infty }{lim}{n}^{2H-{\displaystyle \frac{3}{4}}}{h}^H=\left\{\begin{array}{cc}0,\hfill & 0<\gamma <{\displaystyle \frac{3-4H}{8H-3}},\hfill \\ 1,\hfill & \gamma ={\displaystyle \frac{3-4H}{8H-3}},\hfill \\ \infty ,\hfill & \gamma >{\displaystyle \frac{3-4H}{8H-3}}.\hfill \end{array}\right.$$

for ${\displaystyle \frac{3}{8}}<H<{\displaystyle \frac{3}{4}}$, and moreover

$$\underset{n\to \infty }{lim}{n}^{{\displaystyle \frac{1}{4}}}{h}^{1-H}=\left\{\begin{array}{cc}0,\hfill & 0<\gamma <3-4H,\hfill \\ 1,\hfill & \gamma =3-4H,\hfill \\ \infty ,\hfill & \gamma >3-4H\hfill \end{array}\right.$$

for all $0<H<{\displaystyle \frac{3}{4}}$. It follows that

$$\begin{array}{c}\hfill n^{-{\displaystyle \frac{3}{4}}}{h}^{-H}{Y}_{t_n}^H=\sigma n^{H-{\displaystyle \frac{3}{4}}}{\displaystyle \frac{B_{t_n}^H}{{\left(nh\right)}^H}}+\mu n^{{\displaystyle \frac{1}{4}}}{h}^{1-H}-{\displaystyle \frac{1}{2}}{\sigma }^2{n}^{2H-{\displaystyle \frac{3}{4}}}{h}^H⟶\left\{\begin{array}{cc}0,\hfill & 0<\gamma <3-4H,\hfill \\ \mu ,\hfill & \gamma =3-4H\hfill \end{array}\right.\end{array}$$

for all ${\displaystyle \frac{3}{8}}<H<{\displaystyle \frac{1}{2}}$, and $n^{-{\displaystyle \frac{3}{4}}}{h}^{-H}{Y}_{t_n}^H\stackrel{a.s}{\to }0$ for all $0<H\le {\displaystyle \frac{3}{8}}$, and

$$\begin{array}{c}\hfill n^{-{\displaystyle \frac{3}{4}}}{h}^{-H}{Y}_{t_n}^H=\sigma {\displaystyle \frac{B_{t_n}^H}{n^{\frac{3}{4}}{h}^H}}+\mu n^{{\displaystyle \frac{1}{4}}}{h}^{1-H}-{\displaystyle \frac{1}{2}}{\sigma }^2{n}^{2H-{\displaystyle \frac{3}{4}}}{h}^H⟶\left\{\begin{array}{cc}0,\hfill & 0<\gamma <{\displaystyle \frac{3-4H}{8H-3}},\hfill \\ -{\displaystyle \frac{1}{2}}{\sigma }^2,\hfill & \gamma ={\displaystyle \frac{3-4H}{8H-3}}\hfill \end{array}\right.\end{array}$$

for all ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$, since ${\displaystyle \frac{3-4H}{8H-3}}>1$ for ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{3-4H}{8H-3}}<1$ for all ${\displaystyle \frac{3}{8}}<H<{\displaystyle \frac{1}{2}}$. Combining this with (29), Lemma 9, Lemma 11, and Slutsky’s theorem, we obtain the theorem. □

 Lemma 12. 

Let conditions $\left(C1\right)$ and $\left(C2\right)$ hold with $0<\gamma <{\displaystyle \frac{1-H}{2H-1}}$. For ${\displaystyle \frac{3}{4}}\le H<1$, we have

$${\tilde{T}}_n\left(H\right)·{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}\stackrel{a.s}{⟶}\left({\displaystyle \frac{4H^2}{4H-1}}-1\right){\sigma }^4(n\to \infty ).$$

Proof. 

Similar to the proof of Lemma 11, we get

$${\displaystyle \frac{1}{n{h}^H}}{Y}_{t_n}={\displaystyle \frac{\sigma }{n^{1-H}}}{\displaystyle \frac{B_{t_n}^H}{{\left(nh\right)}^H}}+\mu h^{1-H}-{\displaystyle \frac{1}{2}}{\sigma }^2{n}^{2H-1}{h}^H\stackrel{a.s}{⟶}\left\{\begin{array}{cc}0,\hfill & 0<\gamma <{\displaystyle \frac{1-H}{2H-1}},\hfill \\ -{\displaystyle \frac{1}{2}}{\sigma }^2,\hfill & \gamma ={\displaystyle \frac{1-H}{2H-1}},\hfill \\ \infty ,\hfill & \gamma >{\displaystyle \frac{1-H}{2H-1}},\hfill \end{array}\right.,$$

(31)

for all ${\displaystyle \frac{1}{2}}<H<1$. It follows from Lemma 7 and Lemma 2 that

$$\begin{array}{cc}\hfill {\tilde{T}}_n\left(H\right)·& {\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}\stackrel{a.s}{\sim }{n}^{-2}{h}^{-4H}\left({\displaystyle \frac{4H^2}{4H-1}}-1\right){\left(n{h}^{2H}{N}_n\left(Y^H\right)-{\displaystyle \frac{1}{n}}{\left(Y_{t_n}\right)}^2\right)}^2\hfill \\ & =\left({\displaystyle \frac{4H^2}{4H-1}}-1\right){\left(N_n\left(Y^H\right)-n^{-2}{h}^{-2H}{\left(Y_{t_n}\right)}^2\right)}^2\stackrel{a.s}{⟶}\left({\displaystyle \frac{4H^2}{4H-1}}-1\right){\sigma }^4,\hfill \end{array}$$

as n tends to infinity. □

 Theorem 6. 

Let ${\displaystyle \frac{3}{4}}\le H<1$ and μ be unknown. If conditions $\left(C1\right)$ and $\left(C2\right)$ hold with $0<\gamma <{\displaystyle \frac{2-2H}{5H-2}}$, we then have

$${\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left({\widehat{\theta }}_n-{\sigma }^2\right)\stackrel{d}{⟶}N\left(0,4\beta \left(H\right){\sigma }^6\right)$$

(32)

as n tends to infinity, where $\beta \left(H\right)$ is given in Lemma 9.

Proof. 

Let ${\displaystyle \frac{3}{4}}\le H<1$. By (10), Lemma 4, and the fact that $\sqrt{1+x}-1={\displaystyle \frac{1}{2}}x-{\displaystyle \frac{1}{8}}{x}^2+O\left(x^3\right)$$(x\to 0)$, we get

$$\begin{array}{cc}\hfill {\widehat{\theta }}_n-{\sigma }^2=& \left(N_n\left(Y^H\right)-{\sigma }^2\right)-h^{-2H}{n}^{-2}{\left(Y_{t_n}^H\right)}^2-{\displaystyle \frac{1}{4}}{\tilde{T}}_n\left(H\right)+{\displaystyle \frac{2n{h}^{2H}}{{\rho }_n}}O\left({\tilde{\Delta }}_n{\left(H\right)}^3\right)\hfill \\ \hfill & =(M_n\left(B^H\right)-1){\sigma }^2+2\sigma {\mathsf{\Phi }}_H\left(n\right)+\left({\displaystyle \frac{H^2}{4H-1}}-1\right){\sigma }^4{h}^{2H}{n}^{4H-2}-2\mu \sigma n^{H-1}{h}^{1-H}{\displaystyle \frac{B_{tn}^H}{{\left(nh\right)}^H}}\hfill \\ \hfill & -{\sigma }^2{n}^{2H-2}{\left({\displaystyle \frac{B_{t_n}^H}{{\left(nh\right)}^H}}\right)}^2+{\sigma }^3{n}^{3H-2}{h}^H{\displaystyle \frac{B_{tn}^H}{{\left(nh\right)}^H}}-{\displaystyle \frac{1}{4}}{\tilde{T}}_n\left(H\right)+{\displaystyle \frac{2n{h}^{2H}}{{\rho }_n}}O\left({\tilde{\Delta }}_n{\left(H\right)}^3\right).\hfill \end{array}$$

(33)

Similarly, we can obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left\{{\displaystyle \frac{1}{4}}{\tilde{T}}_n\left(H\right)-\left({\displaystyle \frac{4H^2}{4H-1}}-1\right)n^{4H-2}{h}^H{\sigma }^4\right\}& =\left({\displaystyle \frac{4H^2}{4H-1}}-1\right)\left\{N_n\left(Y^H\right)-n^{-2}{h}^{-2H}{\left(Y_{t_n}\right)}^2+{\sigma }^2\right\}\hfill \\ \hfill & \left\{\left(N_n\left(Y^H\right)-{\sigma }^2\right){\left(nh\right)}^H-n^{-2}{h}^{-2H}{\left(Y_{t_n}\right)}^2{\left(nh\right)}^H\right\}\stackrel{a.s}{⟶}0.\hfill \end{array}$$

(34)

Therefore, using Equations (33) and (34) and Lemma 9, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left({\widehat{\theta }}_n-{\sigma }^2\right)=& {\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left(M_n\left(B^H\right)-1\right){\sigma }^2+2\sigma {\mathsf{\Phi }}_H\left(n\right){\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}-2\mu \sigma {\left(nh\right)}^{1-2H}{\displaystyle \frac{B_{tn}^H}{{\left(nh\right)}^H}}\hfill \\ \hfill & +{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left\{{\displaystyle \frac{1}{4}}{\tilde{T}}_n\left(H\right)-\left({\displaystyle \frac{4H^2}{4H-1}}-1\right)n^{4H-2}{h}^H{\sigma }^4\right\}-{\sigma }^2{\left(nh\right)}^{-H}{\left({\displaystyle \frac{B_{t_n}^H}{{\left(nh\right)}^H}}\right)}^2\hfill \\ \hfill & +{\sigma }^3{h}^{2H}{\displaystyle \frac{B_{tn}^H}{{\left(nh\right)}^H}}+{\displaystyle \frac{2n{h}^{2H}}{{\rho }_n}}O\left({\tilde{\Delta }}_n{\left(H\right)}^3\right){\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\hfill \\ \hfill & \stackrel{d}{⟶}N\left(0,4\beta \left(H\right){\sigma }^6\right)\hfill \end{array}$$

(35)

when $0<\gamma <{\displaystyle \frac{2-2H}{5H-2}}$ and as n tends to infinity. □

4.3. Proofs of Lemmas in Section 4

In this subsection, we complete the proof of Lemma 9.

 Proposition 1. 

Let the conditions in Lemma 1 hold.

 (1) 

For $0<H<{\displaystyle \frac{3}{4}}$, we have

$$\sqrt{n}\left(M_n\left(B^H\right)-1\right)⟶N\left(0,2+{\lambda }_H\right)(n\to \infty ),$$

in distribution, where

$${\lambda }_H=\sum _{n=1}^{\infty }{\left({(n+1)}^{2H}+{(n-1)}^{2H}-2n^{2H}\right)}^2.$$

 (2) 

For $H={\displaystyle \frac{3}{4}}$, we have

$$\sqrt{{\displaystyle \frac{n}{logn}}}\left(M_n\left(B^H\right)-1\right)⟶N\left(0,{\displaystyle \frac{9}{4}}\right)(n\to \infty )$$

in distribution.

 (3) 

For ${\displaystyle \frac{3}{4}}<H<1$, we have

$$n^{2-2H}\left(M_n\left(B^H\right)-1\right)⟶{\displaystyle \frac{2H^2(2H-1)}{4H-3}}{\mathcal{R}}_H(n\to \infty )$$

in $L^2$, where ${\mathcal{R}}_H$ denotes a Rosenblatt random distribution with $E{\left({\mathcal{R}}_H\right)}^2=1$.

The lemma is an insignificant extension for some known results, and its proof is omitted (see, for examples, Theorem 5.4, Proposition 5.4, Theorem 5.5 in Tudor [20]). In fact, for $t_n=nh=T<\infty$, such convergence have been studied and can be found in Breuer and Major [23], Dobrushin and Major [24], Giraitis and Surgailis [25], Nourdin [26], Nourdin and Reveillac [27] and Tudor [20]. On the other hand, for more material on the Rosenblatt distribution and related process, refer to Tudor [20].

Proof of Lemma 9. 

Let $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}})$ be given. We have

$$\underset{n\to \infty }{lim}\sqrt{n}{h}^{2-2H}=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<\gamma <3-4H,\hfill \\ 1,\hfill & \mathrm{if}\gamma =3-4H.\hfill \end{array}\right.$$

(36)

We also have $\underset{n\to \infty }{lim}h{n}^{2H-{\displaystyle \frac{1}{2}}}=0$ for $0<H<{\displaystyle \frac{1}{4}}$ and

$$\underset{n\to \infty }{lim}h{n}^{2H-{\displaystyle \frac{1}{2}}}=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<\gamma <{\displaystyle \frac{3-4H}{4H-1}},\hfill \\ 1,\hfill & \mathrm{if}\gamma ={\displaystyle \frac{3-4H}{4H-1}}\hfill \end{array}\right.(n\to \infty )$$

(37)

for ${\displaystyle \frac{1}{4}}<H<{\displaystyle \frac{3}{4}}$. On the other hand, by Taylor’s expansion, we may prove

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{\sqrt{n}}}{h}^{2H}{K}_n\left(H\right)=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{\sqrt{n}}}{h}^{2H}\sum _{i=1}^n{i}^{4H}{\left(1-{(1-{\displaystyle \frac{1}{i}})}^{2H}\right)}^2={\displaystyle \frac{4H^2}{4H-1}}$$

for ${\displaystyle \frac{1}{4}}<H<{\displaystyle \frac{3}{4}}$ if $\gamma ={\displaystyle \frac{3-4H}{8H-3}}$. It follows from Lemma 7 that $\underset{n\to \infty }{lim}{\displaystyle \frac{1}{\sqrt{n}}}{h}^{2H}{K}_n\left(H\right)=0$ for $0<H\le {\displaystyle \frac{3}{8}}$ and

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{\sqrt{n}}}{h}^{2H}{K}_n\left(H\right)=\underset{n\to \infty }{lim}{n}^{-{\displaystyle \frac{3+\gamma }{2(1+\gamma )}}}{K}_n\left(H\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<\gamma <{\displaystyle \frac{3-4H}{8H-3}},\hfill \\ {\displaystyle \frac{4H^2}{4H-1}},\hfill & \mathrm{if}\gamma ={\displaystyle \frac{3-4H}{8H-3}}\hfill \end{array}\right.$$

(38)

for ${\displaystyle \frac{3}{8}}<H<{\displaystyle \frac{3}{4}}$. Combining the above three convergences and the proof of Lemma 8, we obtain that

$$\begin{array}{cc}\hfill \sqrt{n}{\mathsf{\Psi }}_H\left(n\right)& ={\mu }^2\sqrt{n}{h}^{2-2H}-{\sigma }^2\mu h{n}^{2H-{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{1}{4}}{\sigma }^4{h}^{2H}{\displaystyle \frac{1}{\sqrt{n}}}{K}_n\left(H\right)\hfill \\ \hfill & ⟶\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<\gamma <3-4H,\hfill \\ {\mu }^2,\hfill & \mathrm{if}\gamma =3-4H,\hfill \\ \infty ,\hfill & \mathrm{if}\gamma >3-4H\hfill \end{array}\right.(n\to \infty )\hfill \end{array}$$

(39)

for $0<H<{\displaystyle \frac{1}{2}}$ and

$$\begin{array}{cc}\hfill \sqrt{n}{\mathsf{\Psi }}_H\left(n\right)& ={\mu }^2\sqrt{n}{h}^{2-2H}-{\sigma }^2\mu h{n}^{2H-{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{1}{4}}{\sigma }^4{h}^{2H}{\displaystyle \frac{1}{\sqrt{n}}}{K}_n\left(H\right)\hfill \\ \hfill & ⟶\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<\gamma <{\displaystyle \frac{3-4H}{8H-3}},\hfill \\ {\displaystyle \frac{H^2}{4H-1}}{\sigma }^4,\hfill & \mathrm{if}\gamma ={\displaystyle \frac{3-4H}{8H-3}},\hfill \\ \infty ,\hfill & \mathrm{if}\gamma >{\displaystyle \frac{3-4H}{8H-3}}\hfill \end{array}\right.(n\to \infty )\hfill \end{array}$$

(40)

for ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$. Thus, by (16) and Proposition 1, to end the proof, we check that

$$\sqrt{n}{\mathsf{\Phi }}_H\left(n\right)\stackrel{P}{⟶}0(n\to \infty )$$

(41)

for all $H\in (0,{\displaystyle \frac{1}{2}})\cup ({\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}})$ under some suitable conditions for $\gamma$. By the fact

$$f\left(t_i\right)-f\left(t_{i-1}\right)=\mu h-{\displaystyle \frac{1}{2}}{\sigma }^2{h}^{2H}{\Delta }_i$$

with ${\Delta }_i=i^{2H}-{(i-1)}^{2H}$ and $i\in \{1,2,\dots ,n\}$, we get that

$$\begin{array}{cc}\hfill nE\left[{\mathsf{\Phi }}_H{\left(n\right)}^2\right]& ={\displaystyle \frac{1}{n{h}^{4H}}}E{\left(\sum _{i=1}^n\left[f\left(t_i\right)-f\left(t_{i-1}\right)\right]\left(B_{t_i}^H-B_{t_{i-1}}^H\right)\right)}^2\hfill \\ \hfill & ={\mathsf{\Psi }}_H\left(n\right)+{\displaystyle \frac{1}{n{h}^{2H}}}\sum _{1\le i<j\le n}\left[f\left(t_i\right)-f\left(t_{i-1}\right)\right]\left[f\left(t_j\right)-f\left(t_{j-1}\right)\right]R(j-i)\hfill \\ \hfill & ={\mathsf{\Psi }}_H\left(n\right)+{\displaystyle \frac{1}{n}}{\mu }^2{h}^{2-2H}\sum _{1\le i<j\le n}R(j-i)\hfill \\ \hfill & -{\displaystyle \frac{1}{2n}}\mu {\sigma }^{2}h\sum _{1\le i<j\le n}\left({\Delta }_i+{\Delta }_j\right)R(j-i)+{\displaystyle \frac{1}{4n}}{\sigma }^4{h}^{2H}\sum _{1\le i<j\le n}{\Delta }_i{\Delta }_{j}R(j-i)\hfill \end{array}$$

(42)

for all $n\ge 2$, where $R\left(x\right)={(x+1)}^{2H}+{(x-1)}^{2H}-2x^{2H}$ for $x\ge 1$.

Now, in order to end the proof, we estimate the last three items in (42) in the two cases $0<H<{\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$.

Cases I: $\mathbf{0}<\mathbf{H}<{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}$. Clearly, the sequence

$$\sum _{m=1}^n\left|R\left(m\right)\right|=\sum _{m=1}^n{m}^{2H}\left|{(1+{\displaystyle \frac{1}{m}})}^{2H}+{(1-{\displaystyle \frac{1}{m}})}^{2H}-2\right|$$

converges. It follows that

$${\displaystyle \frac{1}{n}}{h}^{2-2H}\sum _{1\le i<j\le n}R(j-i)={\displaystyle \frac{1}{n}}{h}^{2-2H}\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}R(j-i)={\displaystyle \frac{n-1}{n}}{h}^{2-2H}\sum _{m=1}^{n}R\left(m\right)=O\left(h^{2-2H}\right)⟶0,$$

(43)

$${\displaystyle \frac{1}{n}}h\left|\sum _{1\le i<j\le n}{\Delta }_i·R(j-i)\right|\le {\displaystyle \frac{1}{n}}h\sum _{1\le i<j\le n}\left|R(j-i)\right|=O\left(n^{2H-1}\right)h⟶0,$$

(44)

$${\displaystyle \frac{1}{n}}h\left|\sum _{1\le i<j\le n}{\Delta }_{j}R(j-i)\right|\le {\displaystyle \frac{1}{n}}h\sum _{1\le i<j\le n}\left|R(j-i)\right|=O\left(n^{2H-1}\right)h⟶0$$

(45)

and

$${\displaystyle \frac{1}{n}}{h}^{2H}\left|\sum _{1\le i<j\le n}{\Delta }_i{\Delta }_{j}R(j-i)\right|\le {\displaystyle \frac{1}{n}}{h}^{2H}\sum _{1\le i<j\le n}\left|R(j-i)\right|=h^{2H}O\left(n^{2H-1}\right)⟶0,$$

(46)

as n tends to infinity. Combining these with Lemma 8 and (42), we obtain convergence (41) for all $0<H<{\displaystyle \frac{1}{2}}$. Thus, by Proposition 1, (16), (39), and Slutsky’s theorem, we obtain the desired asymptotic behavior

$$\begin{array}{cc}\hfill \sqrt{n}\left(N_n\left(Y^H\right)-{\sigma }^2\right)& ={\sigma }^2\sqrt{n}\left(M_n\left(B^H\right)-1\right)+\sqrt{n}{\mathsf{\Psi }}_H\left(n\right)+2\sigma \sqrt{n}{\mathsf{\Phi }}_H\left(n\right)\hfill \\ \hfill & ⟶\left\{\begin{array}{cc}N\left(0,(2+{\lambda }_H){\sigma }^4\right),\hfill & \mathrm{if}0<\gamma <3-4H,\hfill \\ N\left({\mu }^2,(2+{\lambda }_H){\sigma }^4\right),\hfill & \mathrm{if}\gamma =3-4H\hfill \end{array}\right.(n\to \infty )\hfill \end{array}$$

for all $0<H<{\displaystyle \frac{1}{2}}$, and statement (1) follows.

Cases II: ${\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}<\mathbf{H}<{\displaystyle \frac{\mathbf{3}}{\mathbf{4}}}$. From

$$\sum _{j=1}^n{j}^{2H-2}=n^{2H-1}\sum _{j=1}^n{\left({\displaystyle \frac{j}{n}}\right)}^{2H-2}·{\displaystyle \frac{1}{n}}\sim {\displaystyle \frac{1}{2H-1}}{n}^{2H-1}(n\to \infty )$$

and Taylor’s expansion, we get that

$$\begin{array}{cc}\hfill \sum _{m=1}^{n}R\left(m\right)& =\sum _{m=1}^n{m}^{2H}\left\{{(1+{\displaystyle \frac{1}{m}})}^{2H}+{(1-{\displaystyle \frac{1}{m}})}^{2H}-2\right\}\sim 2H{n}^{2H-1}\hfill \end{array}$$

as n tends to infinity, which implies that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{n}}{h}^{2-2H}\sum _{1\le i<j\le n}R(j-i)& ={\displaystyle \frac{n-1}{n}}{h}^{2-2H}\sum _{m=1}^{n}R\left(m\right)\hfill \\ \hfill & \sim 2H{n}^{2H-1}{h}^{2-2H}⟶\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<\gamma <{\displaystyle \frac{3-4H}{2H-1}},\hfill \\ 2H,\hfill & \mathrm{if}\gamma ={\displaystyle \frac{3-4H}{2H-1}}\hfill \end{array}\right.\hfill \end{array}$$

(47)

as n tends to infinity. Similarly, we also have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2n}}\mu {\sigma }^{2}h\sum _{1\le i<j\le n}{\Delta }_{i}R(j-i)& ={\displaystyle \frac{1}{2n}}\mu {\sigma }^{2}h\sum _{m=1}^{n}R\left(m\right)\sum _{i=1}^{n-1}{\Delta }_i\hfill \\ \hfill & ={\displaystyle \frac{1}{2n}}\mu {\sigma }^{2}h{(n-1)}^{2H}\sum _{m=1}^{n}R\left(m\right)\sim H\mu {\sigma }^2{n}^{4H-2}h\hfill \\ \hfill & ⟶\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<\gamma <{\displaystyle \frac{3-4H}{4H-2}},\hfill \\ H\mu {\sigma }^2,\hfill & \mathrm{if}\gamma ={\displaystyle \frac{3-4H}{4H-2}}\hfill \end{array}\right.\hfill \end{array}$$

(48)

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2n}}\mu {\sigma }^{2}h\sum _{1\le i<j\le n}& {\Delta }_{j}R(j-i)={\displaystyle \frac{1}{2n}}\mu {\sigma }^{2}h\sum _{m=1}^{n}R\left(m\right)\sum _{j=m+1}^n{\Delta }_j\hfill \\ \hfill & ={\displaystyle \frac{1}{2n}}\mu {\sigma }^{2}h\sum _{m=1}^n\left(n^{2H}-m^{2H}\right)R\left(m\right)\sim \mu {\sigma }^2{\displaystyle \frac{2H^2}{4H-1}}{n}^{4H-2}h\hfill \end{array}$$

(49)

as n tends to infinity. On the other hand, we have

$$\begin{array}{cc}\hfill \sum _{1\le i<j\le n}{\left({\displaystyle \frac{ij}{n^2}}\right)}^{2H-1}{\left({\displaystyle \frac{j}{n}}-{\displaystyle \frac{i}{n}}\right)}^{2H-2}·{\displaystyle \frac{1}{n^2}}⟶{\zeta }_H& ={\int }_0^1{\int }_0^y{\left(xy\right)}^{2H-1}{(y-x)}^{2H-2}dxdy\hfill \\ \hfill & ={\displaystyle \frac{1}{6H-2}}\mathbb{B}(2H,2H-1),\hfill \end{array}$$

as n tends to infinity, where $\mathbb{B}(·,·)$ denotes the classical Beta function. It follows from Taylor’s expansion that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{4n}}{\sigma }^4{h}^{2H}\sum _{1\le i<j\le n}{\Delta }_i{\Delta }_{j}R(j-i)& \sim 2H^3(2H-1){\sigma }^4{h}^{2H}{\displaystyle \frac{1}{n}}\sum _{1\le i<j\le n}{\left(ij\right)}^{2H-1}{(j-i)}^{2H-2}\hfill \\ \hfill & \sim 2H^3(2H-1){\zeta }_H{\sigma }^4{h}^{2H}{n}^{6H-3}\hfill \\ \hfill & ⟶\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<\gamma <{\displaystyle \frac{3-4H}{6H-3}},\hfill \\ 2H^3(2H-1){\zeta }_H{\sigma }^4,\hfill & \mathrm{if}\gamma ={\displaystyle \frac{3-4H}{6H-3}}\hfill \end{array}\right.\hfill \end{array}$$

(50)

for all ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$, as n tends to infinity. Combining these with Lemma 8 and (42), we obtain convergence (41) for all ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$ if $0<\gamma <{\displaystyle \frac{3-4H}{6H-3}}$. Thus, we obtain the desired asymptotic behavior

$$\begin{array}{cc}\hfill \sqrt{n}\left(N_n\left(Y^H\right)-{\sigma }^2\right)& ={\sigma }^2\sqrt{n}\left(M_n\left(B^H\right)-1\right)+\sqrt{n}{\mathsf{\Psi }}_H\left(n\right)+2\sigma \sqrt{n}{\mathsf{\Phi }}_H\left(n\right)\hfill \\ \hfill & ⟶\left\{\begin{array}{cc}N\left(0,(2+{\lambda }_H){\sigma }^4\right),\hfill & \mathrm{if}0<\gamma <{\displaystyle \frac{3-4H}{8H-3}},\hfill \\ N\left({\mu }^2,(2+{\lambda }_H){\sigma }^4\right),\hfill & \mathrm{if}\gamma ={\displaystyle \frac{3-4H}{8H-3}}\hfill \end{array}\right.(n\to \infty )\hfill \end{array}$$

for all ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$ by Proposition 1, (16), and (40), and statement (2) follows.

Now, we verify statement (3). Let ${\displaystyle \frac{3}{4}}\le H<1$. By Lemma 7, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}{\mathsf{\Psi }}_H\left(n\right)& ={\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}\left({\mu }^2{h}^{2-2H}-\mu {\sigma }^{2}h{n}^{2H-1}+{\displaystyle \frac{1}{4n}}{\sigma }^4{h}^{2H}{K}_n\left(H\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{{\mu }^2}{{\left(nh\right)}^{4H-2}}}-{\displaystyle \frac{\mu {\sigma }^2}{{\left(nh\right)}^{2H-1}}}+{\displaystyle \frac{H^2}{4H-1}}{\sigma }^4+{\displaystyle \frac{1}{2n}}{H}^2{\sigma }^4+O\left(n^{-2}\right)\hfill \\ \hfill & ⟶{\displaystyle \frac{H^2}{4H-1}}{\sigma }^4(n\to \infty ),\hfill \end{array}$$

(51)

and moreover, from the proof of statement (2) in Lemma 9, we also have

$$\begin{array}{cc}\hfill {\displaystyle \frac{n^{4-4H}}{{\left(nh\right)}^{2H}}}E\left[{\mathsf{\Phi }}_H{\left(n\right)}^2\right]& ={\displaystyle \frac{n^{3-4H}}{{\left(nh\right)}^{2H}}}{\mathsf{\Psi }}_H\left(n\right)+{\displaystyle \frac{n^{3-4H}}{{\left(nh\right)}^{2H}}}\left\{{\displaystyle \frac{1}{n}}{\mu }^2{h}^{2-2H}\sum _{1\le i<j\le n}R(j-i)\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{2n}}\mu {\sigma }^{2}h\sum _{1\le i<j\le n}\left({\Delta }_i+{\Delta }_j\right)R(j-i)+{\displaystyle \frac{1}{4n}}{\sigma }^4{h}^{2H}\sum _{1\le i<j\le n}{\Delta }_i{\Delta }_{j}R(j-i)\right\}\hfill \\ \hfill & ⟶\beta \left(H\right){\sigma }^4,\hfill \end{array}$$

(52)

where $\beta \left(H\right)={\displaystyle \frac{2H-1}{3H-1}}\mathbb{B}(2H,2H-1)H^3$. Noting that

$${\mathsf{\Phi }}_H\left(n\right)={\displaystyle \frac{1}{n{h}^{2H}}}\sum _{i=1}^n\left[f\left(t_i\right)-f\left(t_{i-1}\right)\right]\left(B_{t_i}^H-B_{t_{i-1}}^H\right)$$

admits a normal distribution for all $n\ge 1$, we see that

$$\begin{array}{c}\hfill 2\sigma {\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}{\mathsf{\Phi }}_H\left(n\right)\stackrel{d}{⟶}N\left(0,4\beta \left(H\right){\sigma }^6\right)\end{array}$$

(53)

from (52). It follows from (16), statement (3) in Proposition 1, and (51) that

$$\begin{array}{cc}\hfill N_n(Y^H;1)& ={\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}\left(N_n\left(Y^H\right)-{\sigma }^2\right)\hfill \\ \hfill & ={\sigma }^2{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}\left(M_n\left(B^H\right)-1\right)+{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}{\mathsf{\Psi }}_H\left(n\right)+2\sigma {\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}{\mathsf{\Phi }}_H\left(n\right)\hfill \\ \hfill & \stackrel{a.s}{⟶}{\displaystyle \frac{H^2}{4H-1}}{\sigma }^4,\hfill \end{array}$$

(54)

as n tends to infinity. Moreover, by (51) we obtain

$$\begin{array}{cc}\hfill {\left(nh\right)}^{2H-1}& \left({\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}{\mathsf{\Psi }}_H\left(n\right)-{\displaystyle \frac{H^2}{4H-1}}{\sigma }^4\right)\hfill \\ & ={\displaystyle \frac{{\mu }^2}{{\left(nh\right)}^{2H-1}}}-\mu {\sigma }^2+{\displaystyle \frac{H^2{\sigma }^4}{2n^{2-2H}}}{h}^{2H-1}+h^{2H-1}O\left(n^{2H-3}\right)\hfill \\ \hfill & ⟶-\mu {\sigma }^2(n\to \infty ).\hfill \end{array}$$

(55)

Combining this with (54), (53), and Proposition 1, we get

$$\begin{array}{cc}\hfill N_n(Y^H;2)& ={\left(nh\right)}^{2H-1}\left(N_n(Y^H;1)-{\displaystyle \frac{H^2{\sigma }^4}{4H-1}}\right)={\sigma }^2{\displaystyle \frac{n^{2-2H}}{nh}}\left(M_n\left(B^H\right)-1\right)+2\sigma {\displaystyle \frac{n^{2-2H}}{nh}}{\mathsf{\Phi }}_H\left(n\right)\hfill \\ \hfill & +\left({\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}{\mathsf{\Psi }}_H\left(n\right)-{\displaystyle \frac{H^2}{4H-1}}{\sigma }^4\right){\left(nh\right)}^{2H-1}\stackrel{a.s}{⟶}-\mu {\sigma }^2,\hfill \end{array}$$

(56)

as n tends to infinity. Finally, by Proposition 1, (53), (55), and Slutsky’s theorem, we obtain

$$\begin{array}{cc}\hfill {\left(nh\right)}^{1-H}\left(N_n(Y^H;2)+\mu {\sigma }^2\right)& ={\sigma }^2{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left(M_n\left(B^H\right)-1\right)+2\sigma {\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}{\mathsf{\Phi }}_H\left(n\right)\hfill \\ \hfill & +\left\{\left({\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^{2H}}}{\mathsf{\Psi }}_H\left(n\right)-{\displaystyle \frac{H^2}{4H-1}}{\sigma }^4\right){\left(nh\right)}^{2H-1}+\mu {\sigma }^2\right\}{\left(nh\right)}^{1-H}\hfill \\ \hfill & ={\sigma }^2{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left(M_n\left(B^H\right)-1\right)+2\sigma {\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}{\mathsf{\Phi }}_H\left(n\right)\hfill \\ \hfill & +{\displaystyle \frac{{\mu }^2}{{\left(nh\right)}^{3H-2}}}+{\displaystyle \frac{H^2{\sigma }^4}{2n^{1-H}}}{h}^H+h^{H}O\left(n^{H-2}\right)\hfill \\ \hfill & \stackrel{d}{⟶}N\left(0,4\beta \left(H\right){\sigma }^6\right).\hfill \end{array}$$

(57)

Thus, the three convergences in statements (2) and (3) follow. □

5. Asymptotic Behavior of Estimator ${\widehat{\mathsf{\mu }}}_{\mathbf{n}}$

In this section, we consider the strong consistency and asymptotic distribution of ${\widehat{\mu }}_n$. We keep the notations from Section 3.

 Theorem 7. 

Let $H\in (0,1)$ and assume that condition (C1) holds. If $\sigma >0$ is known, the estimator given by (4)

$${\widehat{\mu }}_n={\displaystyle \frac{1}{nh}}{Y}_{t_n}+{\displaystyle \frac{1}{2}}{\left(nh\right)}^{2H-1}{\sigma }^2$$

(58)

is strongly consistent and

$${\left(nh\right)}^{1-H}({\widehat{\mu }}_n-\mu )\stackrel{d}{⟶}N(0,{\sigma }^2).$$

Proof. 

The theorem follows from the fact that

$$\begin{array}{cc}\hfill {\widehat{\mu }}_n-\mu & ={\displaystyle \frac{1}{nh}}\left(\mu nh-{\displaystyle \frac{1}{2}}{\sigma }^2{\left(nh\right)}^{2H}+\sigma B_{t_n}^H\right)+{\displaystyle \frac{1}{2}}{\left(nh\right)}^{2H-1}{\sigma }^2-\mu =\sigma {\displaystyle \frac{B_{t_n}^H}{nh}}\hfill \end{array}$$

for all $n\ge 1$. □

 Lemma 13. 

When ${\displaystyle \frac{1}{2}}<H<1$, and assuming that conditions (C1) and (C2) hold, estimator ${\widehat{\theta }}_n$ given by (5) satisfies

$${\left(nh\right)}^{2H-1}\left({\widehat{\theta }}_n-{\sigma }^2\right)\stackrel{a.s}{⟶}0,$$

(59)

if the following conditions are satisfied

 (1) 

${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$ and $0<\gamma <{\displaystyle \frac{3-4H}{8H-3}};$

 (2) 

${\displaystyle \frac{3}{4}}<H<1$ and $0<\gamma <{\displaystyle \frac{2-2H}{5H-3}}.$

Proof. 

When ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$, based on statement (2) of Theorem 5 and Theorem 6, we obtain that when $0<\gamma <{\displaystyle \frac{3-4H}{8H-3}}$,

$$\begin{array}{cc}\hfill {\left(nh\right)}^{2H-1}\left({\widehat{\theta }}_n-{\sigma }^2\right)& =n^{2H-{\displaystyle \frac{3}{2}}}{h}^{2H-1}·\sqrt{n}\left({\widehat{\theta }}_n-{\sigma }^2\right)\hfill \\ \hfill & ={\left(n{h}^{1+\gamma }\right)}^{{\displaystyle \frac{2H-1}{1+\gamma }}}{n}^{-{\displaystyle \frac{1-(4H-3)\gamma }{2(1+\gamma )}}}·\sqrt{n}\left({\widehat{\theta }}_n-{\sigma }^2\right)\stackrel{a.s}{⟶}0\hfill \end{array}$$

as n tends to infinity. Thus, statement (1) is established. When ${\displaystyle \frac{3}{4}}\le H<1$ and $0<\gamma <{\displaystyle \frac{2-2H}{5H-3}}$, we have

$$\begin{array}{cc}\hfill {\left(nh\right)}^{2H-1}\left({\widehat{\theta }}_n-{\sigma }^2\right)& =n^{5H-3}{h}^{3H-1}·{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left({\widehat{\theta }}_n-{\sigma }^2\right)\hfill \\ & ={\left(n{h}^{1+\gamma }\right)}^{{\displaystyle \frac{3H-1}{1+\gamma }}}{n}^{{\displaystyle \frac{(5H-3)\gamma +2H-2}{1+\gamma }}}·{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left({\widehat{\theta }}_n-{\sigma }^2\right)\stackrel{a.s}{⟶}0\hfill \end{array}$$

as n tends to infinity. Thus, statement (2) is established. □

 Theorem 8. 

Let conditions (C1) and (C2) hold and $\sigma >0$ be unknown.

 (1) 

If $0<H<{\displaystyle \frac{1}{2}}$, estimator ${\widehat{\mu }}_n$ given by (5) is a strongly consistent estimator. Furthermore, we have

$${\left(nh\right)}^{1-H}({\widehat{\mu }}_n-\mu )\stackrel{d}{⟶}N(0,{\sigma }^2).$$

 (2) 

If ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$, estimator ${\widehat{\mu }}_n$ given by (5) is a strongly consistent estimator. Furthermore, if $0<\gamma <{\displaystyle \frac{3-4H}{8H-3}}$, we obtain

$${\left(nh\right)}^{1-H}({\widehat{\mu }}_n-\mu )\stackrel{d}{⟶}N(0,{\sigma }^2).$$

 (3) 

If ${\displaystyle \frac{3}{4}}<H<1$, estimator ${\widehat{\mu }}_n$ given by (5) is a strongly consistent estimator. Moreover, if $0<\gamma <{\displaystyle \frac{2-2H}{6H-3}}$, we get

$${\left(nh\right)}^{1-H}({\widehat{\mu }}_n-\mu )\stackrel{d}{⟶}N(0,{\sigma }^2).$$

Proof. 

For $0<H<{\displaystyle \frac{1}{2}}$, applying a transformation to (5), we obtain

$$\begin{array}{cc}\hfill {\widehat{\mu }}_n& ={\displaystyle \frac{1}{nh}}{Y}_{t_n}^H+{\displaystyle \frac{1}{2}}{\left(nh\right)}^{2H-1}{\widehat{\theta }}_n=\mu +{\displaystyle \frac{\sigma B_{t_n}^H}{nh}}-{\displaystyle \frac{1}{2}}{\sigma }^2{\left(nh\right)}^{2H-1}+{\displaystyle \frac{1}{2}}{\left(nh\right)}^{2H-1}{\widehat{\theta }}_n\stackrel{a.s}{⟶}\mu .\hfill \end{array}$$

as n tends to infinity. Moreover,

$$\begin{array}{cc}\hfill {\left(nh\right)}^{1-H}({\widehat{\mu }}_n-\mu )& ={\displaystyle \frac{\sigma B_{t_n}^H}{{\left(nh\right)}^H}}+{\displaystyle \frac{1}{2}}{\left(nh\right)}^H\left({\widehat{\theta }}_n-{\sigma }^2\right)\hfill \\ \hfill & ={\displaystyle \frac{\sigma B_{t_n}^H}{{\left(nh\right)}^H}}+{\displaystyle \frac{1}{2}}{\left(n{h}^{1+\gamma }\right)}^{{\displaystyle \frac{H}{1+\gamma }}}·n^{{\displaystyle \frac{(2H-1)\gamma -1}{2(1+\gamma )}}}·\sqrt{n}\left({\widehat{\theta }}_n-{\sigma }^2\right)\hfill \\ \hfill & \stackrel{d}{⟶}N(0,{\sigma }^2)\hfill \end{array}$$

as n tends to infinity. So statement (1) is proven. When ${\displaystyle \frac{1}{2}}<H<{\displaystyle \frac{3}{4}}$ and $0<\gamma <{\displaystyle \frac{3-4H}{8H-3}}$, by Lemma 13, we have

$$\begin{array}{cc}\hfill {\widehat{\mu }}_n& ={\displaystyle \frac{1}{nh}}{Y}_{t_n}^H+{\displaystyle \frac{1}{2}}{\left(nh\right)}^{2H-1}{\widehat{\theta }}_n=\mu +{\displaystyle \frac{\sigma B_{t_n}^H}{nh}}+{\displaystyle \frac{1}{2}}{\left(nh\right)}^{2H-1}\left({\widehat{\theta }}_n-{\sigma }^2\right)\stackrel{a.s}{⟶}\mu \hfill \end{array}$$

as n tends to infinity. Similarly, using statement (2) of Theorem 5 and the condition ${\displaystyle \frac{3-4H}{8H-3}}<{\displaystyle \frac{1}{2H-1}}$, we get

$$\begin{array}{cc}\hfill {\left(nh\right)}^{1-H}({\widehat{\mu }}_n-\mu )& ={\displaystyle \frac{\sigma B_{t_n}^H}{{\left(nh\right)}^H}}+{\displaystyle \frac{1}{2}}{\left(nh\right)}^H\left({\widehat{\theta }}_n-{\sigma }^2\right)\hfill \\ \hfill & ={\displaystyle \frac{\sigma B_{t_n}^H}{{\left(nh\right)}^H}}+{\displaystyle \frac{1}{2}}{\left(n{h}^{1+\gamma }\right)}^{{\displaystyle \frac{H}{1+\gamma }}}·n^{{\displaystyle \frac{(2H-1)\gamma -1}{2(1+\gamma )}}}·\sqrt{n}\left({\widehat{\theta }}_n-{\sigma }^2\right)\hfill \\ \hfill & \stackrel{d}{⟶}N(0,{\sigma }^2),\hfill \end{array}$$

when $0<\gamma <{\displaystyle \frac{3-4H}{8H-3}}$ and as n tends to infinity. Thus, statement (2) is established. When ${\displaystyle \frac{3}{4}}\le H<1$ and $0<\gamma <{\displaystyle \frac{2-2H}{6H-3}}$, it follows from Lemma 13 that

$$\begin{array}{cc}\hfill {\widehat{\mu }}_n& ={\displaystyle \frac{1}{nh}}{Y}_{t_n}^H+{\displaystyle \frac{1}{2}}{\left(nh\right)}^{2H-1}{\widehat{\theta }}_n=\mu +{\displaystyle \frac{\sigma B_{t_n}^H}{nh}}+{\displaystyle \frac{1}{2}}{\left(nh\right)}^{2H-1}\left({\widehat{\theta }}_n-{\sigma }^2\right)\stackrel{a.s}{⟶}\mu .(n\to \infty )\hfill \end{array}$$

From Theorem 6 and the fact that ${\displaystyle \frac{2-2H}{6H-3}}<{\displaystyle \frac{1-H}{2H-1}}$, for $0<\gamma <{\displaystyle \frac{2-2H}{6H-3}}$, we obtain

$$\begin{array}{cc}\hfill {\left(nh\right)}^{1-H}({\widehat{\mu }}_n-\mu )& ={\displaystyle \frac{\sigma B_{t_n}^H}{{\left(nh\right)}^H}}+{\displaystyle \frac{1}{2}}{\left(nh\right)}^H\left({\widehat{\theta }}_n-{\sigma }^2\right)\hfill \\ \hfill & ={\displaystyle \frac{\sigma B_{t_n}^H}{{\left(nh\right)}^H}}+{\displaystyle \frac{1}{2}}{\left(n{h}^{1+\gamma }\right)}^{{\displaystyle \frac{3H}{1+\gamma }}}·n^{{\displaystyle \frac{(4H-2)\gamma +2H-2}{1+\gamma }}}·{\displaystyle \frac{n^{2-2H}}{{\left(nh\right)}^H}}\left({\widehat{\theta }}_n-{\sigma }^2\right)\hfill \\ \hfill & \stackrel{d}{⟶}N(0,{\sigma }^2).\hfill \end{array}$$

Consequently, statement (3) is established. □

6. Numerical Simulation and Empirical Analysis

In this section, the effectiveness of the proposed estimator is validated through numerical simulations. The results demonstrate that the estimator exhibits strong applicability and reliable performance in practical scenarios. To further assess the precision of the two estimation methods, Monte Carlo simulations were conducted in MATLAB 2017b, where the simulated estimates were compared against the true values, and their mean values and standard deviations were calculated to provide a comprehensive evaluation of the estimator’s performance. In addition, real trading data from the Chinese financial market were retrieved via the Tushare Pro platform using Python 3.10. With the known value of H, the parameters $\mu$ and $\sigma$ were estimated, and track plots were generated in MATLAB and compared with the logarithmic closing prices of the stock, thereby further validating the effectiveness of the pseudo-likelihood estimation.

6.1. Numerical Simulation

First, we emphasize that in all the figures presented below, the sample size was fixed at $n=1000$, and the time step was chosen as $h={\displaystyle \frac{0.01}{\sqrt{n}}}$. The parameters were set to $\mu =4$ and ${\sigma }^2=9$. In the analysis of the asymptotic distribution, the number of replications, i.e., the simulated sample paths, was specified as $nu{m}_{t}rials=1000$. For the sake of notational consistency, we denote $\theta ={\sigma }^2$ throughout the subsequent discussion. To assess the effectiveness and robustness of the proposed estimation method, we designed two primary experimental scenarios:

1.

Case with Partially Known Parameters

• In the case where the parameter $\mu =4$ is known, we estimated the parameter ${\widehat{\theta }}_n$ and further examined its estimation path, quantile–quantile plot, and asymptotic distribution. The corresponding results for the estimator ${\widehat{\theta }}_n$ are presented for $H=0.4$ (Figure 1) and $H=0.6$ (Figure 2).
• In the case where the parameter $\sigma$ is known, we estimated the parameter $\mu$ and examined its estimation path and asymptotic distribution. Similarly, figures present the estimation paths and asymptotic distribution of ${\widehat{\mu }}_n$ when $H=0.4$ (Figure 3) and $H=0.6$ (Figure 4).

2.

Case with Completely Unknown Parameters

• In this scenario, where both $\mu$ and $\sigma$ are unknown, we estimated both parameters simultaneously and analyzed their estimation paths and asymptotic distributions. Figures present the estimation paths and asymptotic distribution of ${\widehat{\mu }}_n$ and ${\widehat{\theta }}_n$ when $H=0.4$ (Figure 5 and Figure 6) and $H=0.6$ (Figure 7 and Figure 8).

Case 1: The asymptotic behavior of the estimators of ${\widehat{\theta }}_n$ and ${\widehat{\mu }}_n$ when $\mu$ is known.

Case 2: The asymptotic behavior of the estimators of ${\widehat{\theta }}_n$ and ${\widehat{\mu }}_n$ when both parameters are unknown.

From the above figures, it can be observed that for different values of H, the numerical simulation results of the convergence and asymptotic properties of the estimators $\mu$ and $\theta$ are largely consistent with the theoretical predictions. The discrepancies are minor, indicating that the obtained estimates exhibit a high degree of accuracy.

In addition, to investigate the asymptotic behavior of the proposed estimators for different sample sizes, we considered three sample sizes: $n=1000$, 2000, and 3000. The comparison of theoretical variance with empirical variance, as well as the corresponding errors, was carried out. The specific experimental design is outlined as follows:

• Table 1. Comparison of theoretical and empirical variances of ${\widehat{\mu }}_n$ under known $\theta =9$ and various Hurst indices.

  H	$\mathit{n}=1000$			$\mathit{n}=2000$			$\mathit{n}=3000$
  	Theoretical	Empirical	Abs. Error	Theoretical	Empirical	Abs. Error	Theoretical	Empirical	Abs. Error
  0.2	1.4264	1.573	0.14663	0.81925	0.90776	0.088501	0.10021	0.10194	0.00174
  0.4	2.2607	2.4086	0.14788	1.4915	1.6008	0.10928	1.1694	1.1947	0.025295
  0.6	3.583	3.6437	0.060714	2.7154	2.6693	0.0461	2.3088	2.3419	0.03308
  0.75	16.0045	16.2563	0.25178	13.4581	13.221	0.2371	12.1608	12.2322	0.071373
  0.8	14.264	14.3885	0.12446	12.4176	12.5023	0.0847	11.4503	11.5162	0.065843

  : Theoretical variance, empirical variance, and their errors for parameter $\mu$ when $\theta =9$ is known.
• Table 2. Comparison of theoretical and empirical variances of ${\widehat{\theta }}_n$ under known $\mu =4$ and various Hurst indices.

  H	$\mathit{n}=1000$			$\mathit{n}=2000$			$\mathit{n}=3000$
  	Theoretical	Empirical	Abs. Error	Theoretical	Empirical	Abs. Error	Theoretical	Empirical	Abs. Error
  0.2	0.20041	0.19597	0.0044404	0.10021	0.096571	0.0036343	0.066803	0.069968	0.0031643
  0.4	0.16829	0.16116	0.0071283	0.084143	0.087156	0.0030125	0.056095	0.055172	0.00092321
  0.6	0.17518	0.18504	0.0098601	0.087612	0.082614	0.0049982	0.058414	0.057796	0.00061751
  0.75	0.51431	0.31398	0.20033	0.27294	0.17281	0.10013	0.18812	0.11809	0.070025
  0.8	0.34117	0.46301	0.12184	0.34117	0.28088	0.060296	0.23582	0.25174	0.01592

  : Theoretical variance, empirical variance, and their errors for parameter $\theta$ when $\mu =4$ is known.
• Table 3. Comparison of theoretical and empirical variances of ${\widehat{\theta }}_n$ and ${\widehat{\mu }}_n$ under various Hurst indices and sample sizes.

  H	Estimator	$\mathit{n}=1000$			$\mathit{n}=2000$			$\mathit{n}=3000$
  		Theoretical	Empirical	Abs. Error	Theoretical	Empirical	Abs. Error	Theoretical	Empirical	Abs. Error
  0.2	${\widehat{\theta }}_n$	0.20041	0.19528	0.0051278	0.10021	0.09621	0.0039946	0.066803	0.065901	0.00090264
  	${\widehat{\mu }}_n$	1.4264	1.4461	0.01974	0.81925	0.80864	0.01061	0.066803	0.070016	0.0032127
  0.4	${\widehat{\theta }}_n$	0.16829	0.15938	0.0089078	0.084143	0.087156	0.0030125	0.056095	0.05876	0.0026642
  	${\widehat{\mu }}_n$	2.2607	2.4223	0.16156	1.4915	1.5998	0.10828	1.1694	1.1819	0.012476
  0.6	${\widehat{\theta }}_n$	0.17518	0.24192	0.066742	0.087612	0.10851	0.020897	0.058414	0.070423	0.012009
  	${\widehat{\mu }}_n$	3.583	3.6574	0.074434	2.7154	2.6748	0.040622	2.3088	2.3434	0.034604
  0.75	${\widehat{\theta }}_n$	0.51431	0.62807	0.11376	0.27294	0.17281	0.10013	0.18812	0.25422	0.066106
  	${\widehat{\mu }}_n$	16.0045	16.3175	0.31298	13.4581	13.2418	0.21634	12.1608	12.2428	0.081992
  0.8	${\widehat{\theta }}_n$	0.48756	1.5388	1.0512	0.48756	1.1108	0.62322	0.48756	1.0498	0.56228
  	${\widehat{\mu }}_n$	14.264	14.4408	0.1768	12.4176	12.5235	0.10598	11.4503	11.5296	0.079271

  : Joint analysis of the variance estimates and errors for both parameters when $\mu$ and $\theta$ are unknown.

The discrepancies are minor, indicating that the obtained estimates exhibit a high degree of accuracy.

6.2. Empirical Analysis

To further evaluate the performance of the proposed model and estimation method in a real-world market setting, we conducted an empirical analysis using Heilan Home Co., Ltd. Jiangyin, Jiangsu Province, China (stock code: 600398), a representative stock from the Chinese A-share market. Daily closing price data were retrieved via the Tushare Pro platform using Python, covering the period from 28 December 2000, to 26 August 2025. Data cleaning and preprocessing were carried out to ensure consistency. As supported by the theoretical results in Section 3, the estimators were consistent as the sample size $Nh\to \infty$; therefore, the full sample period was employed to guarantee robustness. The Hurst exponent of the stock return series was first estimated using the R/S method, yielding $H=0.629$, which suggested the presence of long-memory effects.

Based on this, the key model parameters ${\widehat{\mu }}_n$ and ${\widehat{\theta }}_n$ were estimated within the quasi-likelihood framework proposed in this paper. To provide an intuitive evaluation of model fit, simulated price track were generated in MATLAB using the estimated parameters and compared with the actual closing prices observed. The comparison demonstrated that the model captured the overall price dynamics effectively, thereby confirming both the applicability of the mixed fractional Brownian motion Black–Scholes framework and the reliability of the proposed quasi-likelihood estimation method on real financial data. Furthermore, we simulated stock price tracking using both the fractional Brownian motion model proposed in this study and the classical Black–Scholes model. The comparative results are presented in Figure 9 and Figure 10. As illustrated, our proposed model provides a notably better fit to the observed price dynamics, particularly in capturing volatility clustering and the long-memory behavior inherent in the price process. These results further highlight the advantages and practical applicability of our model in financial data modeling and empirical analysis.

7. Conclusions

In this paper, we studied quasi-likelihood estimation for the fractional Black–Scholes model driven by fractional Brownian motion. Based on discrete observations of the geometric fractional Brownian motion, we constructed the quasi-likelihood function and derived the estimators ${\widehat{\mu }}_n$ and ${\widehat{\theta }}_n$. We further analyzed the asymptotic properties of these estimators, including strong consistency and asymptotic normality, considering both cases where $\mu$ was known or unknown. Numerical simulations and an empirical analysis indicated that the quasi-likelihood estimation method provided accurate parameter estimates under high-frequency observations, while effectively capturing volatility clustering and long-memory characteristics of the price process. These results demonstrate the effectiveness and applicability of the proposed model and estimation methodology in financial data modeling and empirical studies. In summary, this study extended the application of fractional Brownian motion models in financial parameter estimation and provided a feasible methodological framework for estimating parameters in complex stochastic systems, offering both theoretical insights and practical guidance for future research.