1 Introduction
The present paper deals with the fractional Vasicek model of the form
where ${B^{H}}$ is the fractional Brownian motion with the Hurst index $H\in (1/2,1)$. It is a generalization of the classical interest rate model proposed by O. Vasicek [34] in 1977. This generalization enables to study processes with long-range dependence, which arise in financial mathematics and several other areas such as telecommunication networks, investigation of turbulence and image processing. In recent years, many articles on various financial applications of the fractional Vasicek model (1) have appeared, see e.g. [8, 9, 12, 13, 30, 40]). In order to use this model in practice, a theory of parameter estimation is necessary.
(1)
\[ d{X_{t}}=(\alpha -\beta {X_{t}})dt+\gamma d{B_{t}^{H}},\hspace{1em}{X_{0}}={x_{0}}\in \mathbb{R},\]Notice that in the particular case $\alpha =0$, (1) is a so-called fractional Ornstein–Uhlenbeck process, introduced in [7]. The drift parameter estimation for it has been studied since 2002, see the paper [17], where the maximum likelihood estimation was considered. The asymptotic and exact distributions of the maximum likelihood estimator (MLE) were investigated later in [4, 31, 32]. Alternative approaches to the drift parameter estimation were proposed and studied in [3, 6, 14–16, 21]. We refer to the article [28] for a survey on this topic, and to the book [20] for its more detailed presentation.
In the general case, the least squares and ergodic-type estimators of unknown parameters α and β were studied in [27, 38, 39]. The corresponding MLEs of α and β were presented in [25]. Their consistency and asymptotic normality were proved there for the case $\beta >0$. Slightly more general results were proved in [26], where joint asymptotic normality of MLE of the vector parameter $(\alpha ,\beta )$ was established. Recently Tanaka et al. [33] investigated asymptotic behavior of MLEs in the cases $\beta =0$ and $\beta <0$. However, in the latter case the asymptotic distribution was obtained only under assumption that ${x_{0}}=\frac{\alpha }{\beta }$. The study of the case ${x_{0}}\ne \frac{\alpha }{\beta }$ requires a different technique and still remains an open problem. The goal of the present paper is to fill in the gap and to derive asymptotic distributions of the MLEs of α and β for arbitrary ${x_{0}}\in \mathbb{R}$, $\alpha \in \mathbb{R}$ and $\beta <0$. Moreover, we prove that the MLEs for α and β are asymptotically independent.
The asymptotic behavior of the process X and of the estimators substantially depends on the sign of the parameter β. If $\beta <0$, then the process X behaves as ${O_{\mathsf{P}}}({e^{-\beta T}})$ as $T\to \infty $, hence it is non-ergodic. If $\beta >0$, then ${X_{T}}={O_{\mathsf{P}}}(1)$, as $T\to \infty $, and the process has ergodic properties, see, e.g., [27]. The method for the hypothesis testing of the sign of β was developed in [22].
In this article we restrict ourselves to the case $\frac{1}{2}<H<1$. Our proofs are based on the results of the papers [17] and [26], which are valid only for $H\in (\frac{1}{2},1)$ and cannot be immediately extended to the case $H\in (0,\frac{1}{2})$. In particular, the integral representation (7) below, which is the starting point for derivation of moment generating functions (MGFs) in Lemmas 1 and 2, holds for $H\in (0,\frac{1}{2})$ with different (and more complicated) kernel ${K_{H}}$. Therefore, the asymptotic behavior of the MLEs in this case requires a separate study.
The paper is organized as follows. In Section 2 we describe the model and the estimators, and introduce the notation. Section 3 contains the results on distributions and asymptotic behavior of stochastic processes involved into MLEs. In Section 4 we formulate and prove the main results on asymptotic distributions of MLEs. Some auxiliary facts and results concerning modified Bessel functions of the first kind and MGFs related to the normal distribution are collected in the appendices.
2 Preliminaries
Let $(\varOmega ,\mathfrak{F},\{{\mathfrak{F}_{t}}\},\mathsf{P})$ be a complete probability space with filtration. Let ${B^{H}}=\{{B_{t}^{H}},t\ge 0\}$ be a fractional Brownian motion on this probability space, that is, a centered Gaussian process with the covariance function
It follows from (2) that $\mathbb{E}{({B_{t}^{H}}-{B_{s}^{H}})^{2}}=|t-s{|^{2H}}$. Hence, there exists a modification of ${B^{H}}$, which is δ-Hölder continuous for all $\delta \in (0,H)$.
We study the fractional Vasicek model, described by the stochastic differential equation
The main goal is to estimate parameters $\alpha \in \mathbb{R}$ and $\beta <0$ by continuous observations of a trajectory of X on the interval $[0,T]$. We assume that the parameters $\gamma >0$ and $H\in (1/2,1)$ are known. This assumption is natural, because γ and H can be obtained explicitly from the observations by considering realized power variations, see Remark 1 below.
(3)
\[ {X_{t}}={x_{0}}+{\int _{0}^{t}}(\alpha -\beta {X_{s}})ds+\gamma {B_{t}^{H}},\hspace{1em}t\ge 0.\]Equation (3) has a unique solution, which is given by
where ${\textstyle\int _{0}^{t}}{e^{-\beta (t-s)}}d{B_{s}^{H}}$ is a path-wise Riemann–Stieltjes integral. It exists due to [7, Proposition A.1].
(4)
\[ {X_{t}}={x_{0}}{e^{-\beta t}}+\frac{\alpha }{\beta }\big(1-{e^{-\beta t}}\big)+\gamma {\int _{0}^{t}}{e^{-\beta (t-s)}}d{B_{s}^{H}},\hspace{1em}t\ge 0,\]Following [18], for $0<s<t\le T$ we define
\[\begin{array}{l}\displaystyle {\kappa _{H}}=2H\varGamma (3/2-H)\varGamma (H+1/2),\hspace{2em}{\lambda _{H}}=\frac{2H\varGamma (3-2H)\varGamma (H+1/2)}{\varGamma (3/2-H)},\\ {} \displaystyle {k_{H}}(t,s)={\kappa _{H}^{-1}}{s^{1/2-H}}{(t-s)^{1/2-H}},\hspace{2em}{w_{t}^{H}}={\lambda _{H}^{-1}}{t^{2-2H}}.\end{array}\]
We introduce also three stochastic processes
\[\begin{aligned}{}{P_{H}}(t)& =\frac{1}{\gamma }\frac{d}{d{w_{t}^{H}}}{\int _{0}^{t}}{k_{H}}(t,s){X_{s}}\hspace{0.1667em}ds,\\ {} {Q_{H}}(t)& =\frac{1}{\gamma }\frac{d}{d{w_{t}^{H}}}{\int _{0}^{t}}{k_{H}}(t,s)(\alpha -\beta {X_{s}})\hspace{0.1667em}ds,\\ {} {S_{t}}& =\frac{1}{\gamma }{\int _{0}^{t}}{k_{H}}(t,s)\hspace{0.1667em}d{X_{s}}.\end{aligned}\]
Note that by [25, Lemma 4.1]
According to [18, Theorem 1], the process S is an $({\mathfrak{F}_{t}})$-semimartingale with the decomposition
where ${M_{t}^{H}}={\textstyle\int _{0}^{t}}{k_{H}}(t,s)\hspace{0.1667em}d{B_{s}^{H}}$ is a Gaussian martingale with the variance function $\langle {M^{H}}\rangle ={w^{H}}$. The natural filtrations of processes S and X coincide. Moreover, the process X admits the following representation
where ${K_{H}}(t,s)=\gamma H(2H-1){\textstyle\int _{s}^{t}}{r^{H-1/2}}{(r-s)^{H-3/2}}\hspace{0.1667em}dr$.Remark 1.
If we observe the whole path $\{{X_{t}},t\in [0,T]\}$, then the parameters γ and H can be obtained from observations explicitly in the following way. Let $\{{t_{i}^{(n)}}\}$ be a partition of $[0,T]$, such that ${\sup _{i}}|{t_{i+1}^{(n)}}-{t_{i}^{(n)}}|\to 0$, as $n\to \infty $. Denote ${Z_{T}}={\textstyle\int _{0}^{T}}{k_{H}}(T,s)\hspace{0.1667em}d{X_{s}}=\gamma {S_{T}}$. From (6) it follows that ${\langle Z\rangle _{T}}={\gamma ^{2}}{w_{T}^{H}}$ a.s. Hence, the parameter γ is calculated as the limit
The Hurst index H can be evaluated as follows:
\[ H=\frac{1}{2}-\frac{1}{2}\underset{n}{\lim }{\log _{2}}\left(\frac{{\textstyle\textstyle\sum _{i=1}^{2n-1}}{\left({X_{{t_{i+1}^{(2n)}}}}-2{X_{{t_{i}^{(2n)}}}}+{X_{{t_{i-1}^{(2n)}}}}\right)^{2}}}{{\textstyle\textstyle\sum _{i=1}^{n-1}}{\left({X_{{t_{i+1}^{(n)}}}}-2{X_{{t_{i}^{(n)}}}}+{X_{{t_{i-1}^{(n)}}}}\right)^{2}}}\right)\hspace{1em}\text{a.s.,}\]
see, e.g., [20, Sec. 3.1]. There exist several other methods of the Hurst index evaluation based on power variations of X. We refer to the books [5, 20] for further information on this subject.Applying the analog of the Girsanov formula for a fractional Brownian motion ([18, Theorem 3], see also [19]) and (6), one can obtain the likelihood ratio $\frac{d{\mathsf{P}_{\alpha ,\beta }}(T)}{d{\mathsf{P}_{0,0}}(T)}$ for the probability measure ${\mathsf{P}_{\alpha ,\beta }}(T)$ corresponding to our model and the probability measure ${\mathsf{P}_{0,0}}(T)$ corresponding to the model with zero drift [25]:
MLEs of parameters α and β maximize (8) and have the following form [25]:
where
(8)
\[ \begin{aligned}{}\frac{d{\mathsf{P}_{\alpha ,\beta }}(T)}{d{\mathsf{P}_{0,0}}(T)}& =\exp \Bigg\{{\int _{0}^{T}}{Q_{H}}(t)\hspace{0.1667em}d{S_{t}}-\frac{1}{2}{\int _{0}^{T}}{\big({Q_{H}}(t)\big)^{2}}\hspace{0.1667em}d{w_{t}^{H}}\Bigg\}\\ {} & =\exp \left\{\frac{\alpha }{\gamma }{S_{T}}-\beta {\int _{0}^{T}}{P_{H}}(t)\hspace{0.1667em}d{S_{t}}-\frac{{\alpha ^{2}}}{2{\gamma ^{2}}}{w_{T}^{H}}\right.\\ {} & \hspace{1em}+\left.\frac{\alpha \beta }{\gamma }{\int _{0}^{T}}{P_{H}}(t)\hspace{0.1667em}d{w_{t}^{H}}-\frac{{\beta ^{2}}}{2}{\int _{0}^{T}}{\big({P_{H}}(t)\big)^{2}}\hspace{0.1667em}d{w_{t}^{H}}\right\}.\end{aligned}\](9)
\[ {\widehat{\alpha }_{T}}=\frac{{S_{T}}{K_{T}}-{I_{T}}{J_{T}}}{{w_{T}^{H}}{K_{T}}-{J_{T}^{2}}}\gamma ,\hspace{2em}{\widehat{\beta }_{T}}=\frac{{S_{T}}{J_{T}}-{w_{T}^{H}}{I_{T}}}{{w_{T}^{H}}{K_{T}}-{J_{T}^{2}}},\]
\[ {I_{T}}={\int _{0}^{T}}{P_{H}}(t)\hspace{0.1667em}d{S_{t}},\hspace{2em}{J_{T}}={\int _{0}^{T}}{P_{H}}(t)\hspace{0.1667em}d{w_{t}^{H}},\hspace{2em}{K_{T}}={\int _{0}^{T}}{\big({P_{H}}(t)\big)^{2}}\hspace{0.1667em}d{w_{t}^{H}}.\]
It is worth noting that using the definition of ${P_{H}}(t)$ one can easily represent ${J_{T}}$ in the following way
3 Auxiliary results
In this section we find exact and asymptotic distributions of the statistics ${S_{T}}$, ${I_{T}}$, ${J_{T}}$, ${K_{T}}$ and related random variables and vectors.
We start with the bivariate MGF of the vector $({S_{T}},{I_{T}})$. For the case $\beta >0$, it was derived in [26, Lemma 3.3]. However, the same proof is valid for the case $\beta <0$. The following result is a reformulation of [26, Lemma 3.3], obtained by applying the formula (44) from Appendix A.
Lemma 1.
The moment generating function of $({S_{T}},{I_{T}})$ equals
\[\begin{aligned}{}{m_{1}^{(\alpha ,\beta )}}({\xi _{1}},{\xi _{2}})& =\mathbb{E}\big[\exp \{{\xi _{1}}{S_{T}}+{\xi _{2}}{I_{T}}\}\big]\\ {} & ={D^{(\alpha ,\beta )}}{({\xi _{2}})^{-\frac{1}{2}}}\exp \Bigg\{\frac{1}{8{D^{(\alpha ,\beta )}}({\xi _{2}})}{\sum \limits_{i=1}^{4}}{A_{i}^{(\alpha ,\beta )}}({\xi _{1}},{\xi _{2}})-\frac{{\xi _{2}}T}{2}\Bigg\},\end{aligned}\]
where
(10)
\[\begin{aligned}{}{D^{(\alpha ,\beta )}}({\xi _{2}})& ={\bigg(1-\frac{{\xi _{2}}}{2\beta }\bigg)^{2}}+\frac{{\xi _{2}^{2}}}{4{\beta ^{2}}}{e^{-2\beta T}}+\bigg(\frac{{\xi _{2}}}{\beta }-\frac{{\xi _{2}^{2}}}{2{\beta ^{2}}}\bigg)\frac{(-\beta )\pi T}{4\sin \pi H}{e^{-\beta T}}\\ {} & \hspace{1em}\times \bigg[{I_{-H}}\bigg(-\frac{\beta T}{2}\bigg){I_{H-1}}\bigg(-\frac{\beta T}{2}\bigg)+{I_{1-H}}\bigg(-\frac{\beta T}{2}\bigg){I_{H}}\bigg(-\frac{\beta T}{2}\bigg)\bigg],\end{aligned}\](11)
\[\begin{aligned}{}{A_{1}^{(\alpha ,\beta )}}({\xi _{1}},{\xi _{2}})& ={\xi _{2}}\big({c_{1}}\big(\frac{\alpha }{\beta }\big){\xi _{1}}-{c_{2}}\big(\frac{\alpha }{\beta }\big){\xi _{2}}\big){(-\beta )^{H-1}}{T^{1-H}}{e^{-\frac{3\beta T}{2}}}{I_{1-H}}\big(-\frac{\beta T}{2}\big),\end{aligned}\](12)
\[\begin{aligned}{}{A_{2}^{(\alpha ,\beta )}}({\xi _{1}},{\xi _{2}})& =\big({\xi _{1}^{2}}{c_{3}}-{\xi _{1}}{\xi _{2}}{c_{4}}\big(\frac{\alpha }{\beta }\big)+{\xi _{2}^{2}}{c_{5}}\big(\frac{\alpha }{\beta }\big)\big){T^{2-2H}}{e^{-\beta T}}\\ {} & \hspace{1em}\times {I_{1-H}}\big(-\frac{\beta T}{2}\big){I_{H-1}}\big(-\frac{\beta T}{2}\big),\end{aligned}\](13)
\[\begin{aligned}{}{A_{3}^{(\alpha ,\beta )}}({\xi _{1}},{\xi _{2}})& ={\xi _{2}}({\xi _{2}}-2\beta ){c_{6}}\big(\frac{\alpha }{\beta }\big){(-\beta )^{2H-1}}T{e^{-\beta T}}{I_{1-H}}\big(-\frac{\beta T}{2}\big){I_{-H}}\big(-\frac{\beta T}{2}\big),\end{aligned}\](14)
\[\begin{aligned}{}{A_{4}^{(\alpha ,\beta )}}({\xi _{1}},{\xi _{2}})& =\big({c_{1}}\big(\frac{\alpha }{\beta }\big){\xi _{1}}-{c_{2}}\big(\frac{\alpha }{\beta }\big){\xi _{2}}\big)({\xi _{2}}-2\beta ){(-\beta )^{H-1}}\\ {} & \hspace{1em}\times {T^{1-H}}{e^{-\frac{\beta T}{2}}}{I_{1-H}}\big(-\frac{\beta T}{2}\big),\end{aligned}\]
\[\begin{array}{r@{\hskip0pt}l@{\hskip0pt}r@{\hskip0pt}l}\displaystyle {c_{1}}\big(\frac{\alpha }{\beta }\big)& \displaystyle =\big({x_{0}}-\frac{\alpha }{\beta }\big)4{\rho _{H}},\hspace{2em}& \displaystyle {c_{4}}\big(\frac{\alpha }{\beta }\big)& \displaystyle =\big({x_{0}}-\frac{\alpha }{\beta }\big){\rho _{H}}{2^{2H+1}}\varGamma (H),\\ {} \displaystyle {c_{2}}\big(\frac{\alpha }{\beta }\big)& \displaystyle ={\big({x_{0}}-\frac{\alpha }{\beta }\big)^{2}}\frac{{\lambda _{H}^{\ast }}{2^{2H+1}}{\rho _{H}^{2}}}{\varGamma (1-H)},\hspace{2em}& \displaystyle {c_{5}}\big(\frac{\alpha }{\beta }\big)& \displaystyle ={\big({x_{0}}-\frac{\alpha }{\beta }\big)^{2}}\frac{{\lambda _{H}^{\ast }}{2^{4H-1}}{\rho _{H}^{2}}\varGamma (H)}{\varGamma (1-H)},\\ {} \displaystyle {c_{3}}& \displaystyle =\frac{2\varGamma (H)\varGamma (1-H)}{{\lambda _{H}^{\ast }}},\hspace{2em}& \displaystyle {c_{6}}\big(\frac{\alpha }{\beta }\big)& \displaystyle ={\bigg({x_{0}}-\frac{\alpha }{\beta }\bigg)^{2}}2{\lambda _{H}^{\ast }}{\rho _{H}^{2}},\end{array}\]
\[ {\lambda _{H}^{\ast }}=\frac{{\lambda _{H}}}{2-2H},\hspace{2em}{\rho _{H}}=\frac{\sqrt{\pi }\varGamma (3/2-H)}{\gamma {\kappa _{H}}}.\]
The domain of the function ${m_{1}^{(\alpha ,\beta )}}$ equals $\{({\xi _{1}},{\xi _{2}})\in {\mathbb{R}^{2}}:{D^{(\alpha ,\beta )}}({\xi _{2}})>0\}$.
The following lemma gives a joint MGF of $({S_{T}},{I_{T}},{J_{T}},{K_{T}})$.
Lemma 2.
The moment generating function of $({S_{T}},{I_{T}},{J_{T}},{K_{T}})$ equals
\[\begin{aligned}{}{m_{2}}({\theta _{1}},{\theta _{2}},{\theta _{3}},{\theta _{4}})& =\mathbb{E}\big[\exp \{{\theta _{1}}{S_{T}}+{\theta _{2}}{I_{T}}+{\theta _{3}}{J_{T}}+{\theta _{4}}{K_{T}}\}\big]\\ {} & ={m_{1}^{({\alpha _{1}},{\beta _{1}})}}\bigg({\theta _{1}}+\frac{\alpha -{\alpha _{1}}}{\gamma },{\theta _{2}}-\beta +{\beta _{1}}\bigg)\exp \bigg\{\frac{{\alpha _{1}^{2}}-{\alpha ^{2}}}{2{\gamma ^{2}}}{w_{T}^{H}}\bigg\},\end{aligned}\]
where
\[ {\alpha _{1}}=-\frac{\gamma {\theta _{3}}+\alpha \beta }{\sqrt{{\beta ^{2}}-2{\theta _{4}}}},\hspace{2em}{\beta _{1}}=-\sqrt{{\beta ^{2}}-2{\theta _{4}}}.\]
The domain of the function ${m_{2}}$ equals
\[ \left\{({\theta _{1}},{\theta _{2}},{\theta _{3}},{\theta _{4}})\in {\mathbb{R}^{4}}:{\theta _{4}}<{\beta ^{2}}/2,{D^{(\alpha ,\beta )}}\left({\theta _{2}}-\beta -\sqrt{{\beta ^{2}}-2{\theta _{4}}}\right)>0\right\},\]
where ${D^{(\alpha ,\beta )}}$ is defined in (10).
Lemma 3.
Under stated conditions the process ${S_{T}}$ has the normal asymptotic distribution as $T\to \infty $, namely
Proof.
We obtain the distribution via MGF. Using Lemma 1 we have
\[ \mathbb{E}\big[\exp \big\{\theta {T^{H-1/2}}{e^{\beta T}}{S_{T}}\big\}\big]={m_{1}}\big(\theta {T^{H-1/2}}{e^{\beta T}},0\big).\]
Taking each term of the function ${m_{1}}$ separately with ${\xi _{1}}=\theta {T^{H-1/2}}{e^{\beta T}}$ and ${\xi _{2}}=0$ and applying (45) we obtain that $D({\xi _{2}})=1$, ${A_{1}}({\xi _{1}},{\xi _{2}})={A_{3}}({\xi _{1}},{\xi _{2}})=0$,
\[\begin{aligned}{}{A_{2}}({\xi _{1}},{\xi _{2}})& ={\xi _{1}^{2}}{c_{3}}{T^{2-2H}}{e^{-\beta T}}{I_{1-H}}\bigg(-\frac{\beta T}{2}\bigg){I_{H-1}}\bigg(-\frac{\beta T}{2}\bigg)\\ {} & ={\theta ^{2}}{c_{3}}T{e^{\beta T}}{I_{1-H}}\bigg(-\frac{\beta T}{2}\bigg){I_{H-1}}\bigg(-\frac{\beta T}{2}\bigg)\\ {} & \to \frac{{c_{3}}}{\pi (-\beta )}{\theta ^{2}},\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty ,\end{aligned}\]
and
\[\begin{aligned}{}{A_{4}}({\xi _{1}},{\xi _{2}})& ={\xi _{1}}2{c_{1}}\big(\frac{\alpha }{\beta }\big){(-\beta )^{H}}{T^{1-H}}{e^{-\frac{\beta T}{2}}}{I_{1-H}}\bigg(-\frac{\beta T}{2}\bigg)\\ {} & =\theta 2{c_{1}}\big(\frac{\alpha }{\beta }\big){(-\beta )^{H}}{T^{1/2}}{e^{\frac{\beta T}{2}}}{I_{1-H}}\bigg(-\frac{\beta T}{2}\bigg)\\ {} & \to \frac{2{c_{1}}(\frac{\alpha }{\beta }){(-\beta )^{H-1/2}}}{\sqrt{\pi }}\theta ,\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty .\end{aligned}\]
Hence
\[ \mathbb{E}\big[\exp \big\{\theta {T^{H-1/2}}{e^{\beta T}}{S_{T}}\big\}\big]\to \exp \bigg\{\frac{{c_{3}}}{8\pi (-\beta )}{\theta ^{2}}+\frac{{c_{1}}(\frac{\alpha }{\beta }){(-\beta )^{H-1/2}}}{4\sqrt{\pi }}\theta \bigg\},\]
as $T\to \infty $. This means that
\[ {T^{H-1/2}}{e^{\beta T}}{S_{T}}\xrightarrow{d}\mathcal{N}\bigg(\frac{{c_{1}}(\frac{\alpha }{\beta }){(-\beta )^{H-1/2}}}{4\sqrt{\pi }},\frac{{c_{3}}}{4\pi (-\beta )}\bigg),\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty ,\]
which is equivalent to (15). □The following result is crucial for the derivation of the joint asymptotic distribution of MLE.
Lemma 4.
The vector of main components of the MLE has the following behavior
where ξ, η, ζ are independent and $\xi \stackrel{d}{=}\mathcal{N}(0,{\lambda _{H}^{-1}})$, $\eta \stackrel{d}{=}\mathcal{N}(0,1)$,
(16)
\[ \left(\begin{array}{c}{T^{H-1}}({S_{T}}+\beta {J_{T}}-\frac{\alpha }{\gamma }{w_{T}^{H}})\\ {} {e^{\beta T}}({I_{T}}+\beta {K_{T}})\\ {} {e^{2\beta T}}{K_{T}}\end{array}\right)\xrightarrow{d}\left(\begin{array}{c}\xi \\ {} \eta \zeta \\ {} {\zeta ^{2}}\end{array}\right),\hspace{1em}\textit{as}\hspace{2.5pt}T\to \infty ,\]Proof.
We again obtain the stated asymptotic distribution via MGF of the presented vector. It could be easily reduced to already studied MGF. That said, using Lemma 2, we have
where
and
Note that ${\alpha _{1}}(T)\to \alpha $ and ${\beta _{1}}(T)\to \beta $, as $T\to \infty $. Moreover, the arguments of the function ${m_{1}^{({\alpha _{1}}(T),{\beta _{1}}(T))}}$ in (18) have the following asymptotic behavior:
as $T\to \infty $. Further, inserting (22) into (10), and applying the expansion (45) from Appendix A, we obtain
It follows from (21), (22) that
Using this relation, (22) and (45), we get from (11) that
It follows from (21), (22) that
(18)
\[\begin{aligned}{}& \mathbb{E}\bigg[\exp \bigg\{{\theta _{1}}{T^{H-1}}\bigg({S_{T}}+\beta {J_{T}}-\frac{\alpha }{\gamma }{w_{T}^{H}}\bigg)+{\theta _{2}}{e^{\beta T}}({I_{T}}+\beta {K_{T}})+{\theta _{3}}{e^{2\beta T}}{K_{T}}\bigg\}\bigg]\\ {} & \hspace{1em}={m_{2}}\big({\theta _{1}}{T^{H-1}},{\theta _{2}}{e^{\beta T}},{\theta _{1}}\beta {T^{H-1}},{\theta _{2}}\beta {e^{\beta T}}+{\theta _{3}}{e^{2\beta T}}\big)\exp \bigg\{-\frac{{\theta _{1}}\alpha }{\gamma }{T^{H-1}}{w_{T}^{H}}\bigg\}\\ {} & \hspace{1em}={m_{1}^{({\alpha _{1}}(T),{\beta _{1}}(T))}}\bigg({\theta _{1}}{T^{H-1}}+\frac{\alpha -{\alpha _{1}}(T)}{\gamma },{\theta _{2}}{e^{\beta T}}-\beta +{\beta _{1}}(T)\bigg)\\ {} & \hspace{2em}\times \exp \bigg\{\frac{{\alpha _{1}}{(T)^{2}}-{\alpha ^{2}}}{2{\gamma ^{2}}}{w_{T}^{H}}-\frac{{\theta _{1}}\alpha }{\gamma }{T^{H-1}}{w_{T}^{H}}\bigg\},\end{aligned}\]
\[\begin{aligned}{}{\alpha _{1}}(T)& =\frac{\beta \gamma {\theta _{1}}{T^{H-1}}+\alpha \beta }{-\sqrt{{\beta ^{2}}-2({\theta _{2}}\beta {e^{\beta T}}+{\theta _{3}}{e^{2\beta T}})}},\\ {} {\beta _{1}}(T)& =-\sqrt{{\beta ^{2}}-2\big({\theta _{2}}\beta {e^{\beta T}}+{\theta _{3}}{e^{2\beta T}}\big)}.\end{aligned}\]
Applying the Taylor series expansion, we get as $T\to \infty $
(19)
\[\begin{aligned}{}{\alpha _{1}}(T)& =\big(\gamma {\theta _{1}}{T^{H-1}}+\alpha \big){\bigg[1-2\bigg(\frac{{\theta _{2}}}{\beta }{e^{\beta T}}+\frac{{\theta _{3}}}{{\beta ^{2}}}{e^{2\beta T}}\bigg)\bigg]^{-1/2}}\\ {} & =\big(\gamma {\theta _{1}}{T^{H-1}}+\alpha \big)\bigg[1+\bigg(\frac{{\theta _{2}}}{\beta }{e^{\beta T}}+\frac{{\theta _{3}}}{{\beta ^{2}}}{e^{2\beta T}}\bigg)+O\big({e^{2\beta T}}\big)\bigg]\\ {} & =\alpha +\gamma {\theta _{1}}{T^{H-1}}+\frac{{\theta _{2}}\alpha }{\beta }{e^{\beta T}}+\frac{{\theta _{2}}\gamma {\theta _{1}}}{\beta }{T^{H-1}}{e^{\beta T}}+O\big({e^{2\beta T}}\big)\end{aligned}\](20)
\[\begin{aligned}{}{\beta _{1}}(T)& =\beta {\bigg[1-2\bigg({\theta _{2}}\frac{1}{\beta }{e^{\beta T}}+{\theta _{3}}\frac{1}{{\beta ^{2}}}{e^{2\beta T}}\bigg)\bigg]^{1/2}}\\ {} & =\beta \bigg[1-\bigg({\theta _{2}}\frac{1}{\beta }{e^{\beta T}}+{\theta _{3}}\frac{1}{{\beta ^{2}}}{e^{2\beta T}}\bigg)\\ {} & \hspace{2em}-\frac{1}{2}{\bigg({\theta _{2}}\frac{1}{\beta }{e^{\beta T}}+{\theta _{3}}\frac{1}{{\beta ^{2}}}{e^{2\beta T}}\bigg)^{2}}+O\big({e^{3\beta T}}\big)\bigg]\\ {} & =\beta -{\theta _{2}}{e^{\beta T}}+\frac{{\theta _{2}^{2}}+2{\theta _{3}}}{2(-\beta )}{e^{2\beta T}}+O\big({e^{3\beta T}}\big).\end{aligned}\](23)
\[\begin{aligned}{}& {D^{({\alpha _{1}}(T),{\beta _{1}}(T))}}\big({\xi _{2}}(T)\big)\\ {} & \hspace{1em}={\bigg(1-\frac{({\theta _{2}^{2}}+2{\theta _{3}}){e^{2\beta T}}}{4{\beta _{1}}(T)(-\beta )}\bigg)^{2}}+\frac{{({\theta _{2}^{2}}+2{\theta _{3}})^{2}}{e^{4\beta T}}}{16{\beta _{1}}{(T)^{2}}{\beta ^{2}}}{e^{-2{\beta _{1}}(T)T}}\\ {} & \hspace{1em}\hspace{1em}+\bigg(\frac{({\theta _{2}^{2}}+2{\theta _{3}}){e^{2\beta T}}}{2{\beta _{1}}(T)(-\beta )}-\frac{{({\theta _{2}^{2}}+2{\theta _{3}})^{2}}{e^{4\beta T}}}{8{\beta _{1}}{(T)^{2}}{\beta ^{2}}}\bigg)\frac{(-{\beta _{1}}(T))\pi T}{4\sin \pi H}{e^{-{\beta _{1}}(T)T}}\\ {} & \hspace{1em}\hspace{1em}\times \bigg[{I_{-H}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg){I_{H-1}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg)\\ {} & \hspace{1em}\hspace{1em}+{I_{1-H}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg){I_{H}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg)\bigg]+O\big({e^{\beta T}}\big)\\ {} & \hspace{1em}\sim {\bigg(1-\frac{({\theta _{2}^{2}}+2{\theta _{3}}){e^{2\beta T}}}{4{\beta _{1}}(T)(-\beta )}\bigg)^{2}}+\frac{{({\theta _{2}^{2}}+2{\theta _{3}})^{2}}{e^{4\beta T}}}{16{\beta _{1}}{(T)^{2}}{\beta ^{2}}}{e^{-2{\beta _{1}}(T)T}}\\ {} & \hspace{1em}\hspace{1em}+\bigg(\frac{({\theta _{2}^{2}}+2{\theta _{3}}){e^{2\beta T}}}{2{\beta _{1}}(T)(-\beta )}-\frac{{({\theta _{2}^{2}}+2{\theta _{3}})^{2}}{e^{4\beta T}}}{8{\beta _{1}}{(T)^{2}}{\beta ^{2}}}\bigg)\frac{1}{2\sin \pi H}{e^{-2{\beta _{1}}(T)T}}+O\big({e^{\beta T}}\big)\\ {} & \hspace{1em}\to 1-\frac{{\theta _{2}^{2}}+2{\theta _{3}}}{4{\beta ^{2}}\sin \pi H},\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty .\end{aligned}\](24)
\[ {c_{1}}\big(\frac{{\alpha _{1}}(T)}{{\beta _{1}}(T)}\big){\xi _{1}}(T)-{c_{2}}\big(\frac{{\alpha _{1}}(T)}{{\beta _{1}}(T)}\big){\xi _{2}}(T)\sim {c_{1}}\big(\frac{\alpha }{\beta }\big)\frac{{\theta _{2}}\alpha }{-\beta \gamma }{e^{\beta T}},\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty .\](25)
\[\begin{aligned}{}& {A_{1}^{({\alpha _{1}}(T),{\beta _{1}}(T))}}\big({\xi _{1}}(T),{\xi _{2}}(T)\big)\\ {} & \hspace{1em}={\xi _{2}}(T)\big({c_{1}}\big(\frac{{\alpha _{1}}(T)}{{\beta _{1}}(T)}\big){\xi _{1}}(T)-{c_{2}}\big(\frac{{\alpha _{1}}(T)}{{\beta _{1}}(T)}\big){\xi _{2}}(T)\big)\\ {} & \hspace{2em}\times {\big(-{\beta _{1}}(T)\big)^{H-1}}{T^{1-H}}{e^{-\frac{3{\beta _{1}}(T)T}{2}}}{I_{1-H}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg)\\ {} & \hspace{1em}\sim \frac{{\theta _{2}^{2}}+2{\theta _{3}}}{2(-\beta )}{e^{2\beta T}}{c_{1}}\big(\frac{\alpha }{\beta }\big)\frac{\alpha }{(-\beta )\gamma }{\theta _{2}}{e^{\beta T}}{\big(-{\beta _{1}}(T)\big)^{H-1}}{T^{1-H}}\\ {} & \hspace{2em}\times {e^{-\frac{3{\beta _{1}}(T)T}{2}}}{I_{1-H}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg)\\ {} & \hspace{1em}\sim \frac{\alpha {\theta _{2}}({\theta _{2}^{2}}+2{\theta _{3}})}{2{\beta ^{2}}\gamma }{e^{3\beta T}}{c_{1}}\big(\frac{\alpha }{\beta }\big){\big(-{\beta _{1}}(T)\big)^{H-3/2}}\frac{1}{\sqrt{\pi }}{T^{1/2-H}}{e^{-2{\beta _{1}}(T)T}}\\ {} & \hspace{1em}=O\big({T^{1/2-H}}{e^{\beta T}}\big)\to 0,\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty .\end{aligned}\]
\[\begin{array}{l}\displaystyle {\xi _{1}}{(T)^{2}}=\frac{{\theta _{2}^{2}}{\alpha ^{2}}}{{\beta ^{2}}{\gamma ^{2}}}{e^{2\beta T}}+O\big({T^{H-1}}{e^{2\beta T}}\big),\\ {} \displaystyle {\xi _{1}}(T){\xi _{2}}(T)=O\big({e^{3\beta T}}\big),\hspace{2em}{\xi _{2}}{(T)^{2}}=O\big({e^{4\beta T}}\big),\end{array}\]
as $T\to \infty $. Therefore, by (12) and (45) we obtain
(26)
\[\begin{aligned}{}& {A_{2}^{({\alpha _{1}}(T),{\beta _{1}}(T))}}\big({\xi _{1}}(T),{\xi _{2}}(T)\big)\\ {} & \hspace{1em}=\big({\xi _{1}}{(T)^{2}}{c_{3}}-{\xi _{1}}(T){\xi _{2}}(T){c_{4}}\big(\frac{{\alpha _{1}}(T)}{{\beta _{1}}(T)}\big)+{\xi _{2}}{(T)^{2}}{c_{5}}\big(\frac{{\alpha _{1}}(T)}{{\beta _{1}}(T)}\big)\big)\\ {} & \hspace{2em}\times {T^{2-2H}}{e^{-{\beta _{1}}(T)T}}{I_{1-H}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg){I_{H-1}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg)\\ {} & \hspace{1em}\sim {c_{3}}\frac{{\alpha ^{2}}}{{\beta ^{2}}{\gamma ^{2}}}{\theta _{2}^{2}}{e^{2\beta T}}{T^{2-2H}}{e^{-{\beta _{1}}(T)T}}{I_{1-H}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg){I_{H-1}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg)\\ {} & \hspace{1em}\sim {e^{2\beta T}}{c_{3}}\frac{{\alpha ^{2}}}{{\beta ^{2}}{\gamma ^{2}}}{\theta _{2}^{2}}\frac{1}{(-\beta )\pi }{T^{1-2H}}{e^{-2{\beta _{1}}(T)T}}\\ {} & \hspace{1em}=O\big({T^{1-2H}}\big)\to 0,\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty .\end{aligned}\]Note that ${\xi _{2}}(T)-2{\beta _{1}}(T)\to -2\beta $, as $T\to \infty $, by (20) and (22). Hence, by (13) and (45),
Similarly, using (14), (24) and (45), we get
Also, (19) implies
since ${w_{T}^{H}}={\lambda _{H}^{-1}}{T^{2-2H}}$.
(27)
\[\begin{aligned}{}& {A_{3}^{({\alpha _{1}}(T),{\beta _{1}}(T))}}\big({\xi _{1}}(T),{\xi _{2}}(T)\big)\\ {} & \hspace{1em}={\xi _{2}}(T)\big({\xi _{2}}(T)-2{\beta _{1}}(T)\big){c_{6}}\big(\frac{{\alpha _{1}}(T)}{{\beta _{1}}(T)}\big)\\ {} & \hspace{2em}\times {\big(-{\beta _{1}}(T)\big)^{2H-1}}T{e^{-{\beta _{1}}(T)T}}{I_{1-H}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg){I_{-H}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg)\\ {} & \hspace{1em}\sim \frac{{\theta _{2}^{2}}+2{\theta _{3}}}{-2\beta }{e^{2\beta T}}(-2\beta ){c_{6}}\big(\frac{\alpha }{\beta }\big){\big(-{\beta _{1}}(T)\big)^{2H-2}}\frac{1}{\pi }{e^{-2{\beta _{1}}(T)T}}\\ {} & \hspace{1em}\to \frac{{c_{6}}(\frac{\alpha }{\beta }){(-\beta )^{2H-2}}}{\pi }\big({\theta _{2}^{2}}+2{\theta _{3}}\big),\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty .\end{aligned}\](28)
\[\begin{aligned}{}& {A_{4}^{({\alpha _{1}}(T),{\beta _{1}}(T))}}\big({\xi _{1}}(T),{\xi _{2}}(T)\big)\\ {} & \hspace{1em}=\big({c_{1}}\big(\frac{{\alpha _{1}}(T)}{{\beta _{1}}(T)}\big){\xi _{1}}(T)-{c_{2}}\big(\frac{{\alpha _{1}}(T)}{{\beta _{1}}(T)}\big){\xi _{2}}(T)\big)\\ {} & \hspace{2em}\times \big({\xi _{2}}(T)-2{\beta _{1}}(T)\big){\big(-{\beta _{1}}(T)\big)^{H-1}}{T^{1-H}}{e^{-\frac{{\beta _{1}}(T)T}{2}}}{I_{1-H}}\bigg(-\frac{{\beta _{1}}(T)T}{2}\bigg)\\ {} & \hspace{1em}\sim {c_{1}}\big(\frac{\alpha }{\beta }\big)\frac{{\theta _{2}}\alpha }{-\beta \gamma }{e^{\beta T}}(-2\beta ){\big(-{\beta _{1}}(T)\big)^{H-3/2}}\frac{1}{\sqrt{\pi }}{T^{1/2-H}}{e^{-{\beta _{1}}(T)T}}\\ {} & \hspace{1em}=O\big({T^{1/2-H}}\big)\to 0,\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty .\end{aligned}\]
\[ {\alpha _{1}}{(T)^{2}}={\alpha ^{2}}+2\alpha \gamma {\theta _{1}}{T^{H-1}}+{\gamma ^{2}}{\theta _{1}^{2}}{T^{2H-2}}+O\big({e^{\beta T}}\big),\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty .\]
Then, for the expression under the exponential function in (18) we have
(29)
\[\begin{aligned}{}& \frac{{\alpha _{1}}{(T)^{2}}-{\alpha ^{2}}}{2{\gamma ^{2}}}{w_{T}^{H}}-\frac{{\theta _{1}}\alpha }{\gamma }{T^{H-1}}{w_{T}^{H}}=\frac{1}{2}{\theta _{1}^{2}}{T^{2H-2}}{w_{T}^{H}}+O\big({w_{T}^{H}}{e^{\beta T}}\big)\\ {} & \hspace{1em}=\frac{1}{2}{\theta _{1}^{2}}{\lambda _{H}^{-1}}+O\big({w_{T}^{H}}{e^{\beta T}}\big)\to \frac{1}{2}{\theta _{1}^{2}}{\lambda _{H}^{-1}},\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty ,\end{aligned}\]Thus, inserting (23) and (25)–(29) into (18), we arrive at
\[\begin{aligned}{}& \mathbb{E}\bigg[\exp \bigg\{{\theta _{1}}{T^{H-1}}\bigg({S_{T}}+\beta {J_{T}}-\frac{\alpha }{\gamma }{w_{T}^{H}}\bigg)+{\theta _{2}}{e^{\beta T}}({I_{T}}+\beta {K_{T}})+{\theta _{3}}{e^{2\beta T}}{K_{T}}\bigg\}\bigg]\\ {} & \hspace{1em}\to \exp \bigg\{\frac{1}{2}{\theta _{1}^{2}}{\lambda _{H}^{-1}}\bigg\}{\bigg(1-\frac{{\theta _{2}^{2}}+2{\theta _{3}}}{4{\beta ^{2}}\sin \pi H}\bigg)^{-1/2}}\\ {} & \hspace{2em}\times \exp \bigg\{\frac{{c_{6}}(\frac{\alpha }{\beta }){(-\beta )^{2H-2}}({\theta _{2}^{2}}+2{\theta _{3}})}{8\pi \Big(1-\frac{{\theta _{2}^{2}}+2{\theta _{3}}}{4{\beta ^{2}}\sin \pi H}\Big)}\bigg\},\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty .\end{aligned}\]
We see that the limit is a product of MGF of the normal random variable $\xi \stackrel{d}{=}\mathcal{N}(0,{\lambda _{H}^{-1}})$ and MGF of the random vector $\left(\genfrac{}{}{0.0pt}{}{\eta \zeta }{{\zeta ^{2}}}\right)$, where the random variables $\eta \stackrel{d}{=}\mathcal{N}(0,1)$ and $\zeta \stackrel{d}{=}\mathcal{N}\bigg(\frac{\sqrt{{c_{6}}(\frac{\alpha }{\beta })}{(-\beta )^{H-1}}}{2\sqrt{\pi }},\frac{1}{4{\beta ^{2}}\sin \pi H}\bigg)$ are independent, see Lemma 5 in Appendix B. This concludes the proof, since ${c_{6}}(\frac{\alpha }{\beta })={({x_{0}}-\frac{\alpha }{\beta })^{2}}2{\lambda _{H}^{\ast }}{\rho _{H}^{2}}$. □Remark 2.
In fact, $\mathcal{N}(0,{\lambda _{H}^{-1}})$ is an exact distribution of the random variable ${T^{H-1}}({S_{T}}+\beta {J_{T}}-\frac{\alpha }{\gamma }{w_{T}^{H}})$ for any T. It can be easily seen from the above proof by putting ${\theta _{2}}={\theta _{3}}=0$ (then ${\alpha _{1}}(T)=\alpha +\gamma {\theta _{1}}{T^{H-1}}$, ${\beta _{1}}(T)=\beta $, ${\xi _{1}}(T)={\xi _{2}}(T)=0$).
The following series of corollaries will describe asymptotic distributions of minor components of the MLE.
First, by considering the convergence of the first component of the random vector in (16), we immediately get the following result.
Next, we focus on the process ${I_{T}}$. In order to obtain its asymptotic behavior it suffices to write
and then apply (16).
Finally, the asymptotic behavior of ${J_{T}}$ can be easily derived using Lemma 3, Corollary 1 and the identity
\[ -\beta {T^{H-\frac{1}{2}}}{e^{\beta T}}{J_{T}}={T^{H-\frac{1}{2}}}{e^{\beta T}}{S_{T}}-{T^{H-\frac{1}{2}}}{e^{\beta T}}({S_{T}}+\beta {J_{T}}).\]
4 Main results
Now we are ready to prove the main result of the article.
Theorem 1.
Let $\beta <0$, $H\in (1/2,1)$. Then
where $\nu \stackrel{d}{=}\mathcal{N}(0,{\lambda _{H}}{\gamma ^{2}})$, $\eta \stackrel{d}{=}\mathcal{N}(0,1)$, and
are independent random variables. In particular, the estimators ${\widehat{\alpha }_{T}}$ and ${\widehat{\beta }_{T}}$ are asymptotically independent.
(30)
\[ \left(\begin{array}{c}{T^{1-H}}({\widehat{\alpha }_{T}}-\alpha )\\ {} {e^{-\beta T}}({\widehat{\beta }_{T}}-\beta )\end{array}\right)\xrightarrow{d}\left(\begin{array}{c}\nu \\ {} \frac{\eta }{\zeta }\end{array}\right),\hspace{1em}T\to \infty ,\](31)
\[ \zeta \stackrel{d}{=}\mathcal{N}\bigg(\frac{({x_{0}}-\frac{\alpha }{\beta }){\rho _{H}}\sqrt{{\lambda _{H}^{\ast }}}{(-\beta )^{H-1}}}{\sqrt{2\pi }},\frac{1}{4{\beta ^{2}}\sin \pi H}\bigg)\]Proof.
Using (9) and the equality ${T^{1-H}}={\lambda _{H}}{w_{T}^{H}}{T^{H-1}}$, we can write
Note that by Corollary 3 and Lemma 4, we see that ${J_{T}}={O_{\mathsf{P}}}({T^{\frac{1}{2}-H}}{e^{-\beta T}})$, ${I_{T}}+\beta {K_{T}}={O_{\mathsf{P}}}({e^{-\beta T}})$, and ${e^{2\beta T}}{K_{T}}-\frac{1}{{w_{T}^{H}}}{e^{2\beta T}}{J_{T}^{2}}\xrightarrow{d}{\zeta ^{2}}$, as $T\to \infty $. Consequently, the second term in the right-hand side of (32) converges to zero in probability.
(32)
\[\begin{aligned}{}{T^{1-H}}({\widehat{\alpha }_{T}}-\alpha )& ={\lambda _{H}}{w_{T}^{H}}{T^{H-1}}\bigg(\frac{{S_{T}}{K_{T}}-{I_{T}}{J_{T}}}{{w_{T}^{H}}{K_{T}}-{J_{T}^{2}}}\gamma -\alpha \bigg)\\ {} & ={\lambda _{H}}{w_{T}^{H}}{T^{H-1}}\frac{\gamma {S_{T}}{K_{T}}-\gamma {I_{T}}{J_{T}}-\alpha {w_{T}^{H}}{K_{T}}+\alpha {J_{T}^{2}}}{{w_{T}^{H}}{K_{T}}-{J_{T}^{2}}}\\ {} & =\frac{{e^{2\beta T}}{K_{T}}\gamma {\lambda _{H}}{T^{H-1}}({S_{T}}+\beta {J_{T}}-\frac{\alpha }{\gamma }{w_{T}^{H}})}{{e^{2\beta T}}{K_{T}}-\frac{1}{{w_{T}^{H}}}{e^{2\beta T}}{J_{T}^{2}}}\\ {} & \hspace{1em}+\frac{-\gamma {\lambda _{H}}{T^{H-1}}{e^{2\beta T}}{J_{T}}({I_{T}}+\beta {K_{T}})+\alpha {\lambda _{H}}{T^{H-1}}{e^{2\beta T}}{J_{T}^{2}}}{{e^{2\beta T}}{K_{T}}-\frac{1}{{w_{T}^{H}}}{e^{2\beta T}}{J_{T}^{2}}}.\end{aligned}\]Further, by (9),
Corollaries 1 and 3 imply that ${e^{\beta T}}{J_{T}}\frac{1}{{w_{T}^{H}}}({S_{T}}+\beta {J_{T}})$ converges to zero in probability. Then applying Lemma 4 and Slutsky’s theorem, from (32), (33) we get the convergence (30). □
(33)
\[\begin{aligned}{}{e^{-\beta T}}({\widehat{\beta }_{T}}-\beta )& ={e^{-\beta T}}\bigg(\frac{{S_{T}}{J_{T}}-{w_{T}^{H}}{I_{T}}}{{w_{T}^{H}}{K_{T}}-{J_{T}^{2}}}-\beta \bigg)\\ {} & ={e^{-\beta T}}\frac{{S_{T}}{J_{T}}-{w_{T}^{H}}{I_{T}}-\beta {w_{T}^{H}}{K_{T}}+\beta {J_{T}^{2}}}{{w_{T}^{H}}{K_{T}}-{J_{T}^{2}}}\\ {} & =\frac{-{e^{\beta T}}({I_{T}}+\beta {K_{T}})+{e^{\beta T}}{J_{T}}\frac{1}{{w_{T}^{H}}}({S_{T}}+\beta {J_{T}})}{{e^{2\beta T}}{K_{T}}-\frac{1}{{w_{T}^{H}}}{e^{2\beta T}}{J_{T}^{2}}}.\end{aligned}\]Remark 3.
Unlike the ergodic case (studied in [26]), in the non-ergodic case the initial value ${x_{0}}$ affects the asymptotic bias of ${\widehat{\beta }_{T}}$. If $\beta <0$, then the deterministic term $({x_{0}}-\frac{\alpha }{\beta }){e^{-\beta t}}$ in (4) does not converge to zero and, moreover, has the same asymptotic order $O({e^{-\beta T}})$ as the stochastic term $\gamma {\textstyle\int _{0}^{t}}{e^{-\beta (t-s)}}d{B_{s}^{H}}$. This implies that the asymptotic behavior of the statistics ${S_{T}}$, ${I_{T}}$, ${J_{T}}$, and ${K_{T}}$ depends on ${x_{0}}$. A similar dependence on initial condition holds for the non-ergodic Ornstein–Uhlenbeck model driven by the Brownian motion (see [11] and [23, Prop. 3.46]) and some explosive autoregressive models [2, 35, 37].
Remark 4.
The model (1) with ${x_{0}}=\frac{\alpha }{\beta }$ was considered in [33, Th. 5.2]. In this particular case we have $\zeta \stackrel{d}{=}\mathcal{N}(0,\frac{1}{4{\beta ^{2}}\sin \pi H})$. Consequently,
\[ \frac{{e^{-\beta T}}}{2\beta }({\widehat{\beta }_{T}}-\beta )\xrightarrow{d}\frac{X\sqrt{\sin \pi H}}{Y},\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty ,\]
where X and Y are two independent $\mathcal{N}(0,1)$ random variables. This completely agrees with [33, Th. 5.2].Remark 5 (Alternative parameterization).
An alternative specification of the fractional Vasicek model is
For the model (34), the MLEs of the parameters μ and κ have the following form [26]:
where $\widetilde{\nu }\stackrel{d}{=}\mathcal{N}(0,\frac{{\lambda _{H}}{\gamma ^{2}}}{{\kappa ^{2}}})$, $\eta \stackrel{d}{=}\mathcal{N}(0,1)$, and
are independent random variables.
(34)
\[ d{X_{t}}=\kappa (\mu -{X_{t}})dt+\gamma d{B_{t}^{H}},\hspace{1em}t\in [0,T],\hspace{1em}{X_{0}}={x_{0}}.\]
\[ {\widehat{\mu }_{T}}=\frac{{S_{T}}{K_{T}}-{I_{T}}{J_{T}}}{{S_{T}}{J_{T}}-{w_{T}^{H}}{I_{T}}}\gamma ,\hspace{2em}{\widehat{\kappa }_{T}}=\frac{{S_{T}}{J_{T}}-{w_{T}^{H}}{I_{T}}}{{w_{T}^{H}}{K_{T}}-{J_{T}^{2}}}.\]
One can establish the following result: if $\kappa <0$ and $H\in (1/2,1)$, then
(35)
\[ \left(\begin{array}{c}{T^{1-H}}({\widehat{\mu }_{T}}-\mu )\\ {} {e^{-\kappa T}}({\widehat{\kappa }_{T}}-\kappa )\end{array}\right)\xrightarrow{d}\left(\begin{array}{c}\widetilde{\nu }\\ {} \eta /\widetilde{\zeta }\end{array}\right),\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty ,\]The proof of (35) is carried out by the delta-method. By Taylor’s theorem, for the function $g(x,y)=\frac{x}{y}$, we have as $(x,y)\to (\alpha ,\beta )$
Multiplying both sides of (36) by ${T^{1-H}}$, and putting $x={\widehat{\alpha }_{T}}$, $y={\widehat{\beta }_{T}}$, we get
where
as $T\to \infty $, since ${T^{1-H}}({\widehat{\beta }_{T}}-\beta )\xrightarrow{\mathsf{P}}0$ and ${T^{1-H}}({\widehat{\alpha }_{T}}-\alpha )\xrightarrow{d}\nu $ due to (30).
(36)
\[ g(x,y)-g(\alpha ,\beta )=\frac{1}{\beta }(x-\alpha )-\frac{\alpha }{{\beta ^{2}}}(y-\beta )+o\Big(\sqrt{{(x-\alpha )^{2}}+{(y-\beta )^{2}}}\hspace{0.1667em}\Big).\](37)
\[ {T^{1-H}}\bigg(\frac{{\widehat{\alpha }_{T}}}{{\widehat{\beta }_{T}}}-\frac{\alpha }{\beta }\bigg)=\frac{1}{\beta }{T^{1-H}}({\widehat{\alpha }_{T}}-\alpha )+{R_{T}},\](38)
\[ {R_{T}}=-\frac{\alpha }{{\beta ^{2}}}{T^{1-H}}({\widehat{\beta }_{T}}-\beta )+{o_{\mathsf{P}}}\Big({T^{1-H}}\sqrt{{({\widehat{\alpha }_{T}}-\alpha )^{2}}+{({\widehat{\beta }_{T}}-\beta )^{2}}}\hspace{0.1667em}\Big)\xrightarrow{\mathsf{P}}0,\]Finally, by Slutsky’s theorem, from (30), (37) and (38) we obtain the convergence
\[ \left(\begin{array}{c}{T^{1-H}}(\frac{{\widehat{\alpha }_{T}}}{{\widehat{\beta }_{T}}}-\frac{\alpha }{\beta })\\ {} {e^{-\beta T}}({\widehat{\beta }_{T}}-\beta )\end{array}\right)=\left(\begin{array}{c}\frac{1}{\beta }{T^{1-H}}({\widehat{\alpha }_{T}}-\alpha )\\ {} {e^{-\beta T}}({\widehat{\beta }_{T}}-\beta )\end{array}\right)+\left(\begin{array}{c}{R_{T}}\\ {} 0\end{array}\right)\xrightarrow{d}\left(\begin{array}{c}\frac{\nu }{\beta }\\ {} \frac{\eta }{\zeta }\end{array}\right),\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty ,\]
which is equivalent to (35), since ${\widehat{\mu }_{T}}=\frac{{\widehat{\alpha }_{T}}}{{\widehat{\beta }_{T}}}$, ${\widehat{\kappa }_{T}}={\widehat{\beta }_{T}}$, $\mu =\frac{\alpha }{\beta }$, $\kappa =\beta $.Now let us consider the situation when one of the parameters is known. In this case we can obtain strong consistency of the corresponding MLEs (instead of weak one) by applying the strong law of large numbers for martingales, see, e.g., [24, Theorem 2.6.10].
Theorem 2.
Let $\beta <0$ be known and $H\in (1/2,1)$. The MLE for α is
It is unbiased, strongly consistent and normal:
Proof.
The form of the MLE (39) was established in [25, Eq. (3.2)]. The normality follows from Remark 2:
\[ {T^{1-H}}({\widetilde{\alpha }_{T}}-\alpha )={\lambda _{H}}\gamma {T^{H-1}}\bigg({S_{T}}+\beta {J_{T}}-\frac{\alpha }{\gamma }{w_{T}^{H}}\bigg)\stackrel{d}{=}\mathcal{N}\big(0,{\lambda _{H}}{\gamma ^{2}}\big).\]
In order to obtain the strong consistency, we rewrite this equality using the relation
which follows from (6) and (5). Then
Recall that ${M_{T}^{H}}$ is a martingale and ${\langle {M^{H}}\rangle _{T}}={w_{T}^{H}}\to \infty $, as $T\to \infty $. Then $\frac{{M_{T}^{H}}}{{w_{T}^{H}}}\to 0$ a.s., as $T\to \infty $, by the strong law of large numbers for martingales [24, Th. 2.6.10, Cor. 1]. □Theorem 3.
Let α be known, $H\in (1/2,1)$ and $\beta <0$. The MLE for β is
It is strongly consistent and
where η and ζ are the same as in Theorem 1.
(43)
\[ {e^{-\beta T}}({\widetilde{\beta }_{T}}-\beta )\xrightarrow{d}\frac{\eta }{\zeta },\hspace{1em}\textit{as}\hspace{2.5pt}T\to \infty ,\]Proof.
The form of the MLE (42) is found in [25, Eq. (3.3)]. The strong consistency is established in the same way as in [25, Th. 3.2]. It follows from (41) that
\[\begin{aligned}{}\frac{\alpha }{\gamma }{J_{T}}-{I_{T}}-\beta {K_{T}}& =\frac{\alpha }{\gamma }{\int _{0}^{T}}{P_{H}}(t)\hspace{0.1667em}d{w_{T}^{H}}-{\int _{0}^{T}}{P_{H}}(t)\hspace{0.1667em}d{S_{t}}-\beta {\int _{0}^{T}}{\big({P_{H}}(t)\big)^{2}}\hspace{0.1667em}d{w_{t}^{H}}\\ {} & =-{\int _{0}^{T}}{P_{H}}(t)\hspace{0.1667em}d{M_{T}^{H}}.\end{aligned}\]
Whence, (42) implies that
\[ {\widetilde{\beta }_{T}}-\beta =\frac{\frac{\alpha }{\gamma }{J_{T}}-{I_{T}}-\beta {K_{T}}}{{K_{T}}}=-\frac{{\textstyle\textstyle\int _{0}^{T}}{P_{H}}(t)\hspace{0.1667em}d{M_{T}^{H}}}{{\textstyle\textstyle\int _{0}^{T}}{({P_{H}}(t))^{2}}\hspace{0.1667em}d{w_{t}^{H}}}.\]
Since the process ${M^{H}}$ is a martingale with the quadratic variation ${w^{H}}$, the process ${\textstyle\int _{0}^{\cdot }}{P_{H}}(t)\hspace{0.1667em}d{M_{t}^{H}}$ is a martingale with the quadratic variation ${\textstyle\int _{0}^{\cdot }}{({P_{H}}(t))^{2}}\hspace{0.1667em}d{w_{t}^{H}}$. Note that ${\textstyle\int _{0}^{T}}{({P_{H}}(t))^{2}}\hspace{0.1667em}d{w_{t}^{H}}={K_{T}}\to \infty $ in probablility, by Lemma 4. This convergence holds almost surely, because ${\textstyle\int _{0}^{T}}{({P_{H}}(t))^{2}}\hspace{0.1667em}d{w_{t}^{H}}$ is increasing in upper bound T. Therefore, by the strong law of large numbers for martingales [24, Th. 2.6.10, Cor. 1], we get that ${\widetilde{\beta }_{T}}\to \beta $ a.s., as $T\to \infty $.Remark 7.
The particular case when the parameter $\alpha =0$ is known and ${x_{0}}=0$ was studied in [32]. Similarly to Remark 4, we see that in this case the convergence (43) takes the form
\[ \frac{{e^{-\beta T}}}{2\beta }({\widetilde{\beta }_{T}}-\beta )\xrightarrow{d}\frac{X\sqrt{\sin \pi H}}{Y},\hspace{1em}\text{as}\hspace{2.5pt}T\to \infty ,\]
where X and Y are two independent $\mathcal{N}(0,1)$ random variables. This coincides with the result of [32, Th. 2].
