Parameter estimation for ergodic linear SDEs from partial and discrete observations

Abstract

We consider a problem of parameter estimation for the state space model described by linear stochastic differential equations. We assume that an unobservable Ornstein–Uhlenbeck process drives another observable process by the linear stochastic differential equation, and these two processes depend on some unknown parameters. We construct the quasi-maximum likelihood estimator of the unknown parameters and show asymptotic properties of the estimator.

1 Introduction

On the probability space $(\Omega ,{\mathcal {F}},P)$ with a complete and right-continuous filtration $\{{\mathcal {F}}_t\}$, we consider a $(d_1+d_2)$-dimensional Gaussian process $(X_t,Y_t)$ satisfying the following stochastic differential equations:

$$\begin{aligned} dX_t&=-a(\theta _2)X_tdt+b(\theta _2)dW_t^1, \end{aligned}$$

(1.1)

$$\begin{aligned} dY_t&=c(\theta _2)X_tdt+\sigma (\theta _1) dW_t^2, \end{aligned}$$

(1.2)

where $W^1$ and $W^2$ are independent $d_1$ and $d_2$-dimensional $\{{\mathcal {F}}_t\}$-Wiener processes, $(X_0,Y_0)$ is a gaussian random variable independent of $W^1$ and $W^2, \theta _1 \in \Theta _1 \subset {\mathbb {R}}^{m_1}$ and $\theta _2 \in \Theta _2 \subset {\mathbb {R}}^{m_2}$ are unknown parameters, and $a,b:\Theta _2 \rightarrow M_{d_1}({\mathbb {R}}),c:\Theta _2 \rightarrow M_{d_2,d_1}({\mathbb {R}})$ and $\sigma :\Theta _1 \rightarrow M_{d_2}({\mathbb {R}})$ are known functions. Here $M_{m,n}({\mathbb {R}})$ is the set of $m \times n$ matrices over ${\mathbb {R}}$ and $M_{n}({\mathbb {R}})=M_{n,n}({\mathbb {R}})$, $\Theta _1$ and $\Theta _2$ are known parameter spaces. The solution of (1.1) is an Ornstein–Uhlenbeck process and it has an ergodic property.

We assume that the process X is unobservable, and the purpose of this article is to construct estimators of $\theta _1$ and $\theta _2$ based on discrete observations of Y; we assume discrete observations $Y_{t_0}, Y_{t_1}, Y_{t_2}, \cdots , Y_{t_n}$ where $t_i=ih_n$ for some $h_n>0$, instead of considering the continuous observation $\{Y_{t}\}_{0\le t\le T}$. The discrete observation case is much more complicated and interesting because of the construction of the estimator for $\theta _2$, which is the main object of this paper and described in detail in Sect. 3, and also because there is no need to estimate $\theta _1$ (Kutoyants 2004, 2019a).

Note that we can not identify $b(\theta _2)$ and $c(\theta _2)$ simultaneously from observation of $\{Y_t\}$. In fact, the system

$$\begin{aligned} dX_t&=-a(\theta _2)X_tdt+2b(\theta _2)dW_t^1\\ dY_t&=\frac{1}{2}c(\theta _2)X_tdt+\sigma (\theta _1) dW_t^2 \end{aligned}$$

generates the same $\{Y_t\}$ as (1.1) and (1.2). Therefore, we need to impose some restrictions on $a,b,c,\sigma$ and the dimensions of the parameter spaces.

When $\theta _1$ and $\theta _2$ are known, one can estimate the unobservable state $\{X_t\}$ from observations of $\{Y_t\}$ by the following well-known Kalman–Bucy filter.

Theorem 1.1

(Theorem 10.2, Liptser and Shiriaev 2001a) In (1.1) and (1.2), let $\sigma (\theta )\sigma (\theta )'$ be positive definite, where the prime means the transpose. Then $m_t=E[X_t|\{Y_t\}_{0\le s\le t}]$ and $\gamma _t=E[(X_t-m_t)(X_t-m_t)']$ are the solutions of the equations

$$\begin{aligned} dm_t&=-a(\theta _2)m_tdt+\gamma _tc(\theta _2)'\{\sigma (\theta _1)\sigma (\theta _1)'\}^{-1}\{dY_t-c(\theta _2)m_tdt\}, \end{aligned}$$

(1.3)

$$\begin{aligned} \frac{d\gamma _t}{dt}&=-a(\theta _2)\gamma _t-\gamma _ta(\theta _2)'-\gamma _tc(\theta _2)'\{\sigma (\theta _1)\sigma (\theta _1)'\}^{-1}c(\theta _2)\gamma _t+b(\theta _2)b(\theta _2)'. \end{aligned}$$

(1.4)

Equation (1.4) is the matrix Riccati equation, which has been examined in the theory of linear quadratic control (Sontag 2013). It is known that (1.4) has the unique positive-semidefinite solution (Liptser and Shiriaev 2001a). Moreover, under proper conditions, one can show that the corresponding algebraic Riccati equation

$$\begin{aligned} -a(\theta _2)\gamma -\gamma a(\theta _2)'-\gamma c(\theta _2)'\{\sigma (\theta _1)\sigma (\theta _1)'\}^{-1}c(\theta _2)\gamma +b(\theta _2)b(\theta _2)'=O \end{aligned}$$

(1.5)

has the maximal and minimal solutions (Coppel 1974; Zhou et al. 1996), and the solution of (1.4) converges to the maximal solution of (1.5) at an exponential rate (Leipnik 1985). Further details on this topic will be discussed in Sect. 3.

There are already several studies on parameter estimation in the system (1.1) and (1.2) with the Kalman–Bucy filter. For example, Kutoyants (2004) discusses the ergodic case, Kutoyants (1994) and Kutoyants (2019b) small noise cases, and Kutoyants (2019a) the one-step estimator. However, all of them assume $d_1=d_2=1$ and need continuous observation of Y. The continuous observation case is simpler, because we do not have to estimate $\theta _1$. In fact, we have

$$\begin{aligned} {Y_t}^2-{Y_0}^2=2\int _0^tY_sdY_s+\sigma (\theta _1)^2t \end{aligned}$$

by Itô’s formula and (1.1), and therefore we can get the exact value of $\sigma (\theta _1)$.

On the other hand, parametric inference for discretely observed stochastic differential equations without an unobservable process has been studied for decades (for example Sørensen 2002; Shimizu and Yoshida 2006; Yoshida 1992). Especially, Yoshida (2011) developed Ibragimov–Khasminskii theory (Ibragimov and Has’ Minskii 1981) into the quasi-likelihood analysis, and investigated the behavior of the quasi-likelihood estimator and the adaptive Bayes estimator in the ergodic diffusion process. Quasi-likelihood analysis is helpful to discretely observed cases, and many works have been derived from it: see Uchida and Yoshida (2012) for the non-ergodic case, Ogihara and Yoshida (2011) for the jump case, Masuda (2019) for the Lévy driven case, Gloter and Yoshida (2021) for the degenerate case, Kamatani and Uchida (2015) for the multi-step estimator, and Nakakita et al. (2021) for the case with observation noises.

This paper also makes use of quasi-likelihood analysis to investigate the behaviors of our estimators. In Sect. 2, we describe the more precise setup and present asymptotic properties of our estimators, which are main results of this paper. Then we go on to proofs of these results in Sects. 3 and 5. We first discuss the estimation of $\theta _2$ in Sect. 3 because it is the main part of this article, whereas estimation of $\theta _1$ is quite parallel to the usual case without an unobservable variable. We also examine the Riccati differential equation (1.4) and algebraic Riccati equation (1.5) in Sect. 3. In Sect. 4, we discuss the special case where $d_1=d_2=1$. In the one-dimensional case, we can reduce our assumptions to simpler ones. In Sect. 5, we discuss estimation of $\theta _1$. Finally, we show in Sect. 6 the result of computational simulation by YUIMA (Brouste et al. 2014), an package on R, and suggest a way to improve our estimators when the wrong initial value is given.

2 Notations, assumptions and main results

Let $\theta _1^* \in {\mathbb {R}}^{m_1}$ and $\theta _2^*\in {\mathbb {R}}^{m_2}$ be the true values of $\theta _1$ and $\theta _2$, respectively, and define the $(d_1+d_2)$-dimensional Gaussian process $(X_t,Y_t)$ by

$$\begin{aligned} dX_t&=-a(\theta _2^*)X_tdt+b(\theta _2^*)dW_t^1, \end{aligned}$$

(2.1)

$$\begin{aligned} dY_t&=c(\theta _2^*)X_tdt+\sigma (\theta _1^*) dW_t^2, \end{aligned}$$

(2.2)

where $W_1, W_2, X_0, Y_0, a, b, c$ and $\sigma$ are the same as Sect. 1; $a,b:\Theta _2 \rightarrow M_{d_1}({\mathbb {R}}),c:\Theta _2 \rightarrow M_{d_2,d_1}({\mathbb {R}})$ and $\sigma :\Theta _1 \rightarrow M_{d_2}({\mathbb {R}})$. In this article, we have access to the discrete observations $Y_{ih_n}~(i=0,1,\cdots ,n)$, where $h_n$ is some positive constant, and we construct the estimators of $\theta _1$ and $\theta _2$ based on the observations.

We assume that $\Theta _1\subset {\mathbb {R}}^{m_1}$ and $\Theta _2\subset {\mathbb {R}}^{m_2}$ are open bounded subsets and that the Sobolev embedding inequality holds on $\Theta =\Theta _1\times \Theta _2$; for any $p>m_1+m_2$ and $f\in C^1(\Theta )$, there exists some constant C depending only on $\Theta$ such that

$$\begin{aligned} \sup _{\theta \in \Theta }|f(\theta )|\le C\left( \Vert f\Vert _{L^p}+\Vert \partial _{\theta _i}f\Vert _{L^p} \right) . \end{aligned}$$

(2.3)

For example, if each $\Theta _i$ ($i=1,2$) has a Lipchitz boundary, this inequality is valid (Leoni 2017).

Let $Z(\theta )~(\theta \in \Theta =\Theta _1\times \Theta _2)$ be a class of random variables, where $Z(\theta )$ is continuously differentiable with respect to $\theta$. Then by (2.3) and Fubini’s theorem, we get for any $p>m_1+m_2$

$$\begin{aligned} E\left[ \sup _{\theta \in \Theta }|Z(\theta )|^p \right]&\le C2^{p-1}\left( E\left[ \int _{\Theta _i}|Z(\theta )|^pd\theta _i+\int _{\Theta }|\partial _{\theta }Z(\theta )|^pd\theta _i \right] \right) \\&=C2^{p-1}\left( \int _{\Theta _i}E[|Z(\theta )|^p]d\theta +\int _{\Theta }E[|\partial _{\theta }Z(\theta )|^p]d\theta \right) \\&\le C_{p}\sup _{\theta \in \Theta }\left( E[|Z(\theta )|^p]+E[|\partial _{\theta }Z(\theta )|^p] \right) , \end{aligned}$$

where $C_p$ is some constant depending on p and $\Theta$. This result will be frequently referred to in the following sections.

In what follows, we use the following notations:

• ${\mathbb {R}}_+=[0,\infty ),{\mathbb {N}}=\{1,2,\cdots \}$.

• $\Theta =\Theta _1\times \Theta _2,\theta _1=(\theta _1^1,\cdots ,\theta _1^{m_1}),\theta _2=(\theta _2^1,\cdots ,\theta _2^{m_2}),\theta ^*=(\theta _1^*,\theta _2^*).$

• For any subset $\Xi \subset {\mathbb {R}}^m, {\overline{\Xi }}$ is the closure of $\Xi$.

• For every set of matrices A, B and $C, A'$ is the transpose of $A, A^{\otimes 2}=AA', A[B,C]=B'AC$ and $A[B^{\otimes 2}]=B'AB$.

• For every matrix A, |A| is the Frobenius norm of A. Namely, if $A=(a_{ij})_{1\le i\le n,1\le j\le m}, |A|$ is defined by

  $$\begin{aligned} |A|=\sqrt{\sum _{i=1}^n\sum _{j=1}^m a_{ij}^2}. \end{aligned}$$

• For every matrix $A, \lambda _{\min }(A)$ denotes the smallest real part of eigenvalues of matrix A.

• For every symmetric matrix A and $B \in M_{d}({\mathbb {R}}), A> B$ (resp. $A\ge B$) means that $A-B$ is positive (resp. semi-positive) definite.

• For any open subset $\Xi \subset {\mathbb {R}}^m$ and $A:\Xi \rightarrow M_{d}({\mathbb {R}})$ of class $C^k, \partial _{\xi }^kA(\xi )$ denotes the k-dimensional tensor on $M_{d}({\mathbb {R}})$ whose $(j_1,j_2,\cdots ,j_k)$ entry is $\displaystyle \frac{\partial }{\partial \xi _{j_1}}\cdots \frac{\partial }{\partial \xi _{j_k}}A(\theta _i)$, where $1\le j_1,\cdots ,j_k\le m$ and $\xi =(\xi _1,\cdots ,\xi _m)$.

• For every k-dimensional tensor A with $(i_1,i_2,\cdots ,i_k)$ entry $A_{i_1\cdots i_k} \in M_{d}({\mathbb {R}})$ and every matrix $B \in M_d({\mathbb {R}}), AB$ denotes the tensor whose $(i_1,i_2,\cdots ,i_k)$ entry is $A_{i_1\cdots i_k}B$. BA is also defined in the same way.

• For any partially differentiable function $f:\Theta _2 \rightarrow {\mathbb {R}}^{d_2}$ and $S \in M_{d_2}({\mathbb {R}}), S[\partial _{\theta _2}^{\otimes 2}]f(\theta )$ is the matrix whose (i, j)-entry is $\displaystyle \frac{\partial }{\partial {\theta _2^i}}f(\theta _2)S_{ij}\frac{\partial }{\partial {\theta _2^j}}f(\theta _2)$.

• If both A and B are matrices with $M_d({\mathbb {R}})$ entries, AB is the normal product of matrices.

• For every matrix A on $M_d({\mathbb {R}})$ with (i, j) entry $A_{ij} \in M_{d}({\mathbb {R}}), \textrm{Tr}A$ is a matrix on ${\mathbb {R}}$ with (i, j) entry $\textrm{Tr}A_{ij}$.

• For every stochastic process $Z, \Delta _i Z=Z_{t_{i}}-Z_{t_{i-1}}$.

• We write $a^*,b^*,c^*,\sigma ^*$ and $\Sigma ^*$ for $a(\theta _2^*),b(\theta _2^*),c(\theta _2^*),\sigma (\theta _1^*)$ and $\Sigma (\theta _1^*)$.

• We omit the subscript n in $h_n$ and just write h when there is no ambiguity.

• We designate $\sigma (\theta _1)\sigma (\theta _1)'$ as $\Sigma (\theta _1)$.

• C denotes a generic positive constant. When C depends on some parameter p, we might use $C_p$ instead of C.

Moreover, we need the following assumptions:

[A1]:

$nh_n \rightarrow \infty ,~n{h_n}^2 \rightarrow 0$ as $n \rightarrow \infty$. Moreover, we assume $h_n \le 1$ for every $n \in {\mathbb {N}}$.

[A2]:

a, b, c and $\sigma$ are of class $C^4$.

Then we can extend a, b, c and $\sigma$ to continuous functions on ${\overline{\Theta }}_1$ and ${\overline{\Theta }}_2$.

[A3]:

$$\begin{aligned}&\inf _{\theta _2 \in {\overline{\Theta }}_2}\lambda _{\min }(a(\theta _2))>0\\&\inf _{\theta _2 \in {\overline{\Theta }}_2}\lambda _{\min }(b(\theta _2)^{\otimes 2})>0\\&\inf _{\theta _1 \in {\overline{\Theta }}_1}\lambda _{\min }(\Sigma (\theta _1))>0. \end{aligned}$$

[A4]:

For any $\theta _1 \in {\overline{\Theta }}_1$ and $\theta _2 \in {\overline{\Theta }}_2$, the pair of matrix $(a(\theta _2)', \Sigma (\theta _1)[c(\theta _2)^{\otimes 2}])$ is controllable; i.e. the matrix

$$\begin{aligned} \begin{pmatrix} \Sigma (\theta _1)[c(\theta _2)^{\otimes 2}]&a(\theta _2)'\Sigma (\theta _1)[c(\theta _2)^{\otimes 2}]&\cdots&{a(\theta _2)'}^{d_1}\Sigma (\theta _1)[c(\theta _2)^{\otimes 2}] \end{pmatrix} \end{aligned}$$

has full row rank. Moreover, the eigenvalues of the matrix

$$\begin{aligned} H(\theta _1,\theta _2)=\begin{pmatrix} a(\theta _2)'&{}\Sigma (\theta _1)^{-1}[c(\theta _2)^{\otimes 2}]\\ b(\theta _2)^{\otimes 2}&{}-a(\theta _2) \end{pmatrix} \end{aligned}$$

(2.4)

are uniformly bounded away from the imaginary axis; i.e. there are some constant $C>0$ such that for any $\theta _1 \in {\overline{\Theta }}_1$ and $\theta _2 \in {\overline{\Theta }}_2$ and eigenvalue $\lambda$ of $H(\theta _1,\theta _2)$, it holds

$$\begin{aligned} |\textrm{Re}(\lambda )|>C. \end{aligned}$$

By Assumption [A4] and the corollary of Theorem 6 in Coppel (1974), for every $\theta _1 \in {\overline{\Theta }}_1$ and $\theta _2 \in {\overline{\Theta }}_2$, Eq. (1.5) has the maximal solution $\gamma =\gamma _+(\theta _1,\theta _2)$ and minimal solution $\gamma =\gamma _-(\theta _1,\theta _2)$, where $\gamma _+(\theta _1,\theta _2)>\gamma _-(\theta _1,\theta _2)$. The meaning of the maximal and minimal solutions is that for any symmetric solution $\gamma$ of (1.5), it holds $\gamma _-\le \gamma \le \gamma _+$.

Now we define ${\mathbb {Y}}_1$ and ${\mathbb {Y}}_2$ by

$$\begin{aligned}&{\mathbb {Y}}_1(\theta _1)=-\frac{1}{2}\left\{ \textrm{Tr}\Sigma (\theta _1)^{-1}\Sigma (\theta _1^*)-d_1+\log \frac{\textrm{det}\Sigma (\theta _1)}{\textrm{det}\Sigma (\theta _1^*)} \right\} \end{aligned}$$

(2.5)

and

$$\begin{aligned} \begin{aligned} {\mathbb {Y}}_2(\theta _2)&=-\frac{1}{2}\textrm{Tr}\int _{0}^\infty {\Sigma ^*}^{-1}\left. \Biggr [ \left\{ \int _0^{s}c(\theta _2)\exp (-\alpha (\theta _2)u)\gamma _{+}(\theta _1^*,\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\right. \right. \\&\quad \times \exp (-a^*(s-u))\gamma _+(\theta ^*){c^*}'du\\&\quad +c(\theta _2)\exp (-\alpha (\theta _2)s)\gamma _{+}(\theta _1^*,\theta _2)c(\theta _2)'\\&\quad \left. \left. -c^*\exp (-a^*s)\gamma _+(\theta ^*){c^*}' \right. \biggr \}^{\otimes 2} \right. \Biggr ][({{\sigma ^*}'}^{-1})^{\otimes 2}]ds, \end{aligned} \end{aligned}$$

(2.6)

respectively, where

$$\begin{aligned} \alpha (\theta _2)=a(\theta _2)+\gamma _+(\theta _1^*,\theta _2)\Sigma (\theta _1^*)^{-1}[c(\theta _2)^{\otimes 2}], \end{aligned}$$

(2.7)

and assume the following condition.

[A5]:

There is some positive constant $C>0$ satisfying

$$\begin{aligned} {\mathbb {Y}}_1(\theta _1)\le -C|\theta _1-\theta _1^*|^2 \end{aligned}$$

(2.8)

and

$$\begin{aligned} {\mathbb {Y}}_2(\theta _2)\le -C|\theta _2-\theta _2^*|^2. \end{aligned}$$

(2.9)

Remark

By (2.7), it holds

$$\begin{aligned}&\int _0^{s}c^*\exp (-\alpha (\theta _2^*)u)\gamma _{+}(\theta ^*){c^*}'{\Sigma ^*}^{-1}c^*\exp (-a^*(s-u))\gamma _+(\theta ^*){c^*}'du\\&\quad =\int _0^{s}c^*\exp (-\alpha (\theta _2^*)u)\left\{ \alpha (\theta _2^*)-a^* \right\} \exp (-a^*(s-u))\gamma _+(\theta ^*){c^*}'du\\&\quad =c^*\exp (-\alpha (\theta _2^*)s)-c(\theta _2)\exp (-a^*s)\gamma _+(\theta ^*){c^*}', \end{aligned}$$

and therefore ${\mathbb {Y}}_2(\theta _2)$ has the following expression:

$$\begin{aligned} {\mathbb {Y}}_2(\theta _2)&=-\frac{1}{2}\textrm{Tr}\int _{0}^\infty {\Sigma ^*}^{-1}\left. \Biggl [ \left\{ \int _0^{s}\{c(\theta _2)\exp (-\alpha (\theta _2)u)\gamma _{+}(\theta _1^*,\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\right. \right. \\&\quad -c^*\exp (-\alpha (\theta _2^*)u)\gamma _{+}(\theta ^*){c^*}'{\Sigma ^*}^{-1}c^*\}du\\&\quad +c(\theta _2)\exp (-\alpha (\theta _2)s)\gamma _{+}(\theta _1^*,\theta _2)c(\theta _2)'\\&\quad \left. \left. -c^*\exp (-\alpha (\theta _2^*)s)\gamma _{+}(\theta ^*){c^*}' \right. \biggr \}^{\otimes 2} \right] [({{\sigma ^*}'}^{-1})^{\otimes 2}]ds. \end{aligned}$$

In particular, we have ${\mathbb {Y}}_2(\theta _2^*)=0$.

Under these assumptions above, we set

$$\begin{aligned}&{\mathbb {H}}_n^1(\theta _1)=-\frac{1}{2}\sum _{j=1}^n\left\{ \frac{1}{h}\Sigma ^{-1}(\theta _1)[(\Delta _jY)^{\otimes 2}]+\log \det \Sigma (\theta _1) \right\} \end{aligned}$$

(2.10)

and

$$\begin{aligned}&\Gamma ^1=\frac{1}{2}\left[ \textrm{Tr}\{{\Sigma ^*}^{-1}\partial _{\theta _1}\Sigma (\theta _1^*)\}\right] ^{\otimes 2}, \end{aligned}$$

and we define our estimator of $\theta _1$ as the maximizer of ${\mathbb {H}}_n^1(\theta _1)$. Note that $\textrm{Tr}\{{\Sigma ^*}^{-1}\partial _{\theta _1}\Sigma (\theta _1^*)\}$ is a vector whose jth entry is $\displaystyle \textrm{Tr}\left\{ {\Sigma ^*}^{-1}\frac{\partial }{\partial {\theta _1^j}}\Sigma (\theta _1^*)\right\}$. Then the following theorem holds:

Theorem 2.1

We assume [A1]–[A5], and for each $n \in {\mathbb {N}}$, let ${\hat{\theta }}^n_1$ be a random variable satisfying

$$\begin{aligned} {\mathbb {H}}_n^1({\hat{\theta }}^n_1)=\max _{\theta _1 \in {\overline{\Theta }}_1}{\mathbb {H}}_n^1(\theta _1). \end{aligned}$$

Then for every $p>0$ and any continuous function $f:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \limsup _{|x|\rightarrow \infty }\frac{|f(x)|}{|x|^p}<\infty , \end{aligned}$$

it holds that

$$\begin{aligned} E[f(\sqrt{n}({\hat{\theta }}^n_1-\theta _1^*))]\rightarrow E[f(Z)]~(n \rightarrow \infty ), \end{aligned}$$

where $Z \sim N(0,(\Gamma ^1)^{-1})$.

In particular, it holds that

$$\begin{aligned} \sqrt{n}({\hat{\theta }}^n_1-\theta _1^*)\xrightarrow {d}N(0,(\Gamma ^1)^{-1})~(n \rightarrow \infty ). \end{aligned}$$

Next we construct the estimator of $\theta _2$, which is the main object of this article. Recall that under Assumption [A4], (1.5) has the maximal solution $\gamma _+(\theta _1,\theta _2)$. Now we replace $\gamma _t$ with $\gamma _+(\theta _1,\theta _2)$ in (1.3), and define $m_t(\theta _1,\theta _2;m_0)$ by

$$\begin{aligned} {\left\{ \begin{array}{ll} dm_t=-a(\theta _2)m_tdt+\gamma _{+}(\theta _1,\theta _2)c(\theta _2)'\{\sigma (\theta _1)\sigma (\theta _1)'\}^{-1}\{dY_t-c(\theta _2)m_tdt\}\\ m_0(\theta _1,\theta _2;m_0)=m_0, \end{array}\right. } \end{aligned}$$

(2.11)

where $m_0 \in {\mathbb {R}}^{d_1}$ is an arbitrary initial estimated value of $X_0$.

Due to Itô’s formula, the solution of (2.11) can be written as

$$\begin{aligned} \begin{aligned} m_t(\theta _1,\theta _2)&=\exp \left( -\alpha (\theta _1,\theta _2)t\right) m_0\\&\quad +\int _0^t\exp \left( -\alpha (\theta _1,\theta _2)(t-s)\right) \gamma _+(\theta _1,\theta _2)c(\theta _2)'\Sigma (\theta _1)^{-1}dY_s, \end{aligned} \end{aligned}$$

(2.12)

where

$$\begin{aligned} \alpha (\theta _1,\theta _2)=a(\theta _2)+\gamma _+(\theta _1,\theta _2)\Sigma (\theta _1)^{-1}[c(\theta _2)^{\otimes 2}]. \end{aligned}$$

(2.13)

The eigenvalues of $\alpha (\theta _1,\theta _2)$ coincides with those of $H(\theta _1,\theta _2)$ in (2.4) with positive real part (see Zhou et al. 1996), so there exists some constant $C>0$ such that for any $\theta _1 \in \Theta _1$ and $\theta _2 \in \Theta _2$,

$$\begin{aligned} \inf _{\lambda \in \sigma (\alpha (\theta _1,\theta _2))}\textrm{Re}\lambda >C, \end{aligned}$$

where $\sigma (\alpha (\theta _1,\theta _2))$ is the set of all eigenvalues of $\alpha (\theta _1,\theta _2)$.

According to (2.12), we set for $i,n \in {\mathbb {N}}$,

$$\begin{aligned} {\hat{m}}_i^n(\theta _2;m_0)&=\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)t_i\right) m_0\nonumber \\&\quad +\sum _{j=1}^i\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)(t_i-t_{j-1})\right) \gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\Delta _jY, \end{aligned}$$

(2.14)

$$\begin{aligned} {\mathbb {H}}_n^2(\theta _2;m_0)&=\frac{1}{2}\sum _{i=1}^n\left\{ -h\Sigma ({\hat{\theta }}_1^n)^{-1}[(c(\theta _2){\hat{m}}_{j-1}^n(\theta _2))^{\otimes 2}]\right. \nonumber \\&\quad \left. +{\hat{m}}_{j-1}^n(\theta _2)'c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\Delta _jY+\Delta _jY'\Sigma ({\hat{\theta }}_1^n)^{-1}c(\theta _2){\hat{m}}_{j-1}^n(\theta _2)\right\} , \end{aligned}$$

(2.15)

and

$$\begin{aligned} \begin{aligned} \Gamma ^2&=\textrm{Tr}\int _0^\infty {\Sigma ^*}^{-1}[\partial _{\theta _2}^{\otimes 2}]\left\{ \int _0^sc(\theta _2)\exp (-\alpha (\theta _2)u)\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\right. \\&\quad \times \exp (-a^*(s-u))\gamma _+(\theta ^*){c^*}'du\\&\quad \left. \left. +c(\theta _2)\exp (-\alpha (\theta _2)s)\gamma _{+}(\theta _2)c(\theta _2)'\right. \biggr \}\right| _{\theta _2=\theta _2^*}ds, \end{aligned} \end{aligned}$$

(2.16)

where ${\hat{\theta }}_1^n$ is the estimator of $\theta _1$ defined in Theorem 2.1. Then the following theorem holds:

Theorem 2.2

We assume [A1]–[A5], and let $m_0 \in {\mathbb {R}}^{d_1}$ be an arbitrary initial value and ${\hat{\theta }}^n_2={\hat{\theta }}^n_2(m_0)$ be a random variable satisfying

$$\begin{aligned} {\mathbb {H}}_n^2({\hat{\theta }}^n_2)=\max _{\theta _2 \in {\overline{\Theta }}_2}{\mathbb {H}}_n^2(\theta _2) \end{aligned}$$

for each $n \in {\mathbb {N}}$. Moreover, let $\Gamma ^2$ be positive definite. Then for any $p>0$ and continuous function $f:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \limsup _{|x|\rightarrow \infty }\frac{|f(x)|}{|x|^p}<\infty , \end{aligned}$$

it holds that

$$\begin{aligned} E[f(\sqrt{t_n}({\hat{\theta }}^n_2-\theta _2^*))]\rightarrow E[f(Z)]~(n \rightarrow \infty ), \end{aligned}$$

where $Z \sim N(0,(\Gamma ^2)^{-1})$.

In particular, it holds that

$$\begin{aligned} \sqrt{t_n}({\hat{\theta }}^n_2-\theta _2^*)\xrightarrow {d}N(0,(\Gamma ^2)^{-1})~(n \rightarrow \infty ). \end{aligned}$$

Remark

• (1) In order to calculate ${\hat{m}}_i^n$, one can use the autoregressive formula

  $$\begin{aligned} {\hat{m}}_{i+1}^n(\theta _2;m_0)&=\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)h\right) {\hat{m}}_{i}^n(\theta _2;m_0)\\&\quad +\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)h\right) \gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\Delta _{i+1}Y. \end{aligned}$$

• (2) One can obtain $\gamma (\theta _1,\theta _2)$ in the following way (see Zhou et al. 1996 for details). Let $v_1,v_2,\cdots , v_{d_1}$ be generalized eigenvectors of $H(\theta _1,\theta _2)$ in (2.4) with positive real part eigenvalues. Note that $H(\theta _1,\theta _2)$ has $d_1$ eigenvalues (with multiplicity) in the right half-plane and $d_1$ in the left half-plane. We define the matrices $X_1(\theta _1,\theta _2)$ and $X_2(\theta _1,\theta _2)$ by

  $$\begin{aligned} \begin{pmatrix} v_1&v_2&\cdots&v_{d_1} \end{pmatrix}=\begin{pmatrix} X_1(\theta _1,\theta _2)\\ X_2(\theta _1,\theta _2) \end{pmatrix}. \end{aligned}$$

  Then $X_1(\theta _1,\theta _2)$ is invertible and it holds $\gamma _+(\theta _1,\theta _2)=X_2(\theta _1,\theta _2)X_1(\theta _1,\theta _2)^{-1}$.

• (3) ${\mathbb {H}}^2(\theta _2)$ can be interpreted as a approximated log-likelihood function with $\theta _1$ given. In fact, if $X_t=X_t(\theta )$ and $Y_t=Y_t(\theta )$ are generated by (1.1) and (1.2), and we set $m_0=E[X_0|Y_0]$ and $\gamma _0=E[(m_0-X_0)^{\otimes 2}]$, then it follows $m_t(\theta )=E[X_t(\theta )|\{Y_s(\theta )\}_{0\le s\le t}]$ by Theorem 1.1. Thus by the innovation theorem (Kallianpur 1980), we can replace $X_t(\theta )$ with $m_t(\theta )$ in Eq. (1.2), and consider the equation

  $$\begin{aligned} dY_t(\theta )=c(\theta _2)m_t(\theta )dt+\sigma (\theta _1)d{\overline{W}}_t \end{aligned}$$

  where ${\overline{W}}$ is a $d_2$-dimensional Wiener process. We can approximate this equation as

  $$\begin{aligned} \Delta _i Y(\theta )\approx c(\theta _2)m_{t_{i-1}}(\theta )h+\sigma (\theta _1)\Delta _i {\overline{W}}, \end{aligned}$$

  when $h \approx 0$. Then we obtain the approximated likelihood function

  $$\begin{aligned} p(\theta )&\approx \prod _{i=1}^n\frac{1}{(2\pi h)^{\frac{d}{2}}\{\textrm{det}\Sigma (\theta _1)\}^{\frac{1}{2}}}\\&\quad \times \exp \left( -\frac{1}{2h}\Sigma (\theta _1)^{-1}\left[ (\Delta _iY-c(\theta _2)m_{t_{i-1}}(\theta )h)^{\otimes 2} \right] \right) . \end{aligned}$$

• (4) The condition the $W^1$ and $W^2$ are independent is not essential; according to Section 12 of Liptser and Shiriaev (2001b), Kalman–Bucy filter can be extended to the equation of the form

  $$\begin{aligned} dX_t&=\left\{ a_0(\theta _2,Y_t)+a_1(\theta _2,Y_t)X_t \right\} dt+b_1(\theta _2,Y_t)dW_t^1+b_2(\theta _2,Y_t)dW_t^2\\ dY_t&=\left\{ c_0(\theta _2,Y_t)+c_1(\theta _2,Y_t)X_t \right\} dt+\sigma (\theta _1,Y_t)dW_t^2. \end{aligned}$$

  However, this case is more complicated, and thus is left for future research.

3 Proof of Theorem 2.2

In this section, we write $m_t(\theta _2)$, ${\hat{m}}_i^n(\theta _2), {\mathbb {H}}_n^2(\theta _2)$, $\gamma _+(\theta _2)$ and $\alpha (\theta _2)$ instead of $m_t(\theta _1^*,\theta _2;m_0), {\hat{m}}_i^n(\theta _2;m_0)$, ${\mathbb {H}}_n^2(\theta _2;m_0), \gamma _+(\theta _1^*,\theta _2)$ and $\alpha (\theta _1^*,\theta _2)$, respectively, for simplicity.

Moreover, let $m_t^*=E[X_t|\{Y_t\}_{0\le s\le t}]$ and $\gamma _t^*=E[(X_t-m_t)(X_t-m_t)']$. Then by Theorem 1.1, they are the solutions of

$$\begin{aligned} dm_t^*&=-a^*m_t^*dt+\gamma _t^*{c^*}'{\Sigma ^*}^{-1}\{dY_t-c^*m_t^*dt\} \end{aligned}$$

(3.1)

$$\begin{aligned} \frac{d\gamma _t^*}{dt}&=-a^*\gamma _t^*-\gamma _t^*(a^*)'-{\Sigma ^*}^{-1}[(c^*\gamma _t^*)^{\otimes 2}]+{b^*}^{\otimes 2}. \end{aligned}$$

(3.2)

We start with preliminary lemmas, which is frequently referred to in proving inequalities.

Lemma 3.1

Let $\{W_t\}$ be a d-dimensional $\{{\mathcal {F}}_t\}$-Wiener process.

• (1) Let $f:{\mathbb {R}}_+ \rightarrow {\mathbb {R}}^m$ be a measurable function. Then for any $p\ge 1$ and $0\le s\le t$, it holds

  $$\begin{aligned} \left( \int _s^t|f(u)|du \right) ^p\le (t-s)^{p-1}\int _s^t|f(u)|^pdu \end{aligned}$$

• (2) Let $\{A_t\}$ be a $M_{k,d}({\mathbb {R}})$-valued progressively measurable process and $\{W_t\}$ be a d-dimensional Wiener process. Then for every $0\le s\le t\le T$ and $p \ge 2$, it holds

  $$\begin{aligned} E\left[ \sup _{s\le t \le T}\left| \int _s^t A_udW_u\right| ^p \right]&\le C_{p,d,k}E\left[ \left( \int _s^T|A_u|^2du\right) ^{\frac{p}{2}} \right] \\&\le C_{p,d,k}(T-s)^{\frac{p}{2}-1}\int _s^tE[|A_u|^p]du. \end{aligned}$$

Proof

• (1) By Hölder’s inequality, we obtain

  $$\begin{aligned} \int _s^t|f(u)|du&\le \left( \int _s^t|f(u)|^{p}du\right) ^{\frac{1}{p}} \left( \int _s^tdu\right) ^{1-\frac{1}{p}}\\&=(t-s)^{1-\frac{1}{p}}\left( \int _s^t|f(u)|^{p}du\right) ^{\frac{1}{p}}, \end{aligned}$$

  and it shows the desired inequality.

• (2) Let $A_t^{(ij)}$ be the (i, j) entry of $A_t$, and $W_t^{(j)}$ be the jth element of $W_t$. Then the Burkholder-Davis-Gundy inequality gives

  $$\begin{aligned} E\left[ \sup _{s\le t\le T}\left| \int _s^t A_udW_u\right| ^p\right]&=E\left[ \sup _{s\le t\le T}\left\{ \sum _{i=1}^k\left( \sum _{j=1}^d\int _{s}^tA_u^{(ij)}dW_u^{(j)} \right) ^2 \right\} ^\frac{p}{2}\right] \\&\le C_{p,d,k}\sum _{i=1}^k \sum _{j=1}^dE\left[ \sup _{s\le t\le T}\left| \int _{s}^tA_u^{(ij)}dW_u^{(j)} \right| ^p\right] \\&\le C_{p,d,k}\sum _{i=1}^k \sum _{j=1}^dE\left[ \left| \int _{s}^T(A_u^{(ij)})^2du \right| ^{\frac{p}{2}}\right] \\&\le C_{p,d,k}E\left[ \left| \int _{s}^T\sum _{i=1}^k \sum _{j=1}^d(A_u^{(ij)})^2du \right| ^{\frac{p}{2}}\right] \\&=C_{p,d,k}E\left[ \left| \int _{s}^T|A_u|^2du \right| ^{\frac{p}{2}}\right] . \end{aligned}$$

  Hence we have proved the first inequality, and together with (1) we obtain the second one.

$\square$

Lemma 3.2

Let A be a $d\times d$ matrix having eigenvalues $\lambda _1,\cdots ,\lambda _k$. Then for all $\epsilon >0$, there exists some constant $C_{\epsilon ,d}$ depending on $\epsilon$ and d such that

$$\begin{aligned} |\exp (At)|\le C_{\epsilon ,d}(1+|A|^{d-1})e^{(\lambda _{\max }+\epsilon )t}~(t \ge 0), \end{aligned}$$

where

$$\begin{aligned} \lambda _{\max }=\max _{i=1,\cdots ,k}\textrm{Re}\lambda _k. \end{aligned}$$

Proof

Let

$$\begin{aligned} A=U^*(D+N)U,~D=\textrm{diag}(\lambda _1,\lambda _2,\cdots ,\lambda _{d}) \end{aligned}$$

be a Schur decomposition of A, where $\lambda _1,\lambda _2,\cdots ,\lambda _{d}$ are the eigenvalues of A, U is an unitary matrix, and N is a strictly upper triangular matrix. Then we have

$$\begin{aligned} |\exp (At)|&=|\exp ((D+N)t)|=|\exp (Dt)\exp (Nt)|\\&\le |\exp (Dt)||\exp (Nt)|\\&\le C_de^{\lambda _{\max }t}\sum _{k=1}^{d-1}\frac{|N|^{k}}{k!}t^{k}\\&\le C_de^{\lambda _{\max }t}\sum _{k=1}^{d-1}\frac{|A|^{k}}{k!}t^{k}\\&\le C_de^{(\lambda _{\max }+\epsilon )t}\sum _{k=1}^{d-1}\frac{|A|^{k}}{k!}t^{k}e^{-\epsilon t}\\&\le C_{\epsilon ,d}(1+|A|^{d-1})e^{(\lambda _{\max }+\epsilon )t}, \end{aligned}$$

noting that U is unitary, $N^{d}=O$, and $|A|=|D+N|\ge N$. $\square$

Lemma 3.3

For any $s,t\ge 0$ such that $0\le t-s\le 1$ and $p \ge 1$, it holds

$$\begin{aligned}&\sup _{t\ge 0}E[|X_t|^p]\le C_p, \end{aligned}$$

(3.3)

$$\begin{aligned}&E[|Y_s-Y_t|^p]\le C_p|s-t|^{\frac{p}{2}} \end{aligned}$$

(3.4)

and

$$\begin{aligned}&E[|X_s-X_t|^p]\le C_p|s-t|^{\frac{p}{2}}. \end{aligned}$$

(3.5)

Proof

By Itô’s formula, the solution of (2.1) can be expressed as

$$\begin{aligned} X_t=\exp (-a^*t)X_0+\int _0^t\exp (-a^*(t-s))b^*dW_s^1, \end{aligned}$$

(3.6)

where $\exp$ is the matrix exponential. Hence by Lemmas 3.1 and 3.2, we have

$$\begin{aligned} E[|X_t|^p]&\le |\exp (-a^*t)|^pE[|X_0|^p]\\&\quad +C_p\left( \int _0^t|b^*|^2|\exp (-a^*(t-s))|^2ds\right) ^{\frac{p}{2}}\\&\le C_pe^{-\eta pt}+C_p\left( \int _0^te^{-2\eta s}ds\right) ^{\frac{p}{2}}\le C_p \end{aligned}$$

for some constant $\eta >0$. Therefore for $s\le t$ we obtain

$$\begin{aligned} E[|Y_t-Y_s|^p]&=E\left[ \left| c^*\int _s^tX_udu+\sigma ^*(W_t^2-W_s^2)\right| ^p \right] \\&\le C_p\left( (t-s)^{p-1}\int _s^tE[|X_u|^p]du+(t-s)^{\frac{p}{2}} \right) \\&\le C_p\left( (t-s)^{p}+(t-s)^{\frac{p}{2}} \right) \\&\le C_p(t-s)^{\frac{p}{2}}. \end{aligned}$$

We can show (3.5) in the same way. $\square$

We next discuss important properties of $\gamma _+(\theta _1,\theta _2)$ and $\gamma _t^*$.

Proposition 3.4

The maximal solution of (1.5) $\gamma _+(\theta _1,\theta _2)$ is of class $C^4$.

Proof

Let $\theta ^0=(\theta _1^0,\theta _2^0) \in \Theta _1\times \Theta _2$, and we consider the mapping $f:M_{d_1}({\mathbb {R}}) \rightarrow M_{d_1}({\mathbb {R}})$ such that

$$\begin{aligned} f:X\mapsto a(\theta _2^0)X+Xa(\theta _2^0)'+\Sigma (\theta _1^0)^{-1}[(c(\theta _2^0)X)^{\otimes 2}]-b(\theta _2^0)^{\otimes 2}. \end{aligned}$$

Since for every $T \in M_{d_1}({\mathbb {R}})$, we have

$$\begin{aligned} f(X+T)-f(T)&=\left\{ a(\theta _2^0)+X'\Sigma (\theta _1^0)^{-1}[c(\theta _2^0)^{\otimes 2}]\right\} T\\&\quad +T\left\{ a(\theta _2^0)'+\Sigma (\theta _1)^{-1}[c(\theta _2^0)^{\otimes 2}]X\right\} \\&\quad +\Sigma (\theta _1^0)^{-1}[(c(\theta _2^0)T)^{\otimes 2}] \end{aligned}$$

and

$$\begin{aligned} \lim _{|T|\rightarrow 0}\frac{|\Sigma (\theta _1^0)^{-1}[(c(\theta _2^0)T)^{\otimes 2}]|}{|T|}=0, \end{aligned}$$

the differential of f at $X=\gamma _+(\theta ^0)$ is given by

$$\begin{aligned} (df)_{\gamma _+(\theta ^0)}:T \mapsto \alpha (\theta _0)T+T\alpha (\theta _0), \end{aligned}$$

where $\alpha$ is defined by (2.13).

If $(df)_{\gamma _+(\theta ^0)}$ is not injective, $\alpha (\theta _0)$ has eigenvalues $\mu$ and $\lambda$ such that $\mu +{\overline{\lambda }}=0$ (see lemma 2.7 in Zhou et al. 1996). However, noting that $\gamma _+(\theta _1,\theta _2)$ is the unique symmetric solution of $f(X)=O$ such that $-\alpha (\theta _0)$ is stable (Coppel 1974; Zhou et al. 1996), there are no such eigenvalues. Therefore $(df)_{\gamma _+(\theta ^0)}$ is injective, and by the implicit function theorem, there exists a neighborhood $U \subset \Theta _1 \times \Theta _2$ containing $\theta ^0$ and a mapping $\phi :U \rightarrow M_{d_1}({\mathbb {R}})$ of class $C^4$ such that

$$\begin{aligned} \phi (\theta ^0)=\gamma _+(\theta ^0),~~f(\phi (\theta ))=O~(\theta \in U). \end{aligned}$$

Since $-a(\theta _2)-\phi (\theta )\Sigma (\theta _1)^{-1}[c(\theta _2)^{\otimes 2}]$ is stable at $\theta =(\theta _1,\theta _2)=\theta ^0$, it is also stable on a neighborhood of $\theta ^0$. Thus by the uniqueness of $\gamma _+$, we obtain $\gamma _+(\theta )=\phi (\theta )$ on that neighborhood and therefore the desired result. $\square$

By this proposition, Theorem 2.1 and the mean value theorem, we get the following corollary.

Corollary 3.5

For any $p\ge 1$, it holds

$$\begin{aligned}&E\left[ \sup _{\theta _2 \in \Theta _2}|\gamma _+({\hat{\theta }}_1^n,\theta _2)-\gamma _+(\theta _1^*,\theta _2)|^p \right] ^\frac{1}{p}\le Cn^{-\frac{1}{2}} \end{aligned}$$

and

$$\begin{aligned} E\left[ \sup _{\theta _2 \in \Theta _2}|\alpha ({\hat{\theta }}_1^n,\theta _2)-\alpha (\theta _1^*,\theta _2)|^p \right] ^\frac{1}{p}\le Cn^{-\frac{1}{2}}. \end{aligned}$$

Proposition 3.6

For every $\theta _1 \in {\overline{\Theta }}_1$ and $\theta _2 \in {\overline{\Theta }}_2$,

$$\begin{aligned} \gamma _+(\theta _1,\theta _2)>0 \end{aligned}$$

(3.7)

and

$$\begin{aligned} \gamma _-(\theta _1,\theta _2)<0. \end{aligned}$$

(3.8)

Proof

Noting that for A and $\gamma \in M_{d_1}({\mathbb {R}})$,

$$\begin{aligned} \frac{d}{dt}(\exp (At)\gamma \exp (A't))=\exp (At)(A\gamma +\gamma A')\exp (A't), \end{aligned}$$

and the Eq. (1.5) is equivalent to

$$\begin{aligned}&\left\{ a(\theta _2)+\gamma \Sigma (\theta _1)^{-1}[c(\theta _2)^{\otimes 2}]\right\} \gamma +\gamma \left\{ a(\theta _2)+\gamma \Sigma (\theta _1)^{-1}[c(\theta _2)^{\otimes 2}]\right\} '\\&\quad =\gamma \Sigma (\theta _1)^{-1}[c(\theta _2)^{\otimes 2}]\gamma +b(\theta _2)^{\otimes 2}, \end{aligned}$$

we obtain

$$\begin{aligned}&\gamma _+(\theta _1,\theta _2)\\&\quad =\int _{-\infty }^0\exp (\alpha (\theta _1,\theta _2)t)\{\alpha (\theta _1,\theta _2)\gamma +\gamma \alpha (\theta _1,\theta _2)'\}\exp (\alpha (\theta _1,\theta _2)'t)dt\\&\quad =\int _{-\infty }^0\exp (\alpha (\theta _1,\theta _2)t)\left\{ \Sigma (\theta _1)^{-1}[c(\theta _2)^{\otimes 2}][\gamma _+(\theta _1,\theta _2)^{\otimes 2}]+b(\theta _2)^{\otimes 2} \right\} \\&\qquad \times \exp (\alpha (\theta _1,\theta _2)t)dt>0 \end{aligned}$$

by assumption [A3], (2.13) and the stability of $-\alpha (\theta _1,\theta _2)$. In the same way, we can show $\gamma _-(\theta _1,\theta _2)<0.$ $\square$

Combining this result with assumption [A3], (2.13) and Lemma 3.2, we obtain the following corollary.

Corollary 3.7

There exists some constant $C_1>0$ and $C_2>0$ such that

$$\begin{aligned} \sup _{\theta _1\in \Theta _1,\theta _2 \in \Theta _2}|\exp (-\alpha (\theta _1,\theta _2))|\le C_1e^{-C_2t}. \end{aligned}$$

Now we go on to the convergence of $\gamma _t^*$. Concerning the convergence rate of Riccati equations, Leipnik (1985) presents the following result.

Theorem 3.8

(Section 5, Leipnik 1985) Let $A,B,C \in M_d({\mathbb {R}})$ and consider the equation

$$\begin{aligned} \frac{d}{dP}(t)=-A-P(t)B-B'P(t)-P(t)CP(t). \end{aligned}$$

Moreover, assume C is symmetric, $C \le 0, (B,C)$ is controllable and the matrix

$$\begin{aligned} H=\begin{pmatrix} B&{}C\\ -A&{}-B' \end{pmatrix} \end{aligned}$$

has no pure imaginary eigenvalues.

Then if $P_0-P^+$ is non-singular, then it holds for any $\epsilon >0$ that

$$\begin{aligned} |P(t)-P^-| \le Ce^{2(r+\epsilon )t} ~(t\rightarrow \infty ) \end{aligned}$$

and if $P_0-P^-$ is non-singular, then it holds for any $\epsilon >0$ that

$$\begin{aligned} |P(t)-P^+| \le Ce^{-2(r-\epsilon )t} ~(t\rightarrow -\infty ), \end{aligned}$$

where $P^+$ and $P^-$ are the maximal and minimal solutions of the algebraic Riccati equation

$$\begin{aligned} A+PB+B'P+PCP=O \end{aligned}$$

respectively, $r<0$ is the maximum real part of the eigenvalues of $B+CP^+$.

Proposition 3.9

For any $\epsilon >0$, there exists some constant $C>0$ such that

$$\begin{aligned} |\gamma _t^*-\gamma _+(\theta ^*)|\le Ce^{-2\{\lambda _{\min }(\alpha (\theta _2^*))-\epsilon \}t}. \end{aligned}$$

In particular, $|\gamma _t^*|$ is bounded.

Proof

According to (3.2) and Theorem 3.8, it is enough show that $\gamma _0^*-\gamma _-(\theta ^*)$ is non-singular, where $\gamma _-(\theta _1,\theta _2)$ is the minimal solution of (1.5). If we assume $\gamma _0^*-\gamma _-(\theta ^*)$ is singular, there exists $x \in {\mathbb {R}}^{d_1}\backslash \{0\}$ such that $\{\gamma _0^*-\gamma _-(\theta ^*)\}x=0$, and we get $x\gamma _0^*x=x\gamma _-(\theta ^*)x$. However, since $\gamma _0^*\ge 0$ and we have $\gamma _-(\theta ^*)<0$ by Proposition 3.6, that is a contradiction. $\square$

Next we consider the innovation process

$$\begin{aligned} {\overline{W}}_t=(\sigma ^*)^{-1}\left( Y_t-\int _0^tc^*m_s^*ds \right) . \end{aligned}$$

Note that the right-hand side is well-defined since $\{m_t^*\}$ has a progressively measurable modification, and that ${\overline{W}}_t$ is also a Wiener process (Kallianpur 1980). Since $Y_t$ is the solution of

$$\begin{aligned} dY_t=c^*m_t^*dt+\sigma ^*d{\overline{W}}_t, \end{aligned}$$

(3.9)

we obtain together with (3.1)

$$\begin{aligned} dm_t^*=-a^*m_t^*dt+\gamma _t^*{c^*}'{{\sigma ^*}'}^{-1}d{\overline{W}}_t. \end{aligned}$$

Therefore Itô’s formula gives

$$\begin{aligned} m_t^*=\exp (-a^*t)m_0^*+\int _0^t\exp (-a^*(t-s))\gamma _s^*{c^*}'{{\sigma ^*}'}^{-1}d{\overline{W}}_s. \end{aligned}$$

(3.10)

Moreover, using Proposition 3.9, we can show for any $p\ge 1$,

$$\begin{aligned} \sup _{t\ge 0}E[|m_t^*|^p]\le C_p \end{aligned}$$

(3.11)

and

$$\begin{aligned} \sup _{0\le t-s\le 1}E[|m_t^*-m_s^*|^p]\le C_p(t-s)^{\frac{p}{2}} \end{aligned}$$

(3.12)

in the same way as Lemma 3.3.

Lemma 3.10

For $j=0,1,2,\cdots$ and $\theta \in \Theta$, let $Z_j(\theta )$ be a $M_{k,l}({\mathbb {R}})$-valued and ${\mathcal {F}}_{t_j}$-measurable random variable, and $U(\theta )$ be an $M_{l,d}({\mathbb {R}})$-valued random variable. Moreover, we assume $Z_j(\theta )$ is continuously differentiable with respect to $\theta$. Then for any $n \in {\mathbb {N}}$ and $p>m_1+m_2$, it holds

$$\begin{aligned}&E\left[ \sup _{\theta \in \Theta }\left| \sum _{j=1}^nZ_{j-1}(\theta )U(\theta )\Delta _jW\right| ^p \right] \\&\quad \le C_{d,k,l}E\left[ \sup _{\theta \in \Theta }\left| U(\theta )\right| ^{2p}\right] ^\frac{1}{2}\\&\qquad \times \sup _{\theta \in \Theta }\left\{ E\left[ \left\{ \sum _{j=1}^n|Z_{j-1}(\theta )|^2h\right\} ^{p}\right] +E\left[ \left\{ \sum _{j=1}^n\left| \partial _{\theta }Z_{j-1}(\theta )\right| ^2h\right\} ^{p}\right] \right\} ^\frac{1}{2}. \end{aligned}$$

Proof

Let $Z_j^{(ij)}, U^{(ij)}$ and $(Z_jU)^{(ij)}$ be the (i, j) entries of $Z_j, U$ and $Z_jU$, respectively, and $W^{(j)}$ be the jth element of $W^{(j)}$. Then we have

$$\begin{aligned} \begin{aligned}&E\left[ \sup _{\theta \in \Theta }\left| \sum _{j=1}^nZ_{j-1}(\theta )U(\theta )\Delta _jW\right| ^p \right] \\&\quad =E\left[ \sup _{\theta \in \Theta }\left\{ \sum _{p=1}^k\left( \sum _{j=1}^n\sum _{q=1}^d\sum _{r=1}^lZ_{j-1}^{(pr)}(\theta )U^{(rq)}(\theta )\Delta _jW^{(q)}\right) ^2 \right\} ^{\frac{p}{2}}\right] \\&\quad \le C_{d,k,l}E\left[ \sup _{\theta \in \Theta }\left| U(\theta )\right| ^{2p}\right] ^\frac{1}{2}\sum _{p=1}^k\sum _{r=1}^lE\left[ \sup _{\theta \in \Theta }\left| \sum _{j=1}^nZ_{j-1}^{(pr)}(\theta )\Delta _jW^{(q)}\right| ^{2p} \right] ^\frac{1}{2}. \end{aligned} \end{aligned}$$

(3.13)

Moreover, the Sobolev inequality and the Burkholder-Davis-Gundy inequality gives

$$\begin{aligned}&E\left[ \sup _{\theta \in \Theta }\left| \sum _{j=1}^nZ_{j-1}^{(pr)}(\theta )\Delta _jW^{(q)}\right| ^{2p} \right] \nonumber \\&\quad \le C_p\sup _{\theta \in \Theta }\left\{ E\left[ \left| \sum _{j=1}^nZ_{j-1}^{(pr)}(\theta )\Delta _jW^{(q)}\right| ^{2p}\right] +E\left[ \left| \sum _{j=1}^n\frac{\partial }{\partial \theta }Z_{j-1}^{(pr)}(\theta )\Delta _jW^{(q)}\right| ^{2p}\right] \right\} \nonumber \\&\quad \le C_p\sup _{\theta \in \Theta }\left\{ E\left[ \left| \sum _{j=1}^nZ_{j-1}^{(pr)}(\theta )^2h\right| ^{p}\right] +E\left[ \left| \sum _{j=1}^n\left\{ \frac{\partial }{\partial \theta }Z_{j-1}^{(pr)}(\theta )\right\} ^2h\right| ^{p}\right] \right\} \nonumber \\&\quad \le C_p\sup _{\theta \in \Theta }\left\{ E\left[ \left( \sum _{j=1}^n|Z_{j-1}(\theta )|^2h\right) ^{p}\right] +E\left[ \left( \sum _{j=1}^n\left| \partial _{\theta }Z_{j-1}(\theta )\right| ^2h\right) ^{p}\right] \right\} .\nonumber \\ \end{aligned}$$

(3.14)

By (3.13) and (3.14), we obtain the desired result. $\square$

Lemma 3.11

For every $\theta \in \Theta$, let $\{Z_t(\theta )\}$ be a ${\mathbb {M}}_{d,d_1}({\mathbb {R}})$-valued progressively measurable process. Moreover, we assume $Z_t(\theta )$ is differentiable with respect to $\theta$, and for any $T>0, p>0$ and $\theta ,\theta ' \in \Theta$

$$\begin{aligned}&\sup _{0\le t\le T}E\left[ |Z_t(\theta )-Z_t(\theta ')|^p \right] \le C_{T,p}|\theta -\theta '|^p,\\&\sup _{0\le t\le T}E\left[ |\partial _{\theta }Z_t(\theta )-\partial _{\theta }Z_t(\theta ')|^p \right] \le C_{T,p}|\theta -\theta '|^p. \end{aligned}$$

Then $\{\xi _{\cdot }(\theta )\}_{\theta \in \Theta }$ with $\displaystyle \xi _t(\theta )=\int _0^tZ_t(\theta )d{\overline{W}}_s$ has a modification $\{{\tilde{\xi }}_{\cdot }(\theta )\}_{\theta \in \Theta }$ which is continuously differentiable with respect to $\theta$. Moreover, it holds almost surely for any $t\ge 0$ and $\theta \in \Theta$

$$\begin{aligned} \partial _{\theta }{\tilde{\xi }}_t(\theta )=\int _0^t\partial _{\theta }Z_t(\theta )d{\overline{W}}_s. \end{aligned}$$

Proof

For any matrix valued function $\phi$ on ${\mathbb {R}}^{m_1+m_2}$ and $\epsilon >0$, let

$$\begin{aligned} \Delta ^j\phi (\theta ;\epsilon )=\frac{1}{\epsilon }\{\xi _t(\theta +\epsilon e_j)-\xi _t(\theta )\}, \end{aligned}$$

where $e_1,\cdots ,e_{m_1+m_2}$ is the standard basis of ${\mathbb {R}}^{m_1+m_2}$. Then for $\theta ,\theta '\in \Theta ,\epsilon ,\epsilon '>0$ and $p\ge 1$, we have

$$\begin{aligned}&\sup _{0\le t\le T}E\left[ \left| \Delta ^jZ_t(\theta ;\epsilon )-\Delta ^jZ_t(\theta ;\epsilon ')\right| ^p\right] \\&\quad =\sup _{0\le t\le T}E\left[ \left| \int _0^1\frac{\partial }{\partial \theta ^j}Z_t(\theta +u\epsilon e_j)du-\int _0^1\frac{\partial }{\partial \theta ^j}Z_s(\theta +u\epsilon ' e_j)du\right| ^p\right] \\&\quad \le \int _0^1\sup _{0\le t\le T}E\left[ \left| \frac{\partial }{\partial \theta ^j}Z_t(\theta +u\epsilon e_j)-\frac{\partial }{\partial \theta ^j}Z_t(\theta '+u\epsilon ' e_j)\right| \right] du\\&\quad \le C_{p,T}(|\theta -\theta '|+|\epsilon -\epsilon '|), \end{aligned}$$

where $\theta =(\theta ^1,\cdots ,\theta ^{m_1+m_2})$.

Hence by Lemma 3.1, it follows for any $\theta ,\theta ' \in \Theta ,\epsilon ,\epsilon '>0$ and $N \in {\mathbb {N}}$

$$\begin{aligned}&E\left[ \sup _{0\le t \le N}\left| \Delta ^j \xi _t(\theta ;\epsilon )-\Delta ^j \xi _t(\theta ';\epsilon ')\right| ^p \right] \\&\quad =E\left[ \sup _{0\le t \le N}\left| \int _{0}^t\{\Delta ^jZ_t(\theta ;\epsilon )-\Delta ^j Z_t(\theta ';\epsilon ')\}d{\overline{W}}_s\right| ^p \right] \\&\quad \le C_pN^{\frac{p}{2}-1}\int _{0}^NE\left[ \left| \Delta ^jZ_t(\theta ;\epsilon )-\Delta ^j Z_t(\theta ';\epsilon ')\right| ^p\right] ds\\&\quad \le C_{p,N}(|\theta -\theta '|+|\epsilon -\epsilon '|). \end{aligned}$$

Now for this $C_{p,N}$, we take a sequence $\alpha _N>0~(N \in {\mathbb {N}})$ so that

$$\begin{aligned} S_p=\sum _{n=1}^\infty \alpha _NC_{p,N}<\infty ,~~\sum _{n=1}^\infty \alpha _N<\infty , \end{aligned}$$

and define the norm on $C({\mathbb {R}}_+;M_{d,d_1}({\mathbb {R}}))$ by

$$\begin{aligned} \Vert A\Vert =\sum _{N=1}^\infty \alpha _N\left( \sup _{0\le t \le N}|A(s)|\wedge 1 \right) . \end{aligned}$$

Then the topology induced by this norm is equivalent to the topology of uniform convergence, and we have

$$\begin{aligned} E\left[ \left\| \Delta ^j \xi .(\theta ;\epsilon )-\Delta ^j \xi .(\theta ';\epsilon ')\right\| ^p \right] \le C_p(|\theta -\theta '|+|\epsilon -\epsilon '|). \end{aligned}$$

(3.15)

Therefore, by the the Kolmogorov continuity theorem, $\{\Delta ^j \xi .(\theta ;\epsilon )\}_{\theta \in \Theta ,0<|\epsilon |\le 1}$ has a uniformly continuous modification $\{\zeta .(\theta ;\epsilon )\}_{\theta \in \Theta ,0<|\epsilon |\le 1}$. Because of the uniform continuity, $\zeta .(\theta ;\epsilon )$ can be extended to a continuous process on $\theta \in \Theta ,|\epsilon |\le 1$.

On the other hand, we can show in the same way that $\{\xi .(\theta ;\epsilon )\}_{\theta \in \Theta }$ has a continuous modification $\{{\tilde{\xi }}.(\theta ;\epsilon )\}_{\theta \in \Theta }$. Then $\Delta ^j{\tilde{\xi }}.(\theta ;\epsilon )$ and $\zeta .(\theta ;\epsilon )$ are both continuous modifications of $\Delta ^j \xi .(\theta ;\epsilon )$, and thus they are indistinguishable. Therefore almost surely for any $t \ge 0$ and $\theta \in \Theta$,

$$\begin{aligned} \frac{\partial {\tilde{\xi }}_t}{\partial \theta _j}(\theta )=\lim _{\epsilon \rightarrow 0}\frac{\xi _t(\theta +\epsilon e_j)-\xi (\theta )}{\epsilon }=\lim _{\epsilon \rightarrow 0}\Delta ^j \xi _t(\theta ;\epsilon ) \end{aligned}$$

exists. The continuity of $\displaystyle \frac{\partial \xi _t}{\partial \theta _j}(\theta )$ follows from the continuity of $\zeta .(\theta ,\epsilon )$.

Moreover, by the assumption and Lemma 3.1 (2), we have for $p\ge 2$,

$$\begin{aligned}&E\left[ \left| \int _0^t\left\{ \frac{1}{\epsilon }(Z_s(\theta +\epsilon e_j)-Z_s(\theta ))-\frac{\partial }{\partial \theta ^j}Z_s(\theta )\right\} d{\overline{W}}_s\right| ^p \right] \\&\quad =E\left[ \left| \int _0^t\left\{ \frac{\partial Z_s}{\partial \theta _j}(\theta +\eta _s\epsilon e_j)-\frac{\partial }{\partial \theta ^j}Z_s(\theta )\right\} d{\overline{W}}_s\right| ^p \right] \\&\quad \le C_pt^{\frac{p}{2}-1}\int _0^t E\left[ \left| \frac{\partial Z_s}{\partial \theta _j}(\theta +\eta _s\epsilon e_j)-\frac{\partial Z_s}{\partial \theta ^j}(\theta )\right| ^p \right] ds\\&\quad \le C_{p,t}\epsilon \rightarrow 0~(\epsilon \rightarrow 0), \end{aligned}$$

where $0\le \eta _s\le 1$. This means

$$\begin{aligned} \Delta ^j\xi _t(\theta ;\epsilon )\rightarrow \int _0^s\frac{\partial }{\partial \theta ^j}Z_s(\theta )ds~~(\epsilon \rightarrow 0) \end{aligned}$$

in $L^p$, and hence there exists a subsequence $\{\epsilon _n\}_{n \in {\mathbb {N}}}$ such that $\epsilon _n \rightarrow 0$ and

$$\begin{aligned} \Delta ^j\xi _t(\theta ;\epsilon _n)\xrightarrow {\mathrm {a.s.}} \int _0^s\frac{\partial }{\partial \theta ^j}Z_s(\theta )ds~~(n \rightarrow \infty ). \end{aligned}$$

Therefore we obtain almost surely

$$\begin{aligned} \frac{\partial }{\partial \theta ^j}{\tilde{\xi }}_t(\theta )=\Delta ^j{\tilde{\xi }}_t(\theta ;0)=\int _0^s\frac{\partial }{\partial \theta ^j}Z_s(\theta )ds. \end{aligned}$$

$\square$

Lemma 3.12

• (1) For $j \in {\mathbb {N}}$, let $f_j:[t_{j-1},t_j]\times \Theta \rightarrow M_{k,d_2}({\mathbb {R}})$ be of class $C^1$. Then for any $p>m_1+m_2$, it holds

  $$\begin{aligned} \begin{aligned}&E\left[ \sup _{\theta \in \Theta }\left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}f_{j-1}(s,\theta )dY_s\right| ^p \right] ^{\frac{1}{p}}\\&\quad \le C_p\sup _{\theta \in \Theta }\left\{ \sum _{j=1}^i\int _{t_{j-1}}^{t_j}|f_j(s,\theta )|ds+\sum _{j=1}^i\int _{t_{j-1}}^{t_j}|\partial _{\theta }f_j(s,\theta )|ds\right. \\&\qquad \left. +\left( \sum _{j=1}^i\int _{t_{j-1}}^{t_j}|f_{j-1}(s,\theta )|^2ds\right) ^{\frac{1}{2}} +\left( \sum _{j=1}^i\int _{t_{j-1}}^{t_j}|\partial _{\theta }f_{j-1}(s,\theta )|^2ds\right) ^{\frac{1}{2}}\right\} , \end{aligned} \end{aligned}$$

  (3.16)

  where $C_p$ is a constant which depends only on p.

• (2) For $j=0,1,2,\cdots$ and $\theta \in \Theta$, let $Z_j(\theta )$ be a $M_{k,l}({\mathbb {R}})$-valued and ${\mathcal {F}}_{t_j}$-measurable random variable, and $U(\theta )$ be an $M_{l,d}({\mathbb {R}})$-valued random variable. Moreover, we assume $Z_j(\theta )$ is continuously differentiable with respect to $\theta$. Then for any $p>m_1+m_2$, it holds

  $$\begin{aligned} \begin{aligned}&E\left[ \sup _{\theta \in \Theta }\left| \sum _{j=1}^iZ_{j-1}(\theta )U(\theta )\Delta _j Y\right| ^p \right] ^{\frac{1}{p}}\\&\quad \le C_pE\left[ \sup _{\theta \in \Theta }|U(\theta )|^{4p}\right] ^\frac{1}{4p}\\&\qquad \times \sup _{\theta \in \Theta }\left\{ \sum _{j=1}^iE\left[ |Z_{j-1}(\theta )|^{2p}\right] ^\frac{1}{2p}h+\sum _{j=1}^iE\left[ |\partial _{\theta }Z_{j-1}(\theta )|^{2p}\right] ^\frac{1}{2p}h\right\} \\&\qquad +C_{p}E\left[ \sup _{\theta \in \Theta }\left| U(\theta )\right| ^{2p}\right] ^\frac{1}{2p}\\&\qquad \times \sup _{\theta \in \Theta }\left\{ \sum _{j=1}^nE[|Z_{j-1}(\theta )|^{2p}]^\frac{1}{p}h+\sum _{j=1}^nE[\left| \partial _{\theta }Z_{j-1}(\theta )\right| ^{2p}]^\frac{1}{p}h \right\} ^\frac{1}{2}. \end{aligned} \end{aligned}$$

  (3.17)

Proof

(1) By Lemma 3.11, we can assume for every j

$$\begin{aligned} \int _{t_{j-1}}^{t_j}f_{j-1}(s,\theta )dY_s =\int _{t_{j-1}}^{t_j}f_{j-1}(s,\theta )c^*m_s^*ds+\int _{t_{j-1}}^{t_j}f_{j-1}(s,\theta )\sigma ^*{\overline{W}}_s \end{aligned}$$

is continuously differentiable, and

$$\begin{aligned} \partial _{\theta }\int _{t_{j-1}}^{t_j}f_{j-1}(s,\theta )dY_s=\int _{t_{j-1}}^{t_j}\partial _{\theta }f_{j-1}(s,\theta )dY_s. \end{aligned}$$

Therefore by the Sobolev inequality and (3.9),

$$\begin{aligned}&E\left[ \sup _{\theta \in \Theta }\left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}f_{j-1}(s,\theta )dY_s\right| ^p \right] ^{\frac{1}{p}} \nonumber \\&\quad \le C_p\sup _{\theta \in \Theta }\left( E\left[ \left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}f_{j-1}(s,\theta )dY_s\right| ^p \right] ^{\frac{1}{p}}\right. \nonumber \\&\qquad \left. +E\left[ \left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}\partial _{\theta }f_{j-1}(s,\theta )dY_s\right| ^p \right] ^{\frac{1}{p}}\right) \nonumber \\&\quad \le C_p\sup _{\theta \in \Theta }\left( E\left[ \left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}f_{j-1}(s,\theta )c^*m_s^*ds\right| ^p \right] ^{\frac{1}{p}}\right. \nonumber \\&\qquad +E\left[ \left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}f_{j-1}(s,\theta )\sigma ^*d{\overline{W}}_s\right| ^p \right] ^{\frac{1}{p}}\nonumber \\&\qquad +E\left[ \left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}\partial _{\theta }f_{j-1}(s,\theta )c^*m_s^*ds\right| ^p \right] ^{\frac{1}{p}}\nonumber \\&\qquad \left. +E\left[ \left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}\partial _{\theta }f_{j-1}(s,\theta )\sigma ^*d{\overline{W}}_s\right| ^p \right] ^{\frac{1}{p}}\right) . \end{aligned}$$

(3.18)

In order to bound the first term of (3.18), we set

$$\begin{aligned} f(s,\theta )=\sum _{j=1}^i f_j(s,\theta )1_{(t_{j-1},t_j]}(s). \end{aligned}$$

Then we have

$$\begin{aligned} \begin{aligned}&E\left[ \left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}f_{j-1}(s,\theta )m_s^*ds\right| ^p \right] =E\left[ \left| \int _{0}^{t_{i}}f(s,\theta )c^*m_s^*ds\right| ^p \right] \\&\quad \le E\left[ \left( \int _{0}^{t_{i}}|f(s,\theta )c^*m_s^*|ds\right) ^p \right] \\&\quad \le |c^*|^pE\left[ \left| \left( \int _{0}^{t_{i}}|f(s,\theta )|^\frac{1}{p}|m_s^*|^pds\right) ^\frac{1}{p}\left( \int _0^{t_i}|f(s,\theta )|ds\right) ^{1-\frac{1}{p}}\right| ^p \right] \\&\quad \le |c^*|^p\left( \int _0^{t_i}|f(s,\theta )|ds\right) ^{p-1}\int _{0}^{t_{i}}|f(s,\theta )|E[|m_s^*|^p]ds\\&\quad \le C_p\left( \int _0^{t_i}|f(s,\theta )|ds\right) ^{p} =C_p\left( \sum _{j=1}^i\int _{t_{j-1}}^{t_j}|f_j(s,\theta )|ds \right) ^p. \end{aligned} \end{aligned}$$

In the same way, it holds for the third term

$$\begin{aligned} E\left[ \left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}\partial _{\theta }Z_{j-1}(s,\theta )c^*m_s^*ds\right| ^p \right] ^{\frac{1}{p}}\le C_p\left( \sum _{j=1}^i\int _{t_{j-1}}^{t_j}|\partial _{\theta }f_j(s,\theta )|ds \right) ^p. \end{aligned}$$

Next by Lemma 3.1 (2), we obtain for the second term

$$\begin{aligned} \begin{aligned}&E\left[ \left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}f_{j-1}(s,\theta )\sigma ^*d{\overline{W}}_{s}\right| ^p \right] = E\left[ \left| \int _{0}^{t_i}f(s,\theta )\sigma ^*d{\overline{W}}_{s}\right| ^p \right] ^{\frac{1}{p}}\\&\quad \le C_p \left( \int _{0}^{t_i}\sum _{j=1}^i|f(s,\theta )|^2ds\right) ^{\frac{1}{2}} =C_p\left( \sum _{j=1}^i\int _{t_{j-1}}^{t_j}|f_{j-1}(s,\theta )|^2ds\right) ^{\frac{1}{2}} \end{aligned} \end{aligned}$$

and in the same way it holds for the fourth term

$$\begin{aligned} E\left[ \left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}\partial _{\theta }f_{j-1}(s,\theta )d{\overline{W}}_{s}\right| ^p \right] \le C_p\left( \sum _{j=1}^i\int _{t_{j-1}}^{t_j}|\partial _{\theta }f_{j-1}(s,\theta )|^2ds\right) ^{\frac{1}{2}}. \end{aligned}$$

We complete the proof by the above inequalities.

(2) By the Sobolev inequality and (3.9),

$$\begin{aligned}&E\left[ \sup _{\theta \in \Theta }\left| \sum _{j=1}^iZ_{j-1}(\theta )U(\theta )\Delta _j Y\right| ^p \right] ^{\frac{1}{p}}\nonumber \\&\quad \le C_p\left( E\left[ \sup _{\theta \in \Theta } \left| \sum _{j=1}^iZ_{j-1}(\theta )\int _{t_{j-1}}^{t_j}U(\theta )c^*m_s^*ds\right| ^p \right] ^{\frac{1}{p}}\right. \nonumber \\&\qquad \left. +E\left[ \sup _{\theta \in \Theta }\left| \sum _{j=1}^iZ_{j-1}(\theta )U(\theta )\sigma ^*(W_{t_j}-W_{t_{j-1}})\right| ^p \right] ^{\frac{1}{p}}\right) . \end{aligned}$$

(3.19)

For the first term of the right-hand side, it follows from Lemma 3.1 (1), (3.11) and the Sobolev inequality

$$\begin{aligned}&E\left[ \sup _{\theta \in \Theta }\left| \sum _{j=1}^iZ_{j-1}(\theta )\int _{t_{j-1}}^{t_j}U(\theta )c^*m_s^*ds\right| ^p \right] ^{\frac{1}{p}}\\&\quad \le E\left[ \sum _{j=1}^i\left| \sup _{\theta \in \Theta }\int _{t_{j-1}}^{t_j}Z_{j-1}(\theta )U(\theta )c^*m_s^*ds\right| ^p \right] ^{\frac{1}{p}}\\&\quad \le \left( \sum _{j=1}^ih^{p-1}\int _{t_{j-1}}^{t_j}E\left[ \sup _{\theta \in \Theta }|Z_{j-1}(\theta )U(\theta )c^*m_s^*|^p\right] ds \right) ^{\frac{1}{p}}\\&\quad \le |c|^*\left( \sum _{j=1}^ih^{p-1}\int _{t_{j-1}}^{t_j}E\left[ \sup _{\theta \in \Theta }|Z_{j-1}(\theta )|^{2p}\right] ^\frac{1}{2}\right. \\&\qquad \left. \times E\left[ \sup _{\theta \in \Theta }|U(\theta )|^{4p}\right] ^\frac{1}{4}E[|m_s^*|^{4p}]^\frac{1}{4}ds \right) ^{\frac{1}{p}}\\&\quad \le C_pE\left[ \sup _{\theta \in \Theta }|U(\theta )|^{4p}\right] ^\frac{1}{4}\left( \sum _{j=1}^ih^{p}E\left[ \sup _{\theta \in \Theta }|Z_{j-1}(\theta )|^{2p}\right] ^\frac{1}{2} \right) ^{\frac{1}{p}}\\&\quad \le C_pE\left[ \sup _{\theta \in \Theta }|U(\theta )|^{4p}\right] ^\frac{1}{4}\\&\qquad \times \left( \sum _{j=1}^ih^{p}\sup _{\theta \in \Theta }\left\{ E\left[ |Z_{j-1}(\theta )|^{2p}\right] +E\left[ |\partial _{\theta }Z_{j-1}(\theta )|^{2p}\right] \right\} ^\frac{1}{2} \right) ^{\frac{1}{p}}\\&\quad \le C_pE\left[ \sup _{\theta \in \Theta }|U(\theta )|^{4p}\right] ^\frac{1}{4}\\&\qquad \times \sup _{\theta \in \Theta }\sum _{j=1}^i\left\{ E\left[ |Z_{j-1}(\theta )|^{2p}\right] ^\frac{1}{2p}+E\left[ |\partial _{\theta }Z_{j-1}(\theta )|^{2p}\right] ^\frac{1}{2p}\right\} h. \end{aligned}$$

As for the second term, we have by Lemma 3.10

$$\begin{aligned}&E\left[ \sup _{\theta \in \Theta }\left| \sum _{j=1}^iZ_{j-1}(\theta )U(\theta )\sigma ^*(W_{t_j}-W_{t_{j-1}})\right| ^p \right] ^{\frac{1}{p}}\\&\quad \le C_{p}E\left[ \sup _{\theta \in \Theta }\left| U(\theta )\right| ^{2p}\right] ^\frac{1}{2p}\\&\qquad \times \sup _{\theta \in \Theta }\left\{ E\left[ \left( \sum _{j=1}^n|Z_{j-1}(\theta )|^2h\right) ^{p}\right] +E\left[ \left( \sum _{j=1}^n\left| \partial _{\theta }Z_{j-1}(\theta )\right| ^2h\right) ^{p}\right] \right\} ^\frac{1}{2p}\\&\quad \le C_{p}E\left[ \sup _{\theta \in \Theta }\left| U(\theta )\right| ^{2p}\right] ^\frac{1}{2p}\\&\qquad \times \sup _{\theta \in \Theta }\left\{ E\left[ \left( \sum _{j=1}^n|Z_{j-1}(\theta )|^2h\right) ^{p}\right] ^\frac{1}{p}+E\left[ \left( \sum _{j=1}^n\left| \partial _{\theta }Z_{j-1}(\theta )\right| ^2h\right) ^{p}\right] ^\frac{1}{p} \right\} ^\frac{1}{2}\\&\quad \le C_{p}E\left[ \sup _{\theta \in \Theta }\left| U(\theta )\right| ^{2p}\right] ^\frac{1}{2p}\\&\qquad \times \sup _{\theta \in \Theta }\left\{ \sum _{j=1}^nE[|Z_{j-1}(\theta )|^{2p}]^\frac{1}{p}h+\sum _{j=1}^nE[\left| \partial _{\theta }Z_{j-1}(\theta )\right| ^{2p}]^\frac{1}{p}h \right\} ^\frac{1}{2}. \end{aligned}$$

Thus we completed the proof. $\square$

Proposition 3.13

For any $p>m_1+m_2$, it holds

$$\begin{aligned}&\sup _{i \in {\mathbb {N}}}E\left[ \sup _{\theta _2 \in \Theta _2}|{\hat{m}}_i^n(\theta _2)|^{p}\right]<\infty \\&\sup _{i \in {\mathbb {N}}}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \partial _{\theta _2}{\hat{m}}_i^n(\theta _2)\right| ^{p}\right]<\infty \\&\sup _{i \in {\mathbb {N}}}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \partial _{\theta _2}^2{\hat{m}}_i^n(\theta _2)\right| ^{p}\right] <\infty \end{aligned}$$

and

$$\begin{aligned} \sup _{i \in {\mathbb {N}}}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \partial _{\theta _2}^3{\hat{m}}_i^n(\theta _2)\right| ^{p}\right] <\infty . \end{aligned}$$

Proof

We only prove the first one; the rest can be shown in the same way. By (2.14) and the stability of $-\alpha (\theta _1,\theta _2)$, it is enough to show

$$\begin{aligned} \sup _{i \in {\mathbb {N}}}E\left[ \left| \sup _{\theta =(\theta _1,\theta _2)\in \Theta }\sum _{j=1}^i\exp \left( -\alpha (\theta )(t_i-t_{j-1})\right) \right. \right. \\ \left. \left. \gamma _+(\theta )c(\theta _2)'\Sigma (\theta _1)^{-1}\Delta _jY\right. \Biggr | \right. \Biggr ]<\infty . \end{aligned}$$

To accomplish this, it is enough to show

$$\begin{aligned}&\sum _{j=1}^i\left| \exp \left( -\alpha (\theta )(t_i-t_{j-1})\right) \gamma _+(\theta )c(\theta _2)'\Sigma (\theta _1)^{-1}\right| h<C \end{aligned}$$

(3.20)

$$\begin{aligned}&\sum _{j=1}^i\left| \exp \left( -\alpha (\theta )(t_i-t_{j-1})\right) \gamma _+(\theta )c(\theta _2)'\Sigma (\theta _1)^{-1}\right| ^2h<C \end{aligned}$$

(3.21)

$$\begin{aligned}&\sum _{j=1}^i\left| \partial _{\theta }\left\{ \exp \left( -\alpha (\theta )(t_i-t_{j-1})\right) \gamma _+(\theta )c(\theta _2)'\Sigma (\theta _1)^{-1}\right\} \right| h<C \end{aligned}$$

(3.22)

$$\begin{aligned}&\sum _{j=1}^i\left| \partial _{\theta }\left\{ \exp \left( -\alpha (\theta )(t_i-t_{j-1})\right) \gamma _+(\theta )c(\theta _2)'\Sigma (\theta _1)^{-1}\right\} \right| ^2h<C \end{aligned}$$

(3.23)

according to (3.17).

These can be shown by using Corollary 3.7 and noting that it holds by Haber (2018)

$$\begin{aligned}&\left| \partial _{\theta }\exp \left( -\alpha (\theta )(t_i-t_{j-1})\right) \right| \\&\quad =\left| -\int _0^1\exp (-s\alpha (\theta )(t_i-t_{j-1}))\partial _{\theta }\alpha (\theta )(t_i-t_{j-1})\right. \\&\quad \left. \exp (-(1-s)\alpha (\theta )(t_i-t_{j-1}))ds\right. \biggr |\\&\quad \le C(t_i-t_{j-1})e^{-C(t_i-t_{j-1})}. \end{aligned}$$

$\square$

Proposition 3.14

For any $n,i \in {\mathbb {N}}$ and $p>m_1+m_2$,

$$\begin{aligned} E\left[ \sup _{\theta _2 \in \Theta _2}|m_{t_i}(\theta _2)-{\hat{m}}_{i}^n(\theta _2)|^{p}\right] ^\frac{1}{p}\le C_p(n^{-\frac{1}{2}}+h). \end{aligned}$$

Proof

By (2.12) and (2.14), we have

$$\begin{aligned}&E\left[ \sup _{\theta _2 \in \Theta _2}|m_{t_i}(\theta _2)-{\hat{m}}_{i}^n(\theta _2)|^{p}\right] ^\frac{1}{p}\nonumber \\&\quad \le E\left[ \sup _{\theta _2 \in \Theta _2}\left| \left\{ \exp (-\alpha (\theta _1^*,\theta _2)t)-\exp (-\alpha ({\hat{\theta }}_1^n,\theta _2)t)\right\} m_0\right| ^p \right] ^\frac{1}{p}\nonumber \\&\qquad +E\left[ \sup _{\theta _2 \in \Theta _2}\left| \int _0^{t_i}\exp \left( -\alpha (\theta _1^*,\theta _2)(t_i-s)\right) \gamma _+(\theta _1^*,\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}dY_s\right. \right. \nonumber \\&\qquad \left. \left. -\sum _{j=1}^i\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)(t_i-t_{j-1})\right) \gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\Delta _jY\right| ^p \right] ^\frac{1}{p}\nonumber \\&\quad \le E\left[ \sup _{\theta _2 \in \Theta _2}\left| \left\{ \exp (-\alpha (\theta _1^*,\theta _2)t)-\exp (-\alpha ({\hat{\theta }}_1^n,\theta _2)t)\right\} m_0\right| ^p \right] ^\frac{1}{p}\nonumber \\&\qquad +E\left. \Biggl [\sup _{\theta _2 \in \Theta _2}\left. \Biggl |\int _0^{t_i}\exp \left( -\alpha (\theta _1^*,\theta _2)(t_i-s)\right) \gamma _+(\theta _1^*,\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}dY_s\right. \right. \nonumber \\&\qquad \left. \left. -\sum _{j=1}^i\exp \left( -\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1})\right) \gamma _+(\theta _1^*,\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}\Delta _jY\right. \Biggr |^p \right. \Biggr ]^\frac{1}{p}\nonumber \\&\qquad +E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{j=1}^i\exp \left( -\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1})\right) \right. \right. \nonumber \\&\quad \left. \left. \left\{ \gamma _+(\theta _1^*,\theta _2)c(\theta _2)'{\Sigma ^*}^{-1} -\gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\right\} \Delta _jY\right. \Biggr |^p \right. \Biggr ]^\frac{1}{p}\nonumber \\&\qquad +E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{j=1}^i\left\{ \exp \left( -\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1})\right) \right. \right. \right. \nonumber \\&\qquad \left. \left. \left. -\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)(t_i-t_{j-1})\right) \right\} \gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\Delta _jY\right. \Biggr |^p\right. \Biggr ]^\frac{1}{p}. \end{aligned}$$

(3.24)

The first term of the right-hand side can be bounded by the mean value theorem and Theorem 2.1:

$$\begin{aligned} \begin{aligned}&E\left[ \sup _{\theta _2 \in \Theta _2}\left| \left\{ \exp (-\alpha (\theta _1^*,\theta _2)t)-\exp (-\alpha ({\hat{\theta }}_1^n,\theta _2)t)\right\} m_0\right| ^p \right] ^\frac{1}{p}\\&\quad \le CE\left[ |{\hat{\theta }}_1^n-\theta _1^*|^p \right] ^\frac{1}{p}\le Cn^{-\frac{1}{2}}. \end{aligned} \end{aligned}$$

(3.25)

Next we evaluate the second term using (3.16). Noting that by the mean value theorem and Lemma 3.2, we have

$$\begin{aligned}&\left| \exp (-\alpha (\theta _1^*,\theta _2)(t_i-s))-\exp (-\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1}))\right| \\&\quad =\left| \alpha (\theta _1^*,\theta _2)\exp (-\alpha (\theta _1^*,\theta _2)(t_i-u))(s-t_{j-1})\right| \\&\quad \le Ce^{-C(t_i-u)}(s-t_{j-1})\\&\quad \le Ce^{-C(t_i-s)}h \end{aligned}$$

and

$$\begin{aligned}&\left| (t_i-s)\exp (-\alpha (\theta _1^*,\theta _2)(t_i-s))-(t_i-t_{j-1})\exp (-\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1}))\right| \\&\quad \le |(t_{j-1}-s)\exp (-\alpha (\theta _1^*,\theta _2)(t_i-s)|\\&\qquad +(t_i-t_{j-1})\left| \exp (-\alpha (\theta _1^*,\theta _2)(t_i-s))-\exp (-\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1}))\right| \\&\quad \le Ce^{-C(t_i-s)}h, \end{aligned}$$

where $t_{j-1}\le u\le s\le t_j$, it follows from (3.16)

$$\begin{aligned}&E\left. \Biggl [\sup _{\theta _2 \in \Theta _2}\left. \Biggl |\int _0^{t_i}\exp \left( -\alpha (\theta _1^*,\theta _2)(t_i-s)\right) \gamma _+(\theta _1^*,\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}dY_s\right. \right. \nonumber \\&\qquad \left. \left. -\sum _{j=1}^i\exp \left( -\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1})\right) \gamma _+(\theta _1^*,\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}\Delta _jY\right| ^p \right] ^\frac{1}{p}\nonumber \\&\quad =E\left[ \sup _{\theta _2 \in \Theta _2} \left| \sum _{j=1}^i\int _{t_{j-1}}^{t_j}\left\{ \exp (-\alpha (\theta _1^*,\theta _2)(t_i-s))\right. \right. \right. \nonumber \\&\qquad \left. \left. \left. -\exp (-\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1}))\right\} \gamma _+(\theta _1^*,\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}dY_s \right. \Biggr |^p \right. \Biggr ]^\frac{1}{p}\nonumber \\&\quad \le C_p\sup _{\theta _2 \in \Theta _2}\left\{ \sum _{j=1}^i\int _{t_{j-1}}^{t_j}|\exp (-\alpha (\theta _1^*,\theta _2)(t_i-s))\right. \nonumber \\&\qquad -\exp (-\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1}))|ds\nonumber \\&\qquad +\sum _{j=1}^i\int _{t_{j-1}}^{t_j}|\partial _{\theta _2}\alpha (\theta _1^*,\theta _2)(t_i-s)\exp (-\alpha (\theta _1^*,\theta _2)(t_i-s))\nonumber \\&\qquad -\partial _{\theta _2}\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1})\exp (-\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1}))|ds\nonumber \\&\qquad +\left( \sum _{j=1}^i\int _{t_{j-1}}^{t_j}|\exp (-\alpha (\theta _1^*,\theta _2)(t_i-s))-\exp (-\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1}))|^2ds\right) ^\frac{1}{2}\nonumber \\&\qquad +\left( \sum _{j=1}^i\int _{t_{j-1}}^{t_j}|\partial _{\theta _2}\alpha (\theta _1^*,\theta _2)(t_i-s)\exp (-\alpha (\theta _1^*,\theta _2)(t_i-s))\right. \nonumber \\&\qquad \left. \left. -\partial _{\theta _2}\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1})\exp (-\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1}))|^2ds\right. \Biggr )^\frac{1}{2}\right. \Biggr \}\nonumber \\&\quad \le C_p\sum _{j=1}^i\int _{t_{j-1}}^{t_j}e^{-C(t_i-s)}dsh+C_p\left( \sum _{j=1}^ie^{-C(t_i-s)}h^2 \right) ^\frac{1}{2}\nonumber \\&\quad \le C_p\int _0^{t_i}e^{-C(t_i-s)}dsh+C_p\left( \int _0^{t_i}e^{-C(t_i-s)}dsh^2 \right) ^\frac{1}{2}\nonumber \\&\quad \le C_ph. \end{aligned}$$

(3.26)

As for the third term, in the same way as Proposition 3.13, we have

$$\begin{aligned}&E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{j=1}^i\exp \left( -\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1})\right) \right. \right. \\&\quad \left. \left. \left\{ \gamma _+(\theta _1^*,\theta _2)c(\theta _2)'{\Sigma ^*}^{-1} -\gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\right\} \Delta _jY\right. \Biggr |^p \right. \Biggr ]^\frac{1}{p}\le C_pn^{-\frac{1}{2}}, \end{aligned}$$

since it holds

$$\begin{aligned} E\left[ \sup _{\theta _2 \in \Theta _2}\left| \gamma _+(\theta _1^*,\theta _2)c(\theta _2)'{\Sigma ^*}^{-1} -\gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\right| ^p \right] ^\frac{1}{p}\le C_pn^{-\frac{1}{2}} \end{aligned}$$

(3.27)

by the mean value theorem and Theorem 2.1.

Finally, we consider the forth term of (3.24). Noting that it follows from Lemma 3.2 and the stability of $-\alpha (\theta _1,\theta _2)$,

$$\begin{aligned}&\left| \exp \left( -\left[ \alpha (\theta _1,\theta _2)+\left\{ \alpha (\theta _1^*,\theta _2)-\alpha (\theta _1,\theta _2) \right\} u\right] (t_i-t_{j-1})\right) \right| \\&\quad =\left| \exp \left( -\alpha (\theta _1,\theta _2)(1-u)(t_i-t_{j-1})\right) ||\exp \left( \alpha (\theta _1^*,\theta _2)u(t_i-t_{j-1})\right) \right| \\&\quad \le Ce^{-C(1-u)(t_{i}-t_{j-1})}e^{-Cu(t_{i}-t_{j-1})}=Ce^{-C(t_i-t_{j-1})}, \end{aligned}$$

we have

$$\begin{aligned}&\sum _{j=1}^i\left. \biggl |(t_i-t_{j-1})\right. \\&\quad \left. \int _0^1\exp \left( -\left[ \alpha (\theta _1,\theta _2)+\left\{ \alpha (\theta _1^*,\theta _2)-\alpha (\theta _1,\theta _2) \right\} u\right] (t_i-t_{j-1})\right) du\right| h\le C. \end{aligned}$$

In the same way, we obtain the boundedness of

$$\begin{aligned}&\sum _{j=1}^i\left. \biggl |(t_i-t_{j-1})\right. \\&\quad \left. \times \partial _{(\theta _1,\theta _2)}\int _0^1\exp \left( -\left[ \alpha (\theta _1,\theta _2)+\left\{ \alpha (\theta _1^*,\theta _2)-\alpha (\theta _1,\theta _2) \right\} u\right] (t_i-t_{j-1})\right) du\right| h,\\&\sum _{j=1}^i\left. \biggl |(t_i-t_{j-1})\right. \\&\quad \left. \times \int _0^1\exp \left( -\left[ \alpha (\theta _1,\theta _2)+\left\{ \alpha (\theta _1^*,\theta _2)-\alpha (\theta _1,\theta _2) \right\} u\right] (t_i-t_{j-1})\right) du\right| ^2h\\ \end{aligned}$$

and

$$\begin{aligned}&\sum _{j=1}^i\left. \biggl |(t_i-t_{j-1})\right. \\&\quad \left. \times \partial _{(\theta _1,\theta _2)}\int _0^1\exp \left( -\left[ \alpha (\theta _1,\theta _2)+\left\{ \alpha (\theta _1^*,\theta _2)-\alpha (\theta _1,\theta _2) \right\} u\right] (t_i-t_{j-1})\right) du\right| ^2h. \end{aligned}$$

Thus by (3.17) we obtain

$$\begin{aligned}&\sum _{j=1}^iE\left[ \sup _{\theta _2 \in \Theta _2}\left. \biggl |(t_i-t_{j-1})\right. \right. \\&\quad \int _0^1\exp \left( -\left[ \alpha ({\hat{\theta }}_1^n,\theta _2)+\left\{ \alpha (\theta _1^*,\theta _2)-\alpha ({\hat{\theta }}_1^n,\theta _2) \right\} u\right] (t_i-t_{j-1})\right) du\\&\qquad \left. \left. \gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\Delta _jY\right. \biggr |^p \right. \Biggr ]\le C_p. \end{aligned}$$

Therefore it follows

$$\begin{aligned} \begin{aligned}&E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{j=1}^i\left. \Bigl \{\exp \left( -\alpha (\theta _1^*,\theta _2)(t_i-t_{j-1})\right) \right. \right. \right. \\&\qquad \left. \left. \left. -\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)(t_i-t_{j-1})\right) \right\} \gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\Delta _jY\right. \Biggr |^p\right. \Biggr ]^\frac{1}{p}\\&\quad \le C_pn^{-\frac{1}{2}}. \end{aligned} \end{aligned}$$

(3.28)

Now we completed the proof by (3.24)–(3.28). $\square$

Next, we replace $m_0^*$ and $\gamma _s^*$ with $m_0$ and $\gamma _+(\theta ^*)$ in (3.10), and introduce

$$\begin{aligned} {\tilde{m}}_t^*=\exp (-a^*t)m_0+\int _0^t\exp (-a^*(t-s))\gamma _+(\theta ^*){c^*}'{{\sigma ^*}'}^{-1}d{\overline{W}}_s. \end{aligned}$$

(3.29)

Furthermore, we consider for every $n,i \in {\mathbb {N}}$,

$$\begin{aligned} {\tilde{Y}}_t&=Y_0+\int _0^tc^*{\tilde{m}}_s^*ds+\sigma ^*{\overline{W}}_t, \end{aligned}$$

(3.30)

$$\begin{aligned} {\tilde{m}}_i^n(\theta _2)&=\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)t_i\right) m_0\nonumber \\&\quad +\sum _{j=1}^i\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)(t_i-t_{j-1})\right) \gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\Delta _j{\tilde{Y}}. \end{aligned}$$

(3.31)

and

$$\begin{aligned} {\tilde{\Delta }}_iY=c^*{\tilde{m}}_{i-1}(\theta _2^*)h+\sigma ^*\Delta _i{\overline{W}}. \end{aligned}$$

(3.32)

Then in the same way as Proposition 3.13, it holds for any $p>m_1+m_2$

$$\begin{aligned} \sup _{i \in {\mathbb {N}}}E\left[ \sup _{\theta _2 \in \Theta _2}|\partial _{\theta _2}^k{\tilde{m}}_i^n(\theta _2)|^{p}\right] <\infty ~(k=0,1,2,3). \end{aligned}$$

(3.33)

Proposition 3.15

For any $p>0$ and $t\ge 0$, it holds

$$\begin{aligned} E\left[ |m_t^*-{\tilde{m}}_t^*|^p \right] ^\frac{1}{p}\le C_pe^{-Ct}. \end{aligned}$$

Proof

Use (3.10), (3.29), Lemmas 3.1 and 3.2, Proposition 3.9 and the stability of $\alpha$. $\square$

Proposition 3.16

Let $A:\Theta \rightarrow M_{d_1,d_2}({\mathbb {R}})$ be a continuous mapping. Then for any $i,n \in {\mathbb {N}}, p>0$ and $k=0,1,2,\cdots$, it holds

$$\begin{aligned}&E\left[ \sup _{\theta _1\in \Theta _1,\theta _2 \in \Theta _2}\left| \sum _{j=1}^i(t_i-t_{j-1})^k\exp \left( -\alpha (\theta _1,\theta _2)(t_i-t_{j-1})\right) \right. \right. \\&\quad \left. \left. \times A(\theta _1,\theta _2)(\Delta _j{\tilde{Y}}-\Delta _jY)\right. \Biggr |^p \right. \Biggr ]^\frac{1}{p}\le C_{p,k,A}e^{-Ct_i}. \end{aligned}$$

Proof

It can be shown that by Lemma 3.1 and Proposition 3.15, since

$$\begin{aligned} \Delta _j{\tilde{Y}}-\Delta _jY = c^*\int _{t_{j-1}}^{t_j}\{{\tilde{m}}_s^*-m_s^*\}ds. \end{aligned}$$

$\square$

By (2.14) and (3.31), we obtain the following corollaries.

Corollary 3.17

For any $i,n \in {\mathbb {N}}, p>0$ and $k=0,1,2,3,4$, it holds

$$\begin{aligned} E\left[ \sup _{\theta _2\in \Theta _2}\left| \partial _{\theta _2}^k\{{\tilde{m}}_{i}^n(\theta )-{\hat{m}}_{i}^n(\theta )\}\right| ^p \right] ^\frac{1}{p}\le C_pe^{-Ct_i}. \end{aligned}$$

Corollary 3.18

For any $i,n \in {\mathbb {N}}$ and $p>m_1+m_2$, it holds

$$\begin{aligned} E\left[ |\Delta _iY-{\tilde{\Delta }}_iY|^p \right] ^\frac{1}{p}\le C_p(h^{\frac{3}{2}}+n^{-\frac{1}{2}}h+e^{-Ct_i}h). \end{aligned}$$

Proof

This result directly follows from By (3.9), (3.32), (3.12), Lemma 3.1, Propositions 3.14 and 3.15 and Corollary 3.17. $\square$

Proposition 3.19

Let $A:\Theta \rightarrow M_{d_1,d_2}({\mathbb {R}})$ be a continuous mapping. Then for any $n \in {\mathbb {N}}, p>m_1+m_2$ and $k=0,1,2,3$

$$\begin{aligned} E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n\partial _{\theta _2}^k\{{\hat{m}}_{i-1}^n(\theta _2)-{\tilde{m}}_{i-1}^n(\theta _2)\}A(\theta _2)\Delta _i Y\right| ^{p}\right] <C_p. \end{aligned}$$

Proof

By (2.14) and (3.31), we have

$$\begin{aligned}&{\hat{m}}_{i-1}^n(\theta _2)-{\tilde{m}}_{i-1}^n(\theta _2)\\&\quad =\sum _{j=1}^i\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)(t_i-t_{j-1})\right) \gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}(\Delta _jY-\Delta _j{\tilde{Y}}). \end{aligned}$$

Hence for every $k=0,1,2,3$, $\partial _{\theta _2}^k\{{\hat{m}}_{i-1}^n(\theta _2)-{\tilde{m}}_{i-1}^n(\theta _2)\}$ is a sum of the form

$$\begin{aligned} \sum _{j=1}^i\partial _{\theta _2}^l\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)(t_i-t_{j-1})\right) A_1(\theta )(\Delta _jY-\Delta _j{\tilde{Y}})\\ (l=0,1,2,3), \end{aligned}$$

where $A_1$ is a $M_{d_1,d_2}({\mathbb {R}})$-valued k-dimensional tensor of class $C^1$. Thus if we set

$$\begin{aligned} \Phi (\theta )=\sum _{j=1}^i\partial _{\theta _2}^l\exp \left( -\alpha (\theta _1,\theta _2)(t_i-t_{j-1})\right) A_1(\theta )(\Delta _jY-\Delta _j{\tilde{Y}}), \end{aligned}$$

it is enough to show

$$\begin{aligned} E\left[ \sup _{\theta \in \Theta }\left| \sum _{i=1}^n\Phi (\theta )\Delta _i Y\right| ^{p}\right] ^\frac{1}{p}<C_p. \end{aligned}$$

(3.34)

By Haber (2018), we have

$$\begin{aligned} E\left[ \sup _{\theta \in \Theta }|\Phi (\theta )|^p \right] ^\frac{1}{p}\le C_pe^{-Ct_i} \end{aligned}$$

and

$$\begin{aligned} E\left[ \sup _{\theta \in \Theta }|\partial _{\theta }\Phi (\theta )|^p \right] ^\frac{1}{p}\le C_pe^{-Ct_i}. \end{aligned}$$

Thus it holds by (3.17) and Proposition 3.16

$$\begin{aligned} E\left[ \sup _{\theta \in \Theta }\left| \sum _{i=1}^n\Phi (\theta )\Delta _i Y\right| ^{p}\right] ^\frac{1}{p} \le C_p\sum _{i=1}^ne^{-C{t_i}}\le C_p. \end{aligned}$$

Hence we obtain (3.34). $\square$

Proposition 3.20

Let Z be a $M_{d_2}({\mathbb {R}})$-valued random variable. Then for any $n \in {\mathbb {N}}, k=0,1,2,3$ and $p>m_1+m_2$ it holds

$$\begin{aligned}&E\left[ \left| \sup _{\theta _2 \in \Theta _2}\sum _{i=1}^n\partial _{\theta _2}^k\{{\hat{m}}_{i-1}^n(\theta _2)'c(\theta _2)'\}Z\Delta _jY \right| ^p\right] ^\frac{1}{p}\\&\quad \le C_p\left( E\left[ |A|^{4p} \right] ^\frac{1}{4p}nh+E\left[ |A|^{2p} \right] ^\frac{1}{2p}(nh)^\frac{1}{2} \right) . \end{aligned}$$

Proof

By (2.14), $\partial _{\theta _2}^k\{{\hat{m}}_{i}^n(\theta _2)'c(\theta _2)'\}$ is a sum of the form

$$\begin{aligned}&A_i({\hat{\theta }}_1^n,\theta _2)\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)t_i\right) \\&\quad +\sum _{l=0}^k\sum _{j=1}^i\partial _{\theta _2}^l\exp \left( -\alpha ({\hat{\theta }}_1^n,\theta _2)(t_i-t_{j-1})\right) B_i({\hat{\theta }}_1^n,\theta _2)\Delta _jY, \end{aligned}$$

where $A_i$ and $B_i$ are k-dimensional tensor valued continuously differentiable mappings on $\Theta$. Thus if we set

$$\begin{aligned} \Psi _i(\theta )&=\Psi _i(\theta _1,\theta _2)=A_i(\theta )\exp \left( -\alpha (\theta )t_i\right) \\&\quad +\sum _{l=0}^k\sum _{j=1}^i\partial _{\theta _2}^l\exp \left( -\alpha (\theta _1,\theta _2)(t_i-t_{j-1})\right) B_i(\theta )\Delta _jY, \end{aligned}$$

it is enough to show

$$\begin{aligned} E\left[ \left| \sup _{\theta \in \Theta }\sum _{i=1}^n\Psi _{i-1}(\theta )Z\Delta _jY \right| ^p\right] ^\frac{1}{p}\le C_p\left( E\left[ |A|^{4p} \right] ^\frac{1}{4p}nh+E\left[ |A|^{2p} \right] ^\frac{1}{2p}(nh)^\frac{1}{2} \right) . \end{aligned}$$

(3.35)

In the same way as Proposition 3.13, we first obtain

$$\begin{aligned} E\left[ \left| \Psi _i(\theta )\right| ^p \right] \le C_p \end{aligned}$$

and

$$\begin{aligned} E\left[ \left| \partial _{\theta }\Psi _i(\theta )\right| ^p \right] \le C_p. \end{aligned}$$

Therefore noting that $\Psi _i(\theta )$ is ${\mathcal {F}}_{t_{i-1}}$-measurable, we obtain (3.35) by (3.17). $\square$

Next, we define $\tilde{{\mathbb {H}}}_n^2, {\tilde{\Delta }}_n^2$, ${\tilde{\Gamma }}_n^2$ and $\tilde{{\mathbb {Y}}}_n^2$ by

$$\begin{aligned} \tilde{{\mathbb {H}}}_n^2(\theta _2)&=\frac{1}{2}\sum _{i=1}^n\left\{ -h{\Sigma ^*}^{-1}[(c(\theta _2){\tilde{m}}_{i-1}^n(\theta _2))^{\otimes 2}]\right. \nonumber \\&\quad \left. +{\tilde{m}}_{i-1}^n(\theta _2)'c(\theta _2)'{\Sigma ^*}^{-1}{\tilde{\Delta }}_jY+{\tilde{\Delta }}_jY'{\Sigma ^*}^{-1}c(\theta _2){\tilde{m}}_{i-1}^n(\theta _2)\right\} \end{aligned}$$

(3.36)

$$\begin{aligned} \tilde{{\mathbb {Y}}}_n^2(\theta _2)&=\frac{1}{t_n}\{\tilde{{\mathbb {H}}}_n^2(\theta _2)-\tilde{{\mathbb {H}}}_n^2(\theta _2^*)\} \end{aligned}$$

(3.37)

$$\begin{aligned} {\tilde{\Delta }}_n^2&=\frac{1}{\sqrt{t_n}}\partial _{\theta }\tilde{{\mathbb {H}}}_n^2(\theta _2^*), \end{aligned}$$

(3.38)

and

$$\begin{aligned} {\tilde{\Gamma }}_n^2=-\frac{1}{t_n}\partial _{\theta }^2\tilde{{\mathbb {H}}}_n^2(\theta _2^*), \end{aligned}$$

(3.39)

respectively.

Proposition 3.21

For any $n \in {\mathbb {N}}, p>m_1+m_2$ and $k=0,1,2,3$, it holds

$$\begin{aligned} E\left[ \sup _{\theta _2 \in \Theta _2}\left| \partial _{\theta _2}^k\{{\mathbb {H}}_n(\theta _2)-\tilde{{\mathbb {H}}}_n(\theta _2)\}\right| ^p \right] ^\frac{1}{p}\le C_p(nh^{\frac{3}{2}}+n^\frac{1}{2}h+1). \end{aligned}$$

Proof

We only consider the case of $k=0$. The rest is the same. By (2.15) and (3.36),

$$\begin{aligned}&E\left[ \sup _{\theta _2 \in \Theta _2}\left| {\mathbb {H}}_n(\theta _2)-\tilde{{\mathbb {H}}}_n(\theta _2)\right| ^p \right] ^\frac{1}{p}\nonumber \\&\quad \le E\left[ \sup _{\theta _2 \in \Theta _2}\left| \frac{1}{2}h\sum _{i=1}^n\{\Sigma ({\hat{\theta }}_1^n)^{-1}-{\Sigma ^*}^{-1}\}[(c(\theta _2){\hat{m}}_{j-1}^n(\theta _2))^{\otimes 2}]\right| ^p \right] ^\frac{1}{p}\nonumber \\&\qquad +E\left[ \sup _{\theta _2 \in \Theta _2}\left| \frac{1}{2}\sum _{i=1}^n{\hat{m}}_{j-1}^n(\theta _2)'c(\theta _2)'\{\Sigma ({\hat{\theta }}_1^n)^{-1}-{\Sigma ^*}^{-1}\}\Delta _jY\right| ^p \right] ^\frac{1}{p}\nonumber \\&\qquad +E\left[ \sup _{\theta _2 \in \Theta _2}\left| \frac{1}{2}\sum _{i=1}^n\Delta _jY'\{\Sigma ({\hat{\theta }}_1^n)^{-1}-{\Sigma ^*}^{-1}\}c(\theta _2){\hat{m}}_{j-1}^n(\theta _2)\right| ^p \right] ^\frac{1}{p}\nonumber \\&\qquad +E\left[ \sup _{\theta _2 \in \Theta _2}\left| \frac{1}{2}h\sum _{i=1}^n\left\{ {\Sigma ^*}^{-1}[(c(\theta _2){\hat{m}}_{j-1}^n(\theta _2))^{\otimes 2}]\right. \right. \right. \nonumber \\&\qquad \left. \left. \left. -{\Sigma ^*}^{-1}[(c(\theta _2){\tilde{m}}_{j-1}^n(\theta _2))^{\otimes 2}]\right\} \right. \biggr |^p \right] ^\frac{1}{p}\nonumber \\&\qquad +E\left[ \sup _{\theta _2 \in \Theta _2}\left| \frac{1}{2}\sum _{i=1}^n\{{\hat{m}}_{j-1}^n(\theta _2)'c(\theta _2)'{\Sigma ^*}^{-1}\Delta _jY\right. \right. \nonumber \\&\qquad \left. \left. -c(\theta _2){\tilde{m}}_{j-1}^n(\theta _2){\Sigma ^*}^{-1}{\tilde{\Delta }}_jY\}\right. \biggr |^p \right] ^\frac{1}{p}\nonumber \\&\qquad +E\left[ \sup _{\theta _2 \in \Theta _2}\left| \frac{1}{2}\sum _{i=1}^n\{\Delta _jY'{\Sigma ^*}^{-1}c(\theta _2){\hat{m}}_{j-1}^n(\theta _2)\right. \right. \nonumber \\&\qquad \left. \left. -{\tilde{\Delta }}_jY'{\Sigma ^*}^{-1}c(\theta _2){\tilde{m}}_{j-1}^n(\theta _2)\}\right. \biggr |^p \right] ^\frac{1}{p}. \end{aligned}$$

(3.40)

For the first three terms of the right-hand side, we have by Theorem 2.1 and Proposition 3.13

$$\begin{aligned}&E\left[ \sup _{\theta _2 \in \Theta _2}\left| \frac{1}{2}h\sum _{i=1}^n\{\Sigma ({\hat{\theta }}_1^n)^{-1}-{\Sigma ^*}^{-1}\}[(c(\theta _2){\hat{m}}_{j-1}^n(\theta _2))^{\otimes 2}]\right| ^p \right] ^\frac{1}{p}\\&\quad \le C_pn^\frac{1}{2}h, \end{aligned}$$

and by Proposition 3.20

$$\begin{aligned}&E\left[ \sup _{\theta _2 \in \Theta _2}\left| \frac{1}{2}\sum _{i=1}^n{\hat{m}}_{j-1}^n(\theta _2)'c(\theta _2)'\{\Sigma ({\hat{\theta }}_1^n)^{-1}-{\Sigma ^*}^{-1}\}\Delta _jY\right| ^p \right] ^\frac{1}{p}\\&\quad \le C_pn^{-\frac{1}{2}}\{nh+(nh)^\frac{1}{2}\}\le C_p(n^{-\frac{1}{2}}h+h^\frac{1}{2}). \end{aligned}$$

In the same way, the third term can be bounded by $C_p(n^{-\frac{1}{2}}h+h^\frac{1}{2})$.

Furthermore, making use of Proposition 3.13, (3.33) and Corollary 3.17, we can bound the fourth term by $\displaystyle C_p\sum _{i=1}^nhe^{-Ct_i}\le C_ph$, noting that

$$\begin{aligned}&\Sigma ({\hat{\theta }}_1^n)^{-1}[(c(\theta _2){\hat{m}}_{j-1}^n(\theta _2))^{\otimes 2}]-\Sigma ({\hat{\theta }}_1^n)^{-1}[(c(\theta _2){\tilde{m}}_{j-1}^n(\theta _2))^{\otimes 2}]\\&\quad =\{{\hat{m}}_{j-1}^n(\theta _2)+{\tilde{m}}_{j-1}^n(\theta _2)\}'c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}c(\theta _2)\{{\hat{m}}_{j-1}^n(\theta _2)-{\tilde{m}}_{j-1}^n(\theta _2)\}\\&\qquad +\{{\hat{m}}_{j-1}^n(\theta _2)-{\tilde{m}}_{j-1}^n(\theta _2)\}'c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}{\tilde{m}}_{j-1}^n(\theta _2)\\&\qquad +{\tilde{m}}_{j-1}^n(\theta _2)'c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\{{\hat{m}}_{j-1}^n(\theta _2)-{\tilde{m}}_{j-1}^n(\theta _2)\}. \end{aligned}$$

Finally, the last two terms can be bounded by $\displaystyle C_p+C_p\sum _{i=1}^n(h^{\frac{3}{2}}+n^{-\frac{1}{2}}h+e^{-Ct_i}h)\le C_p(1+nh^{\frac{3}{2}}+n^\frac{1}{2}h+h)$ due to the Corollary 3.18, Proposition 3.19 and the identity

$$\begin{aligned}&{\hat{m}}_{j-1}^n(\theta _2)'c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\Delta _jY -{\tilde{m}}_{j-1}^n(\theta _2)'c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}{\tilde{\Delta }}_jY\\&\quad =\{{\hat{m}}_{j-1}^n(\theta _2)-{\tilde{m}}_{j-1}^n(\theta _2)\}'c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\Delta _jY\\&\qquad +{\tilde{m}}_{j-1}^n(\theta _2)'c(\theta _2)'\Sigma ({\hat{\theta }}_1^n)^{-1}\{\Delta _jY-{\tilde{\Delta }}_jY\}. \end{aligned}$$

Putting it all together, we obtain

$$\begin{aligned} E\left[ \sup _{\theta _2 \in \Theta _2}\left| {\mathbb {H}}_n(\theta _2)-\tilde{{\mathbb {H}}}_n(\theta _2)\right| ^p \right] ^\frac{1}{p}&\le C_p(1+nh^{\frac{3}{2}}+n^\frac{1}{2}h+h^\frac{1}{2}+h)\\&\le C_p(1+nh^{\frac{3}{2}}+n^\frac{1}{2}h). \end{aligned}$$

$\square$

Proposition 3.22

For any $p\ge 2$, it holds

$$\begin{aligned} \sup _{n \in {\mathbb {N}}}E\left[ |{\tilde{\Delta }}_n|^p \right] <\infty . \end{aligned}$$

Proof

If we set ${\tilde{M}}_j^n(\theta _2)=c(\theta _2){\tilde{m}}_j^n(\theta )$, we have

$$\begin{aligned} \begin{aligned} {\tilde{\Delta }}_n^2&=\frac{1}{2\sqrt{t_n}}\sum _{i=1}^n\left\{ \partial _{\theta _2}{\tilde{M}}_{i-1}^n(\theta _2)'{{\sigma ^*}'}^{-1}\Delta _i{\overline{W}}+\Delta _i {\overline{W}}'{\sigma ^*}^{-1}\partial _{\theta _2}{\tilde{M}}_{i-1}^n(\theta _2) \right\} \\&=\frac{1}{\sqrt{t_n}}\sum _{i=1}^n\left\{ \partial _{\theta _2}{\tilde{M}}_i^n(\theta _2)'{{\sigma ^*}'}^{-1}\Delta _i {\overline{W}} \right\} . \end{aligned} \end{aligned}$$

(3.41)

by (3.32), (3.38) and (3.51). Thus by Lemma 3.1 and (3.33),

$$\begin{aligned} E\left[ |{\tilde{\Delta }}_n|^p \right] ^\frac{2}{p}&\le \frac{1}{{t_n}^\frac{p}{2}}E\left[ \left( \sum _{i=1}^n|\partial _{\theta _2}{\tilde{M}}_i^n(\theta _2)'{{\sigma ^*}'}^{-1}|^2h\right) ^\frac{p}{2} \right] ^\frac{2}{p}\\&\le \frac{1}{{t_n}^\frac{p}{2}}C_p\sum _{i=1}^nE\left[ |\partial _{\theta _2}{\tilde{M}}_i^n(\theta _2)'{{\sigma ^*}'}^{-1}|^2 \right] ^\frac{2}{p}h\\&\le \frac{1}{{t_n}^\frac{p}{2}}\times C_pnh=C_p. \end{aligned}$$

$\square$

Next, we define the process $\{\mu _t\}$ by replacing Y with ${\tilde{Y}}$ (therefore $m_t^*$ with $m_t(\theta ^*)$ and $\gamma _t^*$ with $\gamma _+(\theta ^*)$) in (2.12);

$$\begin{aligned} \begin{aligned} \mu _t(\theta _2)&=\exp \left( -\alpha (\theta _2)t\right) m_0\\&\quad +\int _0^t\exp \left( -\alpha (\theta _2)(t-s)\right) \gamma _+(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}d{\tilde{Y}}_s. \end{aligned} \end{aligned}$$

(3.42)

Then as $m_t$ is the solution of (2.11), so $\mu _t$ is the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} d\mu _t(\theta _2)=-\alpha (\theta _2)\mu _tdt+\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}d{\tilde{Y}}_t\\ \mu _0(\theta _2)=m_0. \end{array}\right. } \end{aligned}$$

(3.43)

Moreover, it holds $\mu _t(\theta _2^*)={\tilde{m}}_t^*$ since by (3.31) ${\tilde{m}}_t^*$ is the solution of

$$\begin{aligned} d{\tilde{m}}_t^*=-a^*{\tilde{m}}_t^*+\gamma _+(\theta ^*){c^*}'{{\sigma ^*}'}^{-1}d{\overline{W}}_t, \end{aligned}$$

which is equivalent to

$$\begin{aligned} d{\tilde{m}}_t^*=-\alpha (\theta _2^*){\tilde{m}}_t^*dt+\gamma _{+}(\theta ^*){c^*}'{\Sigma ^*}^{-1}d{\tilde{Y}}_t. \end{aligned}$$

Moreover, just as Proposition 3.14, the following proposition holds:

Proposition 3.23

For any $n,i \in {\mathbb {N}}$ and $p>m_1+m_2$, we have

$$\begin{aligned} E\left[ \sup _{\theta _2 \in \Theta _2}|\mu _{t_i}(\theta _2)-{\tilde{m}}_i^n(\theta _2)|^p \right] ^\frac{1}{p}\le C_p(n^{-\frac{1}{2}}+h). \end{aligned}$$

Together with (3.33), we obtain the following corollary.

Corollary 3.24

For any $i \in {\mathbb {N}}$ and $p>m_1+m_2$, we have

$$\begin{aligned} E\left[ \sup _{\theta _2 \in \Theta _2}|\mu _{t_i}(\theta _2)|^p \right] ^\frac{1}{p}\le C_p. \end{aligned}$$

Proposition 3.25

$$\begin{aligned} E[{\Sigma ^*}^{-1}[\{c(\theta _2)\mu _t(\theta _2)-c(\theta _2^*)\mu _t(\theta _2^*)\}^{\otimes 2}]],=-2{\mathbb {Y}}(\theta _2)+O(e^{-t_{i}}) \end{aligned}$$

where $O(e^{-t})$ is some continuous function $r:\Theta \rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} |r(\theta )|\le Ce^{-Ct}. \end{aligned}$$

Proof

By (3.42) and (3.31), we have

$$\begin{aligned} \mu _t(\theta )&=\exp (-\alpha (\theta _2)t)m_0\nonumber \\&\quad +\int _0^t\exp (-\alpha (\theta _2)(t-s))\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*{\tilde{m}}_sds\nonumber \\&\quad +\int _0^t\exp (-\alpha (\theta _2)(t-s))\gamma _{+}(\theta _2)c(\theta _2)'{{\sigma ^*}'}^{-1}d{\overline{W}}_s\nonumber \\&=\exp (-\alpha (\theta _2)t)m_0\nonumber \\&\quad +\int _0^t\exp (-\alpha (\theta _2)(t-s))\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\nonumber \\&\quad \times \left\{ \exp (-a^*s)m_0+\int _0^s\exp (-a^*(s-u))\gamma _+(\theta ^*){c^*}'{{\sigma ^*}'}^{-1}d{\overline{W}}_u\right\} ds\nonumber \\&\quad +\int _0^t\exp (-\alpha (\theta _2)(t-s))\gamma _{+}(\theta _2)c(\theta _2)'{{\sigma ^*}'}^{-1}d{\overline{W}}_s\nonumber \\&=\exp (-\alpha (\theta _2)t)m_0\nonumber \\&\quad +\int _0^t\exp (-\alpha (\theta _2)(t-s))\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\exp (-a^*s)m_0ds\nonumber \\&\quad +\int _0^t\int _0^s\exp (-\alpha (\theta _2)(t-s))\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\nonumber \\&\quad \times \exp (-a^*(s-u))\gamma _+(\theta ^*){c^*}'{{\sigma ^*}'}^{-1}d{\overline{W}}_uds\nonumber \\&\quad +\int _0^t\exp (-\alpha (\theta _2)(t-s))\gamma _{+}(\theta _2)c(\theta _2)'{{\sigma ^*}'}^{-1}d{\overline{W}}_s\nonumber \\&=\exp (-\alpha (\theta _2)t)m_0\nonumber \\&\quad +\int _0^t\exp (-\alpha (\theta _2)(t-s))\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\exp (-a^*s)m_0ds\nonumber \\&\quad +\int _0^t\left\{ \int _s^t\exp (-\alpha (\theta _2)(t-u))\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\right. \nonumber \\&\quad \exp (-a^*(u-s))\gamma _+(\theta ^*){c^*}'{{\sigma ^*}'}^{-1}du\nonumber \\&\quad \left. +\exp (-\alpha (\theta _2)(t-s))\gamma _{+}(\theta _2)c(\theta _2)'{{\sigma ^*}'}^{-1}\right\} d{\overline{W}}_s. \end{aligned}$$

(3.44)

Therefore

$$\begin{aligned}&E[{\Sigma ^*}^{-1}[\{c(\theta _2)\mu _t(\theta _2)-c(\theta _2^*)\mu _t(\theta _2^*)\}^{\otimes 2}]]\\&\quad =E[{\Sigma ^*}^{-1}[\{c(\theta _2)\mu _t(\theta _2)-c(\theta _2^*){\tilde{m}}_t^*\}^{\otimes 2}]]\\&\quad =E\left[ {\Sigma ^*}^{-1}\left[ \left\{ \int _0^t\left\{ \int _s^tc(\theta _2)\exp (-\alpha (\theta _2)(t-u))\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\right. \right. \right. \right. \\&\qquad \times \exp (-a^*(u-s))\gamma _+(\theta ^*){c^*}'du\\&\qquad +c(\theta _2)\exp (-\alpha (\theta _2)(t-s))\gamma _{+}(\theta _2)c(\theta _2)'\\&\qquad \left. \left. \left. \left. -c^*\exp (-a^*(t-s))\gamma _+(\theta ^*){c^*}'\right\} {{\sigma ^*}'}^{-1}d{\overline{W}}_s\right\} ^{\otimes 2} \right] \right] +O(e^{-Ct})\\&\quad =\textrm{Tr}\int _0^t{\Sigma ^*}^{-1}\left[ \left\{ \int _s^tc(\theta _2)\exp (-\alpha (\theta _2)(t-u))\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\right. \right. \\&\qquad \times \exp (-a^*(u-s))\gamma _+(\theta ^*){c^*}'du\\&\qquad +c(\theta _2)\exp (-\alpha (\theta _2)(t-s))\gamma _{+}(\theta _2)c(\theta _2)'\\&\qquad \left. \left. -c^*\exp (-a^*(t-s))\gamma _+(\theta ^*){c^*}' \right\} ^{\otimes 2} \right] [({{\sigma ^*}'}^{-1})^{\otimes 2}]ds+O(e^{-Ct})\\&\quad =\textrm{Tr}\int _0^t{\Sigma ^*}^{-1}\left. \Biggl [ \left\{ \int _0^{s}c(\theta _2)\exp (-\alpha (\theta _2)u)\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\right. \right. \\&\qquad \times \exp (-a^*(s-u))\gamma _+(\theta ^*){c^*}'du\\&\qquad +c(\theta _2)\exp (-\alpha (\theta _2)s)\gamma _{+}(\theta _2)c(\theta _2)'\\&\qquad \left. \left. -c^*\exp (-a^*s)\gamma _+(\theta ^*){c^*}' \right. \biggr \}^{\otimes 2} \right] [({{\sigma ^*}'}^{-1})^{\otimes 2}]ds+O(e^{-Ct}). \end{aligned}$$

Now we have

$$\begin{aligned}&\int _0^s|c(\theta _2)\exp (-\alpha (\theta _2)u)\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\exp (-a^*(s-u))\gamma _+(\theta ^*){c^*}'|du\\&\quad \le \int _0^sC_pe^{-Cu}e^{-C(s-u)}du\le C_pse^{-Cs}\le C_pe^{-Cs} \end{aligned}$$

and thus by (2.6)

$$\begin{aligned}&|E[{\Sigma ^*}^{-1}[\{c(\theta _2)\mu _t(\theta _2)-c(\theta _2^*)\mu _t(\theta _2^*)\}^{\otimes 2}]]+2 {\mathbb {Y}}^2(\theta _2)|\\&\quad =\left| \int _{t}^\infty {\Sigma ^*}^{-1}\left[ \left\{ \int _0^{s}c(\theta _2)\exp (-\alpha (\theta _2)u)\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\right. \right. \right. \\&\qquad \times \exp (-a^*(s-u))\gamma _+(\theta ^*){c^*}'du\\&\qquad +c(\theta _2)\exp (-\alpha (\theta _2)s)\gamma _{+}(\theta _2)c(\theta _2)'\\&\qquad \left. \left. \left. -c^*\exp (-a^*s)\gamma _+(\theta ^*){c^*}' \right\} ^{\otimes 2} \right] [({{\sigma ^*}'}^{-1})^{\otimes 2}]ds\right| \\&\quad \le Ce^{-Ct}. \end{aligned}$$

$\square$

Proposition 3.26

For any $n \in {\mathbb {N}}$ and $p>m_1+m_2$, it holds

$$\begin{aligned} E\left[ \sup _{\theta _2 \in \Theta _2}|\tilde{{\mathbb {Y}}}_n^2(\theta _2)-{\mathbb {Y}}^2(\theta _2)|^p \right] ^\frac{1}{p}\le C_p\left( h+n^{-\frac{1}{2}}+{t_n}^{-\frac{1}{2}} \right) . \end{aligned}$$

Proof

By (3.30) and (3.37)

$$\begin{aligned} \tilde{{\mathbb {Y}}}_n^2(\theta _2)&=\frac{1}{2t_n}\sum _{i=1}^n\left\{ -h{\Sigma ^*}^{-1}[(c(\theta _2){\tilde{m}}_{i-1}^n(\theta _2))^{\otimes 2}]+h{\Sigma ^*}^{-1}[(c^*{\tilde{m}}_{i-1}^n(\theta _2^*))^{\otimes 2}]\right. \\&\quad +\{{\tilde{m}}_{i-1}^n(\theta _2)'c(\theta _2)'-{\tilde{m}}_{i-1}^n(\theta _2^*)'{c^*}'\}{\Sigma ^*}^{-1}(c^*{\tilde{m}}_{i-1}(\theta _2^*)h+\sigma ^*\Delta _j{\overline{W}})\\&\quad +\left. ({\tilde{m}}_{i-1}(\theta _2^*){c^*}'h+\Delta _j{\overline{W}}'{\sigma ^*}'){\Sigma ^*}^{-1}\{c(\theta _2){\tilde{m}}_{i-1}^n(\theta _2)-c^*{\tilde{m}}_{i-1}^n(\theta _2^*)\}\right\} \\&=\frac{1}{2t_n}\sum _{i=1}^n\left\{ -h{\Sigma ^*}^{-1}[(c(\theta _2){\tilde{m}}_{i-1}^n(\theta _2)-c^*{\tilde{m}}_{i-1}^n(\theta _2^*))^{\otimes 2}]\right. \\&\quad +\{{\tilde{m}}_{i-1}^n(\theta _2)'c(\theta _2)'-{\tilde{m}}_{i-1}^n(\theta _2^*)'{c^*}'\}{\Sigma ^*}^{-1}\sigma ^*\Delta _j{\overline{W}})\\&\quad \left. +\Delta _j{\overline{W}}'{\sigma ^*}'{\Sigma ^*}^{-1}\{c(\theta _2){\tilde{m}}_{i-1}^n(\theta _2)-c^*{\tilde{m}}_{i-1}^n(\theta _2^*)\}\right\} . \end{aligned}$$

Thus we have

$$\begin{aligned}&E\left[ \sup _{\theta _2 \in \Theta _2}|\tilde{{\mathbb {Y}}}_n^2(\theta _2)-{\mathbb {Y}}^2(\theta _2)|^p \right] ^\frac{1}{p}\nonumber \\&\quad \le \frac{h}{2t_n}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n{\Sigma ^*}^{-1}[(c(\theta _2){\tilde{m}}_{i-1}^n(\theta _2)-c^*{\tilde{m}}_{i-1}^n(\theta _2^*))^{\otimes 2}]\right. \right. \nonumber \\&\qquad \left. \left. -{\Sigma ^*}^{-1}[(c(\theta _2)\mu _{t_{i-1}}(\theta _2)-c^*\mu _{t_{i-1}}(\theta _2^*))^{\otimes 2}]\right. \Biggr |^p \right] ^\frac{1}{p}\nonumber \\&\qquad +\frac{h}{2t_n}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n\left\{ {\Sigma ^*}^{-1}[(c(\theta _2)\mu _{t_{i-1}}(\theta _2)-c^*\mu _{t_{i-1}}(\theta _2^*))^{\otimes 2}]\right. \right. \right. \nonumber \\&\qquad \left. \left. \left. +2{\mathbb {Y}}^2(\theta )\right\} \right. \Biggr |^p \right] ^\frac{1}{p}\nonumber \\&\qquad +\frac{1}{2t_n}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n\{{\tilde{m}}_{i-1}^n(\theta _2)'c(\theta _2)'-{\tilde{m}}_{i-1}^n(\theta _2^*)'{c^*}'\}{\Sigma ^*}^{-1}\sigma ^*\Delta _j{\overline{W}}\right| ^p \right] ^\frac{1}{p}\nonumber \\&\qquad +\frac{1}{2t_n}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n\Delta _j{\overline{W}}'{\sigma ^*}'{\Sigma ^*}^{-1}\{c(\theta _2){\tilde{m}}_{i-1}^n(\theta _2)-c^*{\tilde{m}}_{i-1}^n(\theta _2^*)\right| ^p \right] ^\frac{1}{p}.\nonumber \\ \end{aligned}$$

(3.45)

For the first term of this, making use of Proposition 3.23, Corollary 3.24 and (3.33), we obtain

$$\begin{aligned} \begin{aligned}&\frac{h}{2t_n}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n{\Sigma ^*}^{-1}[(c(\theta _2){\tilde{m}}_{i-1}^n(\theta _2)-c^*{\tilde{m}}_{i-1}^n(\theta _2^*))^{\otimes 2}]\right. \right. \\&\qquad \left. \left. -{\Sigma ^*}^{-1}[(c(\theta _2)\mu _{t_{i-1}}(\theta _2)-c^*\mu _{t_{i-1}}(\theta _2^*))^{\otimes 2}]\right. \Biggr |^p \right] ^\frac{1}{p}\\&\quad \le C_p\frac{h}{2t_n}\times (n^{-\frac{1}{2}}+h)\times n\le C_p(n^{-\frac{1}{2}}+h), \end{aligned} \end{aligned}$$

(3.46)

just as we evaluated the fourth term of (3.40).

Now we consider the second term. Due to the proof of Proposition 3.25, $c(\theta _2)\mu _{t_i}^n(\theta _2)-c^*\mu _{t_i}^n(\theta _2^*)$ has the form

$$\begin{aligned} c(\theta _2)\mu _{t_i}^n(\theta _2)-c^*\mu _{t}^n(\theta _2^*) =p_i(\theta _2)+\int _0^{t_i}q_i(s;\theta _2)d{\overline{W}}_s \end{aligned}$$

where

$$\begin{aligned} p_i(\theta _2)&=\exp (-\alpha (\theta _2)t_i)m_0-\exp (-\alpha (\theta _2^*)t_i)m_0\\&\quad +\int _0^{t_i}\{\exp (-\alpha (\theta _2)(t_i-s))\gamma _{+}(\theta _2)c(\theta _2)'\\&\quad -\exp (-\alpha (\theta _2^*)(t_i-s))\gamma _{+}(\theta _2^*){c^*}'\}{\Sigma ^*}^{-1}c^*\exp (-a^*s)m_0ds,\\ q_i(s;\theta _2)&=\int _s^{t_i}c(\theta _2)\exp (-\alpha (\theta _2)(t_i-u))\gamma _{+}(\theta _2)c(\theta _2)'\\&\quad \times {\Sigma ^*}^{-1}c^*\exp (-a^*(u-s))\gamma _+(\theta ^*){c^*}'du\\&\quad +c(\theta _2)\exp (-\alpha (\theta _2)(t_i-s))\gamma _{+}(\theta _2)c(\theta _2)'\\&\quad -c^*\exp (-a^*(t_i-s))\gamma _+(\theta ^*){c^*}'. \end{aligned}$$

Then if we set $\displaystyle \nu ^i_t(\theta _2)=p_i(\theta _2)+\int _0^{t}q_i(s;\theta _2)d{\overline{W}}_s$, Itô’s formula gives

$$\begin{aligned}&{\Sigma ^*}^{-1}\left[ \left\{ c(\theta _2)\mu _{t_i}^n(\theta _2)-c^*\mu _{t}^n(\theta _2^*) \right\} ^{\otimes 2}\right] \\&\quad ={\Sigma ^*}^{-1}[( \nu ^i_{t_i}(\theta _2) )^{\otimes 2}] =\int _0^{t_i}{\Sigma ^*}^{-1}[( \nu ^i_{t_i}(\theta _2) )^{\otimes 2}]\\&\quad ={\Sigma ^*}^{-1}[p_i(\theta _2)^{\otimes 2}]+2\int _{0}^{t_i}{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)d{\overline{W}}_s\\&\qquad +\textrm{Tr}\int _0^{t_i}{\Sigma ^*}^{-1}[q_i(s;\theta _2)^{\otimes 2}]ds\\&\quad =E\left[ {\Sigma ^*}^{-1}\left[ \left\{ c(\theta _2)\mu _{t_i}^n(\theta _2)-c^*\mu _{t}^n(\theta _2^*) \right\} ^{\otimes 2}\right] \right] \\&\qquad +2\int _{0}^{t_i}{\nu ^i_{s}(\theta _2)(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)d{\overline{W}}_s\\&\quad =-2{\mathbb {Y}}^2(\theta _2)+2\int _{0}^{t_i}{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)d{\overline{W}}_s+O(e^{-Ct_i}). \end{aligned}$$

Therefore

$$\begin{aligned} \begin{aligned}&\frac{h}{2t_n}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n\left\{ {\Sigma ^*}^{-1}[(c(\theta _2)\mu _{t_{i-1}}^n(\theta _2)-c^*\mu _{t_{i-1}}^n(\theta _2^*))^{\otimes 2}]+2{\mathbb {Y}}^2(\theta )\right\} \right| ^p \right] ^\frac{1}{p}\\&\quad \le \frac{h}{t_n}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n\int _{0}^{t_i}{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)d{\overline{W}}_s\right| ^p \right] ^\frac{1}{p}+\frac{1}{2t_n}\sum _{i=1}^nCe^{-Ct_i}h\\&\quad \le \frac{h}{t_n}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n\int _{0}^{t_i}{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)d{\overline{W}}_s\right| ^p \right] ^\frac{1}{p}+\frac{C}{t_n}. \end{aligned} \end{aligned}$$

(3.47)

Now by Lemma 3.11 and the continuos differentiability of $p_i$ and $q_i$, we can assume $\displaystyle \nu ^i_t(\theta _2)$ is continuously differentiable with respect to $\theta _2$ and almost surely

$$\begin{aligned} \partial _{\theta _2}\nu ^i_t(\theta _2)=\partial _{\theta _2}p_i(\theta _2)+\int _0^t\partial _{\theta _2}q_i(s;\theta _2)ds. \end{aligned}$$

Thus by Lemma 3.1 (2) we obtain for any $T>0,p\ge 2$ and $\theta _2,\theta _2' \in \Theta _2$

$$\begin{aligned} \sup _{0\le t\le T}E\left[ |\nu _t^i(\theta _2)-\nu _t^i(\theta _2')|^p \right] \le C_p|\theta _2-\theta _2'|^p \end{aligned}$$

and

$$\begin{aligned} \sup _{0\le t\le T}E\left[ |\partial _{\theta _2}\nu _t^i(\theta _2)-\partial _{\theta _2}\nu _t^i(\theta _2')|^p \right] \le C_p|\theta _2-\theta _2'|^p. \end{aligned}$$

Then again by Lemma 3.11, $\displaystyle \int _{0}^{t_i}{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)d{\overline{W}}_s$ is continuously differentiable and we have almost surely

$$\begin{aligned} \partial _{\theta _2}\int _{0}^{t_i}{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)d{\overline{W}}_s =\int _{0}^{t_i}\partial _{\theta _2}\{{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)\}d{\overline{W}}_s. \end{aligned}$$

Therefore the Sobolev inequality gives for any $p>m_1+m_2$

$$\begin{aligned} \begin{aligned}&E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n\int _{0}^{t_i}{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)d{\overline{W}}_s\right| ^p \right] ^\frac{1}{p}\\&\quad =E\left[ \sup _{\theta _2 \in \Theta _2}\left| \int _{0}^{t_n}\sum _{i=1}^n{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)1_{[0,t_i]}(s)d{\overline{W}}_s\right| ^p \right] ^\frac{1}{p}\\&\quad \le C_p\sup _{\theta _2 \in \Theta _2}E\left[ \left| \int _{0}^{t_n}\sum _{i=1}^n{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)1_{[0,t_i]}(s)d{\overline{W}}_s\right| ^p \right] ^\frac{1}{p}\\&\qquad +C_p\sup _{\theta _2 \in \Theta _2}E\left[ \left| \int _{0}^{t_n}\sum _{i=1}^n\partial _{\theta _2}\{{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)\}1_{[0,t_i]}(s)d{\overline{W}}_s\right| ^p \right] ^\frac{1}{p}. \end{aligned} \end{aligned}$$

(3.48)

Now we have $|p_t(\theta _2)|\le Ce^{-Ct_i},|q_i(s;\theta _2)|\le Ce^{-C(t_i-s)}$ and hence

$$\begin{aligned} E\left[ |{\nu ^i_{s}(\theta _2)}|^p \right] \le C_p. \end{aligned}$$

Thus we obtain

$$\begin{aligned}&E\left[ \left| \sum _{i=1}^n{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)1_{[0,t_i]}(s)\right| ^p\right] ^\frac{1}{p}\le \frac{C_p}{h}, \end{aligned}$$

and therefore by Lemma 3.1

$$\begin{aligned}&E\left[ \left| \int _0^{t_n}\sum _{i=1}^n{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)1_{[0,t_i]}(s)d{\overline{W}}_s\right| ^p\right] \le \frac{C_p}{h}{t_n}^{\frac{p}{2}}. \end{aligned}$$

In the same way, we obtain

$$\begin{aligned} E\left[ \left| \int _{0}^{t_n}\sum _{i=1}^n\partial _{\theta _2}\{{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)\}1_{[0,t_i]}(s)d{\overline{W}}_s\right| ^p \right] ^\frac{1}{p}\le \frac{C_p}{h}{t_n}^{\frac{p}{2}}. \end{aligned}$$

Hence by (3.48), it follows

$$\begin{aligned} E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n\int _{0}^{t_i}{\nu ^i_{s}(\theta _2)}'{\Sigma ^*}^{-1}q_i(s;\theta _2)d{\overline{W}}_s\right| ^p \right] ^\frac{1}{p} \le \frac{C_p}{h}{t_n}^{\frac{p}{2}}, \end{aligned}$$

and therefore by (3.47)

$$\begin{aligned} \begin{aligned}&\frac{h}{2t_n}E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n{\Sigma ^*}^{-1}[(c(\theta _2)\mu _{t_{i-1}}^n(\theta _2)-c^*\mu _{t_{i-1}}^n(\theta _2^*))^{\otimes 2}]-{\mathbb {Y}}^2(\theta )\right| ^p \right] ^\frac{1}{p}\\&\quad \le C_p\frac{h}{t_n}\frac{{t_n}^{\frac{p}{2}}}{h}+\frac{C}{t_n} \le C_p\frac{1}{{t_n}^\frac{1}{2}}. \end{aligned} \end{aligned}$$

(3.49)

Finally, as for the third and fourth terms in (3.45), by the Sobolev inequality, Lemma 3.1 and (3.33) it holds

$$\begin{aligned} \begin{aligned}&E\left[ \sup _{\theta _2 \in \Theta _2}\left| \sum _{i=1}^n\{{\tilde{m}}_{i-1}^n(\theta _2)'c(\theta _2)'-{\tilde{m}}_{i-1}^n(\theta _2^*)'{c^*}'\}{\Sigma ^*}^{-1}\sigma ^*\Delta _j{\overline{W}}\right| ^p \right] \\&\quad \le C_p{t_n}^{\frac{p}{2}}. \end{aligned} \end{aligned}$$

(3.50)

We obtain the desired result by putting (3.45), (3.46), (3.49) and (3.50) together. $\square$

Now we set

$$\begin{aligned} {\tilde{M}}_j^n(\theta _2)=c(\theta _2){\tilde{m}}_j^n(\theta ). \end{aligned}$$

(3.51)

Then by (3.32) and (3.39), we have

$$\begin{aligned} {\tilde{\Gamma }}_n^2&=\frac{1}{t_n}\sum _{i=1}^n\left\{ {\Sigma ^*}^{-1}[\partial _{\theta _2}^{\otimes 2}]{\tilde{M}}_{i}^n(\theta ^*)h\right. \\&\quad \left. -\partial _{\theta _2}^2{\tilde{M}}_{i-1}^n(\theta ^*)'{{\sigma ^*}'}^{-1}\Delta _j{\overline{W}}-\Delta _j{\overline{W}}'{\sigma ^*}^{-1}\partial _{\theta _2}^2{\tilde{M}}_{i-1}^n(\theta ^*) \right\} . \end{aligned}$$

Moreover, by (2.16) and (3.44), we obtain the following results in the same way as Propositions 3.25 and 3.26:

$$\begin{aligned}&E\left[ {\Sigma ^*}^{-1}[\partial _{\theta _2}^{\otimes 2}]{\tilde{M}}_{i}^n(\theta ^*) \right] =\Gamma ^2+O(e^{-Ct_i}) \end{aligned}$$

(3.52)

$$\begin{aligned}&E\left[ \left| \frac{1}{t_n}\sum _{i=1}^n{\Sigma ^*}^{-1}[\partial _{\theta _2}^{\otimes 2}]{\tilde{M}}_{i}^n(\theta ^*)h-\Gamma ^2\right| ^p \right] \le C_p\left( h^p+n^{-\frac{1}{2}p}+\frac{1}{{t_n}^{\frac{p}{2}}} \right) \end{aligned}$$

(3.53)

$$\begin{aligned}&E\left[ |{\tilde{\Gamma }}_n-\Gamma ^2|^p \right] \le C_p\left( h^p+n^{-\frac{1}{2}p}+\frac{1}{{t_n}^{\frac{p}{2}}} \right) . \end{aligned}$$

(3.54)

Proposition 3.27

It holds

$$\begin{aligned} {\tilde{\Delta }}_n^2 \xrightarrow {d} N(0,\Gamma ^2). \end{aligned}$$

Proof

Since ${\tilde{\Delta }}_n^2$ is given by the formula (3.41), we set

$$\begin{aligned} \xi _i^n=\frac{1}{\sqrt{t_n}}\partial _{\theta _2}{\tilde{M}}_{i-1}^n(\theta _2^*)'{{\sigma ^*}'}^{-1}\Delta _{i} {\overline{W}}. \end{aligned}$$

Then $(\xi _i^n)^{\otimes 2}$ is the matrix whose (i, j) entry is

$$\begin{aligned}&\frac{1}{t_n}\frac{\partial }{\partial \theta _2^{i}}{\tilde{M}}_{i-1}^n(\theta _2^*)'{{\sigma ^*}'}^{-1}\Delta _{i} {\overline{W}}\frac{\partial }{\partial \theta _2^{j}}{\tilde{M}}_{i-1}^n(\theta _2^*)'{{\sigma ^*}'}^{-1}\Delta _{i} {\overline{W}}\\&\quad =\frac{1}{t_n}\frac{\partial }{\partial \theta _2^{i}}{\tilde{M}}_{i-1}^n(\theta _2^*)'{{\sigma ^*}'}^{-1}\Delta _{i} {\overline{W}}\Delta _{i} {\overline{W}}'{{\sigma ^*}}^{-1}\frac{\partial }{\partial \theta _2^{j}}{\tilde{M}}_{i-1}^n(\theta _2^*). \end{aligned}$$

Hence it follows from (3.53)

$$\begin{aligned} \sum _{i=1}^nE\left[ (\xi _i^n)^{\otimes 2}|{\mathcal {F}}_{t_{i-1}} \right] =\sum _{i=1}^nE\left[ {\Sigma ^*}^{-1}[\partial _{\theta _2}^{\otimes 2}]{\tilde{M}}_{i-1}^n(\theta _2)|{\mathcal {F}}_{t_{i-1}} \right] \xrightarrow {P}\Gamma ^2~(n \rightarrow \infty ). \end{aligned}$$

Moreover, we have for $\epsilon >0$

$$\begin{aligned}&\sum _{i=1}^nE[|\xi _i^n|^21_{\{|\xi _i^n|>\epsilon \}}|{\mathcal {F}}_{t_{i-1}}]\\&\quad \le \sum _{i=1}^nE[|\xi _i^n|^4|{\mathcal {F}}_{t_{i-1}}]^{\frac{1}{2}}P(|\xi _i^n|>\epsilon |{\mathcal {F}}_{t_{i-1}})^{\frac{1}{2}}\\&\quad \le \sum _{i=1}^nE[|\xi _i^n|^4|{\mathcal {F}}_{t_{i-1}}]^{\frac{1}{2}}\times \frac{1}{\epsilon ^2}E[|\xi _i^n|^4|{\mathcal {F}}_{t_{i-1}}]^{\frac{1}{2}}\\&\quad =\sum _{i=1}^n\frac{|{\sigma ^*}^{-1}|^4}{\epsilon ^2{t_n}^2}|\partial _{\theta }{\tilde{M}}_{i-1}^n(\theta ^*)|^4E[(\Delta _i{\overline{W}})^4]\\&\quad \le \frac{|{\sigma ^*}^{-1}|^4}{\epsilon ^2{t_n}^2}\sum _{i=1}^n|\partial _{\theta }{\tilde{M}}_{i-1}^n(\theta ^*)|^4h^2, \end{aligned}$$

and hence

$$\begin{aligned} E\left[ \sum _{i=1}^nE[|\xi _i^n|^21_{\{|\xi _i^n|>\epsilon \}}|{\mathcal {F}}_{t_{i-1}}] \right]&\le \sum _{i=1}^n\frac{|{\sigma ^*}^{-1}|^4}{\epsilon ^2{t_n}^2}E[|\partial _{\theta }{\tilde{M}}_{i-1}^n(\theta ^*)|^4]h^2\\&\le C_{\epsilon }\sum _{i=1}^n\frac{1}{{t_n}^2}h^2=\frac{C_{\epsilon }}{n}\rightarrow 0~(n \rightarrow \infty ). \end{aligned}$$

Therefore we obtain the desired result by the martingale central limit theorem. $\square$

Proposition 3.28

For any $p>m_1+m_2$, it holds

$$\begin{aligned} \sup _{n \in {\mathbb {N}}}E\left[ \sup _{\theta _2\in \Theta _2}\left| \frac{1}{t_n}\partial _{\theta _2}^3 {\mathbb {H}}_n^2(\theta _2)\right| ^p \right] ^\frac{1}{p}<\infty . \end{aligned}$$

Proof

Use (2.15) to divide the left-hand side into three parts, and then apply (3.13) and Proposition 3.20. $\square$

Proof of Theorem 2.2

We set $\Delta _n^2, \Gamma _n^2$ and ${\mathbb {Y}}_n^2$ by

$$\begin{aligned} {\mathbb {Y}}_n^2(\theta _2)&=\frac{1}{t_n}\{{\mathbb {H}}_n^2(\theta _2)-{\mathbb {H}}_n^2(\theta _2^*)\} \end{aligned}$$

(3.55)

$$\begin{aligned} \Delta _n^2&=\frac{1}{\sqrt{t_n}}\partial _{\theta }{\mathbb {H}}_n^2(\theta _2^*) \end{aligned}$$

(3.56)

$$\begin{aligned} \Gamma _n^2&=-\frac{1}{t_n}\partial _{\theta }^2{\mathbb {H}}_n^2(\theta _2^*). \end{aligned}$$

(3.57)

Then by Proposition 3.21 for any $n \in {\mathbb {N}}$ and $p>m_1+m_2$, it holds

$$\begin{aligned}&E\left[ \left| \Delta _n^2-{\tilde{\Delta }}_n^2\right| ^p \right] ^\frac{1}{p}\le C_p\left( n^\frac{1}{2}+h^\frac{1}{2}+(nh)^{-1} \right) \end{aligned}$$

(3.58)

$$\begin{aligned}&E\left[ \left| \Gamma _n^2-{\tilde{\Gamma }}_n^2\right| ^p \right] ^\frac{1}{p}\le C_p\left( h^\frac{1}{2}+n^{-\frac{1}{2}}+(nh)^{-1} \right) \end{aligned}$$

(3.59)

and

$$\begin{aligned} E\left[ \sup _{\theta _2 \in \Theta _2}\left| {\mathbb {Y}}_n^2(\theta _2)-\tilde{{\mathbb {Y}}}_n^2(\theta _2)\right| ^p \right] ^\frac{1}{p}\le C_p\left( h^\frac{1}{2}+n^{-\frac{1}{2}}+(nh)^{-1} \right) . \end{aligned}$$

Together with Proposition 3.22, (3.54) and Proposition 3.26, we have for any $p>m_1+m_2$ (therefore for any $p>0$)

$$\begin{aligned}&\sup _{n \in {\mathbb {N}}}E\left[ |\Delta _n^2|^p \right] ^\frac{1}{p}<\infty , \end{aligned}$$

(3.60)

$$\begin{aligned}&\sup _{n \in {\mathbb {N}}}E\left[ \left| {t_n}^\frac{1}{2}(\Gamma _n^2-\Gamma ^2)\right| ^p \right] ^\frac{1}{p}<\infty \end{aligned}$$

(3.61)

and

$$\begin{aligned} \sup _{n \in {\mathbb {N}}}E\left[ \sup _{\theta _2 \in \Theta _2}\left| {t_n}^\frac{1}{2}({\mathbb {Y}}_n^2(\theta )-{\mathbb {Y}}^2(\theta _2))\right| ^p \right] ^\frac{1}{p}<\infty . \end{aligned}$$

(3.62)

Moreover, by Proposition 3.27 and (3.58) we obtain

$$\begin{aligned} \Delta _n \xrightarrow {d} N(0,\Gamma ^2). \end{aligned}$$

(3.63)

Then we have proved the theorem by the assumption [A5], Proposition 3.28, (3.60)–(3.63) and Theorem 5 in Yoshida (2011). $\square$

4 One-dimensional case

In this section, we consider the special case where $d_1=d_2=1$. In this case, $a(\theta _2),b(\theta _2),c(\theta _2)$ and $\sigma (\theta _1)$ are scalar valued, so we set $m_1=1$ and $\sigma (\theta _1)=\theta _1$. Moreover, we assume $\Theta _1 \subset (\epsilon ,\infty )$ for some $\epsilon >0$. In the one-dimensional case, (1.5) is reduced to

$$\begin{aligned} \frac{c(\theta _2)^2}{{\theta _1}^2}\gamma ^2+2a(\theta _2)\gamma +b(\theta _2)^2=0, \end{aligned}$$

and the larger solution of this is

$$\begin{aligned} \gamma _+(\theta _1,\theta _2)=\frac{{\theta _1}^2a(\theta _2)}{c(\theta _2)^2}\left( \sqrt{1+\frac{b(\theta _2)^2c(\theta _2)^2}{{\theta _1}^2a(\theta _2)^2}}-1 \right) . \end{aligned}$$

Thus we have

$$\begin{aligned} \alpha (\theta _1,\theta _2)=\sqrt{a(\theta _2)^2+\frac{b(\theta _2)^2c(\theta _2)^2}{{\theta _1}^2}} \end{aligned}$$

(4.1)

by (2.13). Furthermore, the eigenvalues of $H(\theta _1,\theta _2)$ in Assumption [A4] is $\pm \alpha (\theta _1,\theta _2)$ and hence one can remove Assumption [A4].

As for the estimation of $\theta _1$, one can obtain the explicit expression of ${\hat{\theta }}_1^n$. In fact, we have

$$\begin{aligned} {\mathbb {H}}_n^1(\theta _1)=-\frac{1}{2}\sum _{j=1}^n\left\{ \frac{1}{h{\theta _1}^2}(\Delta _jY)^{2}+2\log \theta _1 \right\} \end{aligned}$$

and hence

$$\begin{aligned} \frac{d}{d\theta _1}{\mathbb {H}}_n^1(\theta _1)=\frac{1}{h{\theta _1}^3}\sum _{j=1}^n(\Delta _jY)^{2}-\frac{n}{\theta _1}. \end{aligned}$$

Thus we obtain the formula

$$\begin{aligned} {\hat{\theta }}_1^n=\left( \frac{1}{t_n}\sum _{j=1}^n(\Delta _jY)^{2} \right) ^\frac{1}{2}. \end{aligned}$$

Moreover, ${\mathbb {Y}}_1(\theta _1)$ and $\Gamma ^1$ can be written as

$$\begin{aligned} {\mathbb {Y}}_1(\theta _1)=-\frac{1}{2}\left( \frac{{\theta _1^*}^2}{{\theta _1}^2}-1-2\log \frac{{\theta _1^*}}{{\theta _1}} \right) \end{aligned}$$

and

$$\begin{aligned} \Gamma ^1=\frac{1}{2}\left( \frac{2\theta _1^*}{{\theta _1^*}^2} \right) ^2=\frac{2}{{\theta _1^*}^2}. \end{aligned}$$

Therefore noting that $\displaystyle x^2-1-2\log x\ge (x-1)^2~(x\ge 0)$ we have

$$\begin{aligned} {\mathbb {Y}}_1(\theta _1)\le -\frac{1}{2}\left( \frac{{\theta _1^*}}{{\theta _1}}-1 \right) ^2\le -\frac{(\theta _1-\theta _1^*)^2}{2\epsilon ^2} \end{aligned}$$

and hence (2.8) holds.

As for the estimation of $\theta _2$, since we have

$$\begin{aligned} \gamma (\theta _1,\theta _2)=\frac{{\theta _1}^2}{c(\theta _2)^2}\left\{ \alpha (\theta _1,\theta _2)-a(\theta _2) \right\} \end{aligned}$$

(4.2)

by (2.13), we obtain for $\alpha (\theta _2)\ne a^*$

$$\begin{aligned} \begin{aligned} {\mathbb {Y}}_2(\theta _2)&=-\frac{1}{2}\int _0^\infty \left\{ -\frac{\{a(\theta _2)-a^*\}(\alpha (\theta _2^*)-a^*)}{\alpha (\theta _2)-a^*}e^{-a^*s}\right. \\&\quad \left. +\frac{\{\alpha (\theta _2)-\alpha (\theta _2^*)\}\{\alpha (\theta _2)-a(\theta _2)\}}{\alpha (\theta _2)-a^*}e^{-\alpha (\theta _2)s} \right\} ^2ds\\&=-\frac{1}{4a^*\alpha (\theta _2)\{a^*+\alpha (\theta _2)\}}\\&\quad \times \left[ \{a^*\alpha (\theta _2)-a(\theta _2)\alpha (\theta _2^*)\}^2 +a^*\alpha (\theta _2)\left\{ \alpha (\theta _2)-a(\theta _2)-\alpha (\theta _2^*)+a^* \right\} ^2 \right] \\&=-\frac{a^*a(\theta )^2}{4\alpha (\theta _2)\{a^*+\alpha (\theta _2)\}}\left\{ \frac{\alpha (\theta _2)}{a(\theta _2)}-\frac{\alpha (\theta _2^*)}{a(\theta _2^*)} \right\} ^2\\&\quad -\frac{1}{4a^*\alpha (\theta _2)\{a^*+\alpha (\theta _2)\}}\left\{ \alpha (\theta _2)-a(\theta _2)-\alpha (\theta _2^*)+a^* \right\} ^2, \end{aligned} \end{aligned}$$

(4.3)

making use of (2.6) and the identity

$$\begin{aligned} \int _0^\infty \left( pe^{-\alpha t}+qe^{-\beta t} \right) ^2dt&=\frac{p^2}{2\alpha }+\frac{2pq}{\alpha +\beta }+\frac{q^2}{2\beta }\\&=\frac{1}{2\alpha \beta }\left\{ (\alpha q+\beta p)^2+\alpha \beta (p-q)^2 \right\} , \end{aligned}$$

where $\alpha ,\beta >0$ and $p,q \in {\mathbb {R}}$. Even if $\alpha (\theta _2)=a^*$, we obtain the same formula by letting $a^* \rightarrow \alpha (\theta _2)$ in (4.3).

Now we obtain a sufficient condition for (2.9) by the following proposition.

Proposition 4.1

Assume [A3], $\displaystyle \inf _{\theta _2 \in \Theta _2}|c(\theta _2)|>C$ and

$$\begin{aligned} |a(\theta _2)-a(\theta _2^*)|+|\alpha (\theta _2)-\alpha (\theta _2^*)|\ge C|\theta _2-\theta _2^*|. \end{aligned}$$

(4.4)

Then it holds

$$\begin{aligned} Y(\theta _2)\le -C|\theta _2-\theta _2^*|^2. \end{aligned}$$

(4.5)

Proof

Let us assume there is no constant C satisfying (4.5). Then there exists some sequence $\theta _2^{(n)} \in \Theta _2~(n \in {\mathbb {N}})$ such that

$$\begin{aligned} \left| \frac{\alpha (\theta _2^{(n)})}{a(\theta _2^{(n)})}-\frac{\alpha (\theta _2^*)}{a(\theta _2^*)}\right| \le \frac{1}{n}|\theta _2^{(n)}-\theta _2^*| \end{aligned}$$

and

$$\begin{aligned} \left| \alpha (\theta _2^{(n)})-a(\theta _2^{(n)})-\alpha (\theta _2^*)+a^*\right| \le \frac{1}{n}|\theta _2^{(n)}-\theta _2^*|. \end{aligned}$$

Thus if we set

$$\begin{aligned} A(\theta _2)=\alpha (\theta _2)-a(\theta _2) \end{aligned}$$

and

$$\begin{aligned} B(\theta _2)=\frac{\alpha (\theta _2)}{a(\theta _2)}, \end{aligned}$$

it follows that

$$\begin{aligned} |a(\theta _2^{(n)})-a(\theta _2^*)|&=\left| \frac{A(\theta _2^{(n)})}{B(\theta _2^{(n)})-1}-\frac{A(\theta _2^*)}{\displaystyle B(\theta _2^*)-1}\right| \\&\le \frac{|A(\theta _2^{(n)})-A(\theta _2^*)|}{B(\theta _2^{(n)})-1}+\frac{|A(\theta _2^*)||B(\theta _2^{(n)})-B(\theta _2^*)|}{\{ B(\theta _2^{(n)})-1 \}\{ B(\theta _2^*)-1 \}}\\&\le \frac{C}{n}|\theta _2^{(n)}-\theta _2^*|, \end{aligned}$$

noting that it holds $B(\theta _2)-1>C$ by the assumptions and (4.1). In the same way, we have

$$\begin{aligned} |\alpha (\theta _2^{(n)})-\alpha (\theta _2^*)|\le \frac{C}{n}|\theta _2^{(n)}-\theta _2^*|, \end{aligned}$$

but these contradict (4.4). $\square$

We similarly obtain the explicit expression of $\Gamma ^2$ by (4.2):

$$\begin{aligned} \Gamma ^2&=\frac{1}{{\theta _1^*}^2}\int _0^\infty [\partial _{\theta _2}^{\otimes 2}]\left\{ \int _0^sc(\theta _2)\exp (-\alpha (\theta _2)u)\gamma _{+}(\theta _2)c(\theta _2)'{\Sigma ^*}^{-1}c^*\right. \\&\quad \exp (-a^*(s-u))\gamma _+(\theta ^*){c^*}'du\\&\quad \left. \left. +c(\theta _2)\exp (-\alpha (\theta _2)s)\gamma _{+}(\theta _2)c(\theta _2)'\right. \biggr \}\right| _{\theta _2=\theta _2^*}ds\\&=\int _0^\infty \{\partial _\theta \alpha (\theta ^*)e^{-\alpha ^*s}-\partial _\theta a(\theta ^*)e^{-a^*s}\}^{\otimes 2}ds\\&=\frac{\{\partial _{\theta _2}\alpha (\theta ^*)\}^{\otimes 2}}{2\alpha ^*}+\frac{\{\partial _{\theta _2}a(\theta ^*)\}^{\otimes 2}}{2a^*}\\&\quad -\frac{\partial _{\theta _2}\alpha (\theta ^*)\otimes \partial _{\theta _2}a(\theta ^*)+\partial _{\theta _2} a(\theta ^*)\otimes \partial _{\theta _2}\alpha (\theta ^*)}{\alpha ^*+a^*}\\&=\frac{1}{2\alpha ^*}\left( \partial _{\theta _2}\alpha (\theta ^*)-\frac{2\alpha ^*}{\alpha ^*+a^*}\partial _{\theta _2}a(\theta ^*) \right) ^{\otimes 2}+\frac{(\alpha ^*)^2+(a^*)^2}{2(\alpha ^*+a^*)a^*}\{\partial _{\theta _2}a(\theta ^*)\}^{\otimes 2}. \end{aligned}$$

Hence $\Gamma ^2$ is positive definite if and only if $\{\partial _{\theta _2}a(\theta ^*)\}^{\otimes 2}$ or $\{\partial _{\theta _2}\alpha (\theta ^*)\}^{\otimes 2}$ is positive definite. This does not happen if $m_2\ge 3$; in fact, one can take $x \in {\mathbb {R}}^{m_2}$ so that $x'\partial _{\theta _2}a(\theta ^*)=x'\partial _{\theta _2}\alpha (\theta ^*)$ if $m_2\ge 3$. Thus we need to assume $m_2 \le 2$ in the one-dimensional case.

Putting it all together, we obtain the following result.

Theorem 4.2

Let $m_1=1, m_2 \le 2, \sigma (\theta _1)=\theta _2$ and $\Theta _1 \subset (\epsilon ,\infty )$ for some $\epsilon >0$. Moreover, we assume [A1], [A2] and the following conditions:

[B1]:

$$\begin{aligned}&\inf _{\theta _2 \in \Theta _2}a(\theta _2)>0\\&\inf _{\theta _2 \in \Theta _2}|b(\theta _2)|>0\\&\inf _{\theta _2 \in \Theta _2}|c(\theta _2)|>0 \end{aligned}$$

[B2]:

For any $\theta _1 \in \Theta _1$ and $\theta _2,\theta _2^* \in \Theta _2$,

$$\begin{aligned} |a(\theta _2,\theta _1)-a(\theta _2^*,\theta _1)|+|\alpha (\theta _2,\theta _1)-\alpha (\theta _2^*,\theta _1)|\ge C_{\theta _1}|\theta _2-\theta _2^*|. \end{aligned}$$

[B3]:

For any $\theta \in \Theta , \{\partial _{\theta _2}a(\theta )\}^{\otimes 2}$ or $\{\partial _{\theta _2}\alpha (\theta )\}^{\otimes 2}$ is positive definite.

(1) If we set

$$\begin{aligned} {\hat{\theta }}_1^n=\left( \frac{1}{t_n}\sum _{j=1}^n(\Delta _jY)^{2} \right) ^\frac{1}{2}, \end{aligned}$$

then for every $p>0$ and any continuous function $f:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \limsup _{|x|\rightarrow \infty }\frac{|f(x)|}{|x|^p}<\infty , \end{aligned}$$

it holds that

$$\begin{aligned} E[f(\sqrt{n}({\hat{\theta }}^n_1-\theta _1^*))]\rightarrow E[f(Z)]~(n \rightarrow \infty ), \end{aligned}$$

where $\displaystyle Z \sim N\left( 0,\frac{{\theta _1^*}^2}{2}\right)$.

In particular, it holds that

$$\begin{aligned} \sqrt{n}({\hat{\theta }}^n_1-\theta _1^*)\xrightarrow {d}N\left( 0,\frac{{\theta _1^*}^2}{2}\right) ~(n \rightarrow \infty ). \end{aligned}$$

(2) Let us define $\gamma _+(\theta _1,\theta _2)$ and $\alpha (\theta _1,\theta _2)$ by

$$\begin{aligned} \gamma _+(\theta _1,\theta _2)=\frac{{\theta _1}^2a(\theta _2)}{c(\theta _2)^2}\left( \sqrt{1+\frac{b(\theta _2)^2c(\theta _2)^2}{{\theta _1}^2a(\theta _2)^2}}-1 \right) . \end{aligned}$$

and

$$\begin{aligned} \alpha (\theta _1,\theta _2)=\sqrt{a(\theta _2)^2+\frac{b(\theta _2)^2c(\theta _2)^2}{{\theta _1}^2}}, \end{aligned}$$

respectively, and set

$$\begin{aligned} \begin{aligned} {\hat{m}}_i^n(\theta _2;m_0)&=e^{-\alpha ({\hat{\theta }}_1^n,\theta _2)t_i}m_0\\&\quad +\frac{1}{({\hat{\theta }}_1^n)^2}\sum _{j=1}^ie^{-\alpha ({\hat{\theta }}_1^n,\theta _2)(t_i-t_{j-1})}\gamma _+({\hat{\theta }}_1^n,\theta _2)c(\theta _2)\Delta _jY, \\ {\mathbb {H}}_n^2(\theta _2;m_0)&=\frac{1}{2}\sum _{i=1}^n\left\{ -h(c(\theta _2){\hat{m}}_{j-1}^n(\theta _2))^{2}+2{\hat{m}}_{j-1}^n(\theta _2)c(\theta _2)\Delta _jY\right\} , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \Gamma ^2&=\frac{\{\partial _{\theta _2}\alpha (\theta ^*)\}^{\otimes 2}}{2\alpha ^*}+\frac{\{\partial _{\theta _2}a(\theta ^*)\}^{\otimes 2}}{2a(\theta ^*)}\\&\quad -\frac{\partial _{\theta _2}\alpha (\theta ^*)\otimes \partial _{\theta _2}a(\theta ^*)+\partial _{\theta _2} a(\theta ^*)\otimes \partial _{\theta _2}\alpha (\theta ^*)}{\alpha (\theta ^*)+a(\theta ^*)}, \end{aligned} \end{aligned}$$

where $m_0 \in {\mathbb {R}}^{d_1}$ is an arbitrary initial value.

Then, if ${\hat{\theta }}^n_2={\hat{\theta }}^n_2(m_0)$ is a random variable satisfying

$$\begin{aligned} {\mathbb {H}}_n^2({\hat{\theta }}^n_2)=\max _{\theta _2 \in {\overline{\Theta }}_2}{\mathbb {H}}_n^2(\theta _2) \end{aligned}$$

for each $n \in {\mathbb {N}}$, then for any $p>0$ and continuous function $f:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \limsup _{|x|\rightarrow \infty }\frac{|f(x)|}{|x|^p}<\infty , \end{aligned}$$

it holds that

$$\begin{aligned} E[f(\sqrt{t_n}({\hat{\theta }}^n_2-\theta _2^*))]\rightarrow E[f(Z)]~(n \rightarrow \infty ), \end{aligned}$$

where $Z \sim N(0,(\Gamma ^2)^{-1})$.

In particular, it holds that

$$\begin{aligned} \sqrt{t_n}({\hat{\theta }}^n_2-\theta _2^*)\xrightarrow {d}N(0,(\Gamma ^2)^{-1})~(n \rightarrow \infty ). \end{aligned}$$

5 Proof of Theorem 2.1

In this section, we prove Theorem 2.1, which can be proved in the same way as the diffusion case in Yoshida (2011).

Lemma 5.1

For every $p\ge 2$ and $A \in M_{d_1}({\mathbb {R}})$, it holds

$$\begin{aligned} E\left[ \left| \sum _{j=1}^nA[(\Delta _jY)^{\otimes 2}]-\sum _{j=1}^nA[(\sigma ^*\Delta _jW^2)^{\otimes 2}]\right| ^p \right] ^\frac{1}{p}\le C_p|A|(nh^2+n^\frac{1}{2}h^\frac{3}{2}). \end{aligned}$$

Proof

First we get

$$\begin{aligned}&E\left[ \left| \sum _{j=1}^nA[(\Delta _jY)^{\otimes 2}]-\sum _{j=1}^nA[(\sigma ^*\Delta _jW^2)^{\otimes 2}]\right| ^p \right] ^\frac{1}{p}\nonumber \\&\quad =E\left[ \left| \sum _{j=1}^nA\left[ \left( c^*\int _{t_{j-1}}^{t_j}X_tds+\sigma ^*\Delta _jW^2\right) ^{\otimes 2}\right] -\sum _{j=1}^nA[(\sigma ^*\Delta _jW^2)^{\otimes 2}]\right| ^p \right] ^\frac{1}{p}\nonumber \\&\quad \le E\left[ \left| \sum _{j=1}^nA\left[ \left( c^*\int _{t_{j-1}}^{t_j}X_tds\right) ^{\otimes 2}\right] \right| ^p \right] ^\frac{1}{p}\nonumber \\&\qquad +E\left[ \left| \sum _{j=1}^nA\left[ c^*\int _{t_{j-1}}^{t_j}X_tds,\sigma ^*\Delta _jW^2\right] \right| ^p \right] ^\frac{1}{p}\nonumber \\&\qquad +E\left[ \left| \sum _{j=1}^nA\left[ \sigma ^*\Delta _jW^2,c^*\int _{t_{j-1}}^{t_j}X_tds\right] \right| ^p \right] ^\frac{1}{p}. \end{aligned}$$

For the first term of the rightest-hand side, we obtain by Lemmas 3.1 and 3.3

$$\begin{aligned}&E\left[ \left| \sum _{j=1}^nA\left[ \left( c^*\int _{t_{j-1}}^{t_j}X_tds\right) ^{\otimes 2}\right] \right| ^p \right] ^\frac{1}{p}\\&\quad \le |A||c^*| \sum _{j=1}^nE\left[ \left| \int _{t_{j-1}}^{t_j}X_tds\right| ^{2p} \right] ^\frac{1}{p}\\&\quad \le |A||c^*|\sum _{j=1}^nh^{2p-1}\left( \int _{t_{j-1}}^{t_j}E[ \left| X_t\right| ^{2p} ]\right) ^\frac{1}{p}ds\\&\quad \le C_p|A|nh^2. \end{aligned}$$

For the second and third terms, it holds

$$\begin{aligned}&E\left[ \left| \sum _{j=1}^nA\left[ c^*\int _{t_{j-1}}^{t_j}X_tds,\sigma ^*\Delta _jW^2\right] \right| ^p \right] ^\frac{1}{p}\\&\quad \le E\left[ \left| \sum _{j=1}^nA\left[ c^*\int _{t_{j-1}}^{t_j}(X_t-X_{t_{j-1}})ds,\sigma ^*\Delta _jW^2\right] \right| ^p \right] ^\frac{1}{p}\\&\qquad +E\left[ \left| \sum _{j=1}^nA\left[ c^*X_{t_{j-1}}h,\sigma ^*\Delta _jW^2\right] \right| ^p \right] ^\frac{1}{p}\\&\quad \le |A||c^*||\sigma ^*|\sum _{j=1}^nE\left[ \left| \int _{t_{j-1}}^{t_j}(X_t-X_{t_{j-1}})ds\right| ^p|\Delta _jW^2|^p\right] ^\frac{1}{p}\\&\qquad +hE\left[ \left| \sum _{j=1}^nX_{t_{j-1}}'{c^*}'A\sigma ^*\Delta _jW^2\right| ^p \right] ^\frac{1}{p}\\&\quad \le |A||c^*||\sigma ^*|\sum _{j=1}^nE\left[ \left| \int _{t_{j-1}}^{t_j}(X_t-X_{t_{j-1}})ds\right| ^{2p}\right] ^\frac{1}{2p}E\left[ |\Delta _jW^2|^{2p}\right] ^\frac{1}{2p}\\&\qquad +hE\left[ \left| \sum _{j=1}^nX_{t_{j-1}}'{c^*}'A\sigma ^*\Delta _jW^2\right| ^p \right] ^\frac{1}{p}\\&\quad \le C_p|A|\sum _{j=1}^n\left( h^{2p-1}\int _{t_{j-1}}^{t_j}E[|X_u-X_{t_{j-1}}|^{2p}]du \right) ^\frac{1}{2p}h^\frac{1}{2}\\&\qquad +C_p|A|h\left( (nh)^{\frac{p}{2}-1}\sum _{j=1}^nE[|X_{t_{j-1}}|^p]h\right) ^\frac{1}{p}\\&\quad \le C_p|A|(nh^2+n^\frac{1}{2}h^\frac{3}{2}). \end{aligned}$$

Therefore we get the desired result. $\square$

Lemma 5.2

Let $A_k \in M_{d_1}({\mathbb {R}})~(k=1,2,\cdots ,d)$, $A=(A_1,\cdots ,A_d)$ and

$$\begin{aligned} M_n(A)=\sum _{j=1}^n\left\{ \frac{1}{h}A[(\Delta _jW^2)^{\otimes 2}]-\textrm{Tr}A\right\} . \end{aligned}$$

Then it holds that

$$\begin{aligned} E[|M_n(A)|^p]\le C_p|A|\sqrt{n} \end{aligned}$$

(5.1)

and

$$\begin{aligned} \frac{1}{\sqrt{n}}M_n \xrightarrow {d} N(0,2(\textrm{Tr}A)^{\otimes 2})~(n\rightarrow \infty ). \end{aligned}$$

(5.2)

Proof

On account of

$$\begin{aligned} E\left[ \frac{1}{h}A[(\Delta _jW^2)^{\otimes 2}]-\textrm{Tr}A\bigg |{\mathcal {F}}_{t_{j-1}} \right] =\frac{1}{h}E\left[ A[(\Delta _jW^2)^{\otimes 2}]\right] -\textrm{Tr}A=0, \end{aligned}$$

$\{A[(\Delta _jW^2)^{\otimes 2}]/h-\textrm{Tr}A\}_j$ is a martingale difference sequence with respect to $\{{\mathcal {F}}_{t_j}\}$. Hence the Burkholder inequality gives

$$\begin{aligned} E[|M_n|^p]^\frac{p}{2}&\le C_pE\left[ \left| \sum _{j=1}^n\frac{1}{h}A[(\Delta _jW^2)^{\otimes 2}]-\textrm{Tr}A \right| ^{2p} \right] ^\frac{1}{2p}\\&\le C_p\sum _{j=1}^nE\left[ \left| \frac{1}{h}A[(\Delta _jW^2)^{\otimes 2}]-\textrm{Tr}A \right| ^{2p} \right] ^\frac{1}{2p}\\&\le C_pn\left\{ |A|^{2p}E\left[ \left| \frac{1}{h}W_h^2\right| ^{4p}+|A|^2p \right] \right\} \\&\le C_p|A|n, \end{aligned}$$

and we obtain (5.1).

Moreover, due to the fact that $\{A[(\Delta _jW^2)^{\otimes 2}]/h-\textrm{Tr}A\}_j$ is independent and identically distributed, we have

$$\begin{aligned}&E\left[ \left( \frac{1}{h}A[(\Delta _jW^2)^{\otimes 2}]-\textrm{Tr}A \right) ^{\otimes 2} \right] \\&\quad =\frac{1}{h^2}E\left[ \{A[(\Delta _jW^2)^{\otimes 2}]\}^{\otimes 2} \right] -\frac{1}{h}E[A[(\Delta _jW^2)^{\otimes 2}]]'\textrm{Tr}A\\&\qquad -(\textrm{Tr}A)'\frac{1}{h}E[A[(\Delta _jW^2)^{\otimes 2}]]+(\textrm{Tr}A)^{\otimes 2}\\&\quad =3(\textrm{Tr}A)^{\otimes 2}-2(\textrm{Tr}A)^{\otimes 2}+(\textrm{Tr}A)^{\otimes 2}=2(\textrm{Tr}A)^{\otimes 2}. \end{aligned}$$

Thus we obtain (5.2). $\square$

By Lemmas 5.1 and 5.2, we get the following lemma.

Lemma 5.3

Let $A_k \in M_{d_1}({\mathbb {R}})~(k=1,2,\cdots ,d)$, $A=(A_1,\cdots ,A_d)$ and

$$\begin{aligned} L_n(A)=\sum _{j=1}^n\left\{ \frac{1}{h}A[(\Delta _jY)^{\otimes 2}]-\textrm{Tr}A\Sigma \right\} . \end{aligned}$$

Then it holds that

$$\begin{aligned} E[|L_n(A)|^p]\le C_p|A|(nh+n^\frac{1}{2}h^\frac{1}{2}+n^\frac{1}{2}) \end{aligned}$$

(5.3)

and

$$\begin{aligned} \frac{1}{\sqrt{n}}L_n \xrightarrow {d} N(0,2(\textrm{Tr}A\Sigma )^{\otimes 2})~(n\rightarrow \infty ). \end{aligned}$$

(5.4)

Lemma 5.4

For every $p>0$, it holds

$$\begin{aligned} \sup _{n \in {\mathbb {N}}}E\left[ \left| \frac{1}{n}\sup _{\theta _1 \in \Theta _1}\partial _{\theta _1}^3 {\mathbb {H}}_n^1(\theta _1)\right| ^p \right] <\infty \end{aligned}$$

Proof

It is enough to prove the inequality for sufficiently large p. By Lemmas 5.2 and 5.1 and Assumptions [A2] and [A4] we get

$$\begin{aligned}&E\left[ \left| \frac{1}{n}\partial _{\theta _1}^3{\mathbb {H}}_n^1(\theta _1)\right| ^p\right] ^\frac{1}{p}\\&\quad =E\left[ \left| \frac{1}{2n}\sum _{j=1}^n\left\{ \frac{1}{h}\partial _{\theta _1}^3\Sigma ^{-1}(\theta _1)[(\Delta _jY)^{\otimes 2}]+\partial _{\theta _1}^3\log \det \Sigma (\theta _1) \right\} \right| ^p \right] ^\frac{1}{p}\\&\quad \le E\left[ \left| \frac{1}{2nh}\sum _{j=1}^n\left\{ \partial _{\theta _1}^3\Sigma ^{-1}(\theta _1)[(\Delta _jY)^{\otimes 2}-\textrm{Tr}\partial _{\theta _1}^3\Sigma ^{-1}(\theta _1) \right\} \right| ^p \right] ^\frac{1}{p}\\&\qquad +\frac{1}{2}\left\{ |\textrm{Tr}\partial _{\theta _1}^3\Sigma ^{-1}(\theta _1)|+|\partial _{\theta _1}^3\log \det \Sigma (\theta _1)| \right\} \\&\quad \le C_p|\partial _{\theta _1}^3\Sigma ^{-1}(\theta _1)|(h+n^{-\frac{1}{2}}h^\frac{1}{2}+n^{-\frac{1}{2}})\\&\qquad +\frac{1}{2}\left\{ |\textrm{Tr}\partial _{\theta _1}^3\Sigma ^{-1}(\theta _1)|+|\partial _{\theta _1}^3\log \det \Sigma (\theta _1)| \right\} \\&\quad \le C_p, \end{aligned}$$

and similarly

$$\begin{aligned} E\left[ \left| \frac{1}{n}\partial _{\theta _1}^4{\mathbb {H}}_n^1(\theta _1)\right| ^p\right] ^\frac{1}{p}\le C_p. \end{aligned}$$

Thus we get the desired result for $p>d_1$ by the Sobolev inequality. $\square$

Proof of Theorem 2.1

Let

$$\begin{aligned} \Delta _n^1&=\frac{1}{\sqrt{n}}\partial _{\theta _1}{\mathbb {H}}_n^1(\theta _1^*),\\ \Gamma _n^1&=-\frac{1}{n}\partial _{\theta _1}^2{\mathbb {H}}_n^1(\theta _1^*) \end{aligned}$$

and

$$\begin{aligned} {\mathbb {Y}}_n^1(\theta _1)=\frac{1}{n}\{{\mathbb {H}}_n^1(\theta _1)-{\mathbb {H}}_n^1(\theta _1^*)\}. \end{aligned}$$

Then

$$\begin{aligned} \Delta _n^1&=-\frac{1}{2\sqrt{n}}\sum _{j=1}^n\left\{ \frac{1}{h}\partial _{\theta _1}\Sigma ^{-1}(\theta _1^*)[(\Delta _jY)^{\otimes 2}]+\frac{\partial _{\theta _1}\det \Sigma (\theta _1^*)}{\det \Sigma ^*} \right\} \\&=\frac{1}{2\sqrt{n}}\sum _{j=1}^n\left\{ \frac{1}{h}{\Sigma ^*}^{-1}\partial _{\theta _1}\Sigma (\theta _1^*){\Sigma ^*}^{-1}[(\Delta _jY)^{\otimes 2}]-\textrm{Tr}{\Sigma ^*}^{-1}\partial _{\theta _1}\Sigma (\theta _1^*)\right\} , \end{aligned}$$

and hence by Lemma 5.3, we obtain

$$\begin{aligned} E[|\Delta _n^1|^p]\le C_p \end{aligned}$$

(5.5)

and

$$\begin{aligned} \Delta _n^1 \xrightarrow {d} N(0,(\Gamma ^1)^{-1}) ~(n \rightarrow \infty ). \end{aligned}$$

(5.6)

By the same lemma, it follows that

$$\begin{aligned} \begin{aligned}&E\left[ \left| n^\frac{1}{2}(\Gamma _n^1-\Gamma ^1)\right| ^p \right] ^\frac{1}{p}\\&\quad =E\left[ \left| n^\frac{1}{2}\left[ \frac{1}{n}\sum _{j=1}^n\left\{ \frac{1}{h}\partial _{\theta _1}^2\Sigma ^{-1}(\theta _1^*)[(\Delta _jY)^{\otimes 2}]+\partial _{\theta _1}^2\log \det \Sigma (\theta _1^*) \right\} -\Gamma ^1\right] \right| ^p \right] ^\frac{1}{p}\\&\quad \le C_p(n^\frac{1}{2}h^2+h^\frac{3}{2}+1)=O(1), \end{aligned} \end{aligned}$$

(5.7)

noting that

$$\begin{aligned}&\textrm{Tr}\partial _{\theta _1}^2\Sigma ^{-1}(\theta _1^*)\Sigma ^*\\&\quad =\textrm{Tr}\left\{ 2\{{\Sigma ^*}^{-1}\partial _{\theta _1}\Sigma (\theta _1^*)\}^{\otimes 2}-\Sigma ^{-1}\frac{\partial }{\partial \theta _{1}^i}\frac{\partial }{\partial \theta _{1}^j}\Sigma (\theta _1^*)\right\} \end{aligned}$$

is equal to $-\partial _{\theta _1}^2\log \det \Sigma (\theta _1^*)+\Gamma ^1$.

Moreover, we can show

$$\begin{aligned}&\sup _{\theta _1 \in \Theta _1}E\left[ (n^\frac{1}{2}|{\mathbb {Y}}_n^1(\theta _1)-{\mathbb {Y}}^1(\theta _1)|)^p \right] ^\frac{1}{p}\\&\quad = \sup _{\theta _1 \in \Theta _1}E\left[ \left| -\frac{1}{2\sqrt{n}}\sum _{j=1}^n\left\{ \frac{1}{h}\{\Sigma ^{-1}(\theta _1)-\Sigma ^{-1}(\theta _1^*)\}[(\Delta _jY)^{\otimes 2}]\right. \right. \right. \\&\qquad \left. \left. \left. -\textrm{Tr}\{\Sigma (\theta _1)^{-1}-I\}\right\} \right| ^p \right] ^\frac{1}{p}\\&\quad \le C_p(n^\frac{1}{2}h+h+1)=O(1) \end{aligned}$$

and in the same way

$$\begin{aligned} \sup _{\theta _1 \in \Theta _1}E\left[ (n^\frac{1}{2}|\partial _{\theta _1}\{{\mathbb {Y}}_n^1(\theta _1)-{\mathbb {Y}}^1(\theta _1)\}|)^p \right] ^\frac{1}{p} =O(1). \end{aligned}$$

Thus by the Sobolev inequality, it holds for $p>d_1$

$$\begin{aligned} E\left[ \sup _{\theta _1 \in \Theta _1}(n^\frac{1}{2}|{\mathbb {Y}}_n^1(\theta _1)-{\mathbb {Y}}^1(\theta _1)|)^p \right] ^\frac{1}{p} =O(1). \end{aligned}$$

(5.8)

Then we have proved the theorem by the assumption [A5], Lemma 5.4, (5.5), (5.6), (5.7), (5.8) and Theorem 5 in Yoshida (2011). $\square$

6 Simulations

In this section, we will verify the results of the previous sections by computational simulations. We set $d_1=d_2=1$ and consider the equations

$$\begin{aligned} {\left\{ \begin{array}{ll} dX_t=-aX_tdt+bdW_t^1\\ dY_t=X_tdt+\sigma dW_t^2 \end{array}\right. } \end{aligned}$$

with $X_0=Y_0=0$, where we want to estimate $\theta _1=\sigma$ and $\theta _2=(a,b)$ from observations of $Y_t$.

We generated sample data $Y_{t_i}~(i=0,1,\cdots ,n)$ with $n=10^6$, $h=0.0001$ and true value of $(a^*,b^*,\sigma )=(1.5,0.3,0.02)$, and performed 10000 Monte Carlo replications. Recall that for the estimation of $\theta _2$, we first calculate ${\hat{m}}$ by (2.14), and we have to choose its initial value $m_0$. Although we proved that Theorem 2.2 holds for an arbitrary choice of $m_0$, this value is a substitute for $E[X_0|Y_0]$, and thus in practice the choice of $m_0$ is very important as will be shown in the following. Also, the choice of the number of terms to drop, which is explained below, is relevant.

Taking these facts into account, we calculated the estimator of $\theta _2$ in the following ways for each simulated data.

Estimation (i):

$m_0=0$.

Estimation (ii):

$m_0=1$.

Estimation (iii):

$m_0=1$ and removed first 100 terms of ${\hat{m}}_i^n$; i.e. we replaced ${\mathbb {H}}_n^2(\theta _2;m_0)$ with

$$\begin{aligned} \frac{1}{2}\sum _{i=101}^n\left\{ -h(c(\theta _2){\hat{m}}_{j-1}^n(\theta _2))^{2}+2{\hat{m}}_{j-1}^n(\theta _2)c(\theta _2)\Delta _jY\right\} . \end{aligned}$$

Estimation (iv):

$m_0=1$ and removed first 1000 terms of ${\hat{m}}_i^n$; i.e. we replaced ${\mathbb {H}}_n^2(\theta _2;m_0)$ with

$$\begin{aligned} \frac{1}{2}\sum _{i=1001}^n\left\{ -h(c(\theta _2){\hat{m}}_{j-1}^n(\theta _2))^{2}+2{\hat{m}}_{j-1}^n(\theta _2)c(\theta _2)\Delta _jY\right\} . \end{aligned}$$

Table 1 shows the means and standard deviations of each estimators, and one can observe asymptotic normalities of them in Fig. 1. Note that the difference of four estimations are not relevant to the estimation of $\theta _1$.

Table 1 The summary of the simulation results with $n=10^6$, $h=0.0001$

We can see from the results of first estimation and the corresponding histograms that the estimators behaved in accordance with the theory. At the same time, the following results shows that the wrong value of $m_0$ can significantly impact the accuracy of our estimator, but it can be improved by leaving out first several terms of ${\hat{m}}_i^n$. One can figure out the reason why this modification of removing first terms works by looking at Fig. 2; it shows that ${\hat{m}}_i^n(\theta ^*)$ with $m_0=1$ well approximate $X_{t_i}$ except at the beginning. However, according to the result of Estimations (iii) and (iv), removing first 100 terms is not enough to improve estimators, whereas significant improvement is made by removing 1,000 terms. Thus, it is important choose $m_0$ which is closer to $E[X_0|Y_0]$ when some information of $X_0$ is available. Note that if $X_0$ and $Y_0$ are independent, then the best choice of $m_0$ is $E[X_0|Y_0]=E[X_0]$.

On the other hand, when you have no information of $X_0$, it will be interesting to consider the way to decide how many terms of ${\hat{m}}_i^n(\theta )$ should be removed. One possible way is to increase the removing number and calculate estimators until they converge. However, this method is computationally intensive, and more theoretical way will be needed.

The data and script that supports the findings of this study are available in the supplementary material of this article.

Fig. 1

Histograms of normalized estimators in Estimation (i). The red lines are the density of the normal distributions with means and standard deviations of data

Fig. 2

A path of $X_t$ and ${\hat{m}}_i^n(\theta ^*)$ with $m_0=1$