Threshold estimation for jump-diffusions under small noise asymptotics

Abstract

We consider parameter estimation of stochastic differential equations driven by a Wiener process and a compound Poisson process as small noises. The goal is to give a threshold-type quasi-likelihood estimator and show its consistency and asymptotic normality under new asymptotics. One of the novelties of the paper is that we give a new localization argument, which enables us to avoid truncation in the contrast function that has been used in earlier works and to deal with a wider class of jumps in threshold estimation than ever before.

1 Introduction

This paper is concerned with the following stochastic differential equation (SDE):

$$\begin{aligned} \textrm{d}{X^\varepsilon _t} = a(X^\varepsilon _t,\mu _0) \textrm{d}{t} + \varepsilon b(X^\varepsilon _t, \sigma _0) \textrm{d}{W_t} + \varepsilon c(X^\varepsilon _{t-}, \alpha _0) \textrm{d}{Z_t^{\lambda _\varepsilon }}, \quad X_0^{\varepsilon } = x_0 \in \mathbb {R}, \end{aligned}$$

(1.1)

where $\varepsilon >0$, and $\Theta _i$ ($i=1,2,3$) are smooth bounded open convex sets in $\mathbb {R}^{d_i}$ with $d_i\in \mathbb {N}$ ($i=1,2,3$), respectively, and $\theta _0=(\mu _0,\sigma _0,\alpha _0)\in \Theta _0:=\Theta _1\times \Theta _2\times \Theta _3\subset \mathbb {R}^d$ with $d:=d_1+d_2+d_3$ with $\Theta :=\bar{\Theta }_0$, and each domain of a, b, c is $\mathbb {R}\times \bar{\Theta }_i$ ($i=1,2,3$), respectively. Also, $Z^{\lambda _\varepsilon }=(Z_t^{\lambda _\varepsilon })_{t\ge 0}$ is a compound Poisson process given by

$$\begin{aligned} Z^{\lambda _\varepsilon }_t = \sum _{i=1}^{N_t^{\lambda _\varepsilon }} V_i, \quad Z^{\lambda _\varepsilon }_0 = 0, \end{aligned}$$

where $N^{\lambda _\varepsilon }=(N^{\lambda _\varepsilon }_t)_{t\ge 0}$ is a Poisson process with intensity $\lambda _\varepsilon >0$, and $V_i$’s are i.i.d. random variables with common probability density function $f_{\alpha _0}$, and are independent of $N^{\lambda _\varepsilon }$ [cf. Example 1.3.10 in Applebaum (2009)]. $W=(W_t)_{t\ge 0}$ is a Wiener process. Here, we denote the filtered probability space by $(\Omega ,\mathcal {F}, (\mathcal {F}_t)_{t\ge 0}, P)$. Suppose that we have discrete data $X^\varepsilon _{t_0}, \dots , X^\varepsilon _{t_n}$ from (1.1) for $0=t_0<\dots <t_n=1$ with $t_i - t_{i-1} = 1/n$. We consider the problem of estimating the true $\theta _0\in \Theta _0$ under $n\rightarrow \infty$ and $\varepsilon \rightarrow 0$. We also define $x_t$ as the solution of the corresponding deterministic differential equation

$$\begin{aligned} \frac{\textrm{d}x_t}{\textrm{d}t} = a(x_t, \mu _0) \end{aligned}$$

with the initial condition $x_0$.

In the ergodic case, threshold estimation for SDEs with Lévy noise is proposed in Shimizu and Yoshida (2006), and has been considered so far by various researchers [see, e.g., Amorino and Gloter 2019; Gloter et al. 2018; Ogihara and Yoshida 2011; Shimizu 2017, and other references are given in Amorino and Gloter (2021)]. On the other hand, in the small noise case, no one has succeeded in giving a proof for such joint threshold estimation of the parameter relative to drift, diffusion and jumps. So in this paper, we give a framework and a proof for the threshold estimation in the small noise case.

As an essential part of our framework for estimation, we suppose not only $n\rightarrow \infty$ and $\varepsilon \rightarrow 0$ but $\lambda _\varepsilon \rightarrow \infty$, while the intensity $\lambda _\varepsilon$ is fixed, $n\rightarrow \infty$ and $\varepsilon \rightarrow 0$ in the previous works of estimations for SDEs with small noise [see, e.g., Gloter and Sørensen 2009; Kobayashi and Shimizu 2022; Long et al. 2013; Sørensen and Uchida 2003, and references are given in Long et al. (2017)]. The asymptotics with $\lambda _\varepsilon \rightarrow \infty$ would be the first and new attempt in many works of literature, and enables us to deal with the joint estimation of the parameter $(\mu ,\sigma ,\alpha )$ relative to drift, diffusion and jumps, while the papers above deal with only the estimation of drift and diffustion parameters (or in some papers drift parameter only). Indeed, one can immediately notice that if the intensity $\lambda _\varepsilon$ is constant, then the number of large jumps never goes to infinity in probability, and so we would never establish a consistent estimator of jump size density. Therefore, we suppose that $\lambda _\varepsilon \rightarrow \infty$ as $\varepsilon \downarrow 0$ ($\lambda _\varepsilon$ is not necessary to depend on $\varepsilon$ as in Remark 2.4). Also, the assumption $\lambda _\varepsilon \rightarrow \infty$ seems natural when we deal with data obtained in the long term with the pitch of observations shortened, which is familiar in both cases of ergodic and small noise. Thus, one can agree with our proposal.

Another attempt in this paper is to give a proof by using localization argument [as in, e.g., Remark 1 in Sørensen and Uchida (2003)] in the entire context, though the argument is usually omitted, or instead, Propostion 1 in Gloter and Sørensen (2009) is just referred. As to the proof, we prepare the localization assumptions for jump size densities, i.e., Assumptions 2.9 to 2.12, together with usual localization assumptions for coefficient functions in (1.1), i.e., Assumptions 2.5 and 2.6. Owing to prepare Assumptions 2.9 to 2.12, this paper has more examples of jump size densities than the papers (Ogihara and Yoshida 2011; Shimizu and Yoshida 2006) (see Sect. 5 in this paper, and see, e.g., Ogihara and Yoshida 2011, Example). On the other hand, Assumptions 2.9 to 2.12 are too complicated for us to omit the localization argument. Thus, we show our main results under the localization argument in the entirety of our proof, which is one of the novelties of our paper.

A further attempt of this paper is to simplify the contrast functions used in earlier works (Ogihara and Yoshida 2011; Shimizu and Yoshida 2006) by removing $\varphi _n$ defined in Ogihara and Yoshida (2011) and Shimizu and Yoshida (2006) from their contrast functions. As we mentioned above, the class of jump size densities is wide and includes unbounded densities [e.g., log-normal distribution) which are not included in Ogihara and Yoshida (2011) and Shimizu and Yoshida (2006). Note that the class of jump size densities in Shimizu (2006) is also wide (Shimizu 2006 does not assume the twice differentiability of jump size densities, while conversely this paper does not assume $\int |z|^p\frac{\partial }{\partial \alpha _j} f_\alpha (z)\textrm{d}{z}$ ($p\ge 1$) as in the assumption A5 in Shimizu (2006)], but (Shimizu 2006) is concerned with moment estimators in the ergodic case.

In order to see the behavior of our estimator in numerical experiments, we give Table 1 under the assumption that $\lambda _\varepsilon$ is known. Of course, this assumption is impractical when we deal with only observations, and how to choose threshold $v_{nk}/n^\rho$ in filters $1_{C^{n,\varepsilon ,\rho }_k}$ and $1_{D^{n,\varepsilon ,\rho }_k}$ defined in Notation 2.7 is one of the crucial points for estimation with jumps, but it is not within the scope of this paper (see, e.g., Shimizu 2008, 2010 for the readers who are interested in the techniques of the way to choose such threshold, and then Lemma 4.8 may also help you estimate the intensity $\lambda _\varepsilon$). Instead of this discussion, we give another experiment as in Table 2 to see what will occur by using different thresholds.

In Sect. 2, we set up some assumptions and notations. In Sect. 3, we state our main results, i.e., the consistency and the asymptotic normality of our estimator. In Sect. 4, we give a proof of our main results. In Sect. 5, we give some examples of the jump size density for compound Poisson processes in our model. In Sect. 6, we give two numerical experiments to see the finite sample performance of our estimator. In “Appendix A”, we state and prove some slightly different versions of well-known results.

2 Assumptions and notations

This section is devoted to prepare some notations and assumptions. Before going to see our assumptions, we begin by setting up the following two notations:

Notation 2.1

Let $I_{x_0}$ be the image of $t\mapsto x_t$ on [0, 1], and set

$$\begin{aligned} I_{x_0}^{\delta } := \left\{ { y\in \mathbb {R} \,\Big |\, {{\,\textrm{dist}\,}}(y,I_{x_0}) = \inf _{x\in I_{x_0}} |x-y| < \delta }\right\} . \end{aligned}$$

Notation 2.2

A function $\psi$ on $\mathbb {R}\times \mathbb {R}\times \Theta _3$ is of the form

$$\begin{aligned} \psi (x,y,\alpha ) := \left\{ \begin{array}{ll} \log \left| {\frac{1}{c(x,\alpha )} f_\alpha \left( \frac{y}{c(x,\alpha )} \right) }\right| &{} \text {if } c(x,\alpha ) \ne 0 \text { and } f_{\alpha } \left( \frac{y}{c(x,\alpha )} \right) >0, \\ 0 &{} \text {otherwise}. \end{array} \right. \end{aligned}$$

Then, we prepare the following assumptions:

Assumption 2.1

$a(\cdot ,\mu _0)$, $b(\cdot ,\sigma _0)$ and $c(\cdot ,\alpha _0)$ are Lipschitz continuous on $\mathbb {R}$.

Assumption 2.2

The functions a, b, c are differentiable with respect to $\theta$ on $I_{x_0}^\delta \times \Theta$ for some $\delta >0$, and the families $\left\{ { \frac{\partial a}{\partial \theta _{j}} \left( \cdot ,\mu \right) }\right\} _{\mu \in \Theta _1}$, $\left\{ { \frac{\partial b}{\partial \theta _j} \left( \cdot ,\sigma \right) }\right\} _{\sigma \in \Theta _2}$, $\left\{ { \frac{\partial c}{\partial \theta _j} \left( \cdot ,\alpha \right) }\right\} _{\alpha \in \Theta _3}$ $(j=1,\dots ,d)$ are equi-Lipschitz continuous on $I_{x_0}^\delta$.

Assumption 2.3

For any $p\ge 0$, let $f_{\alpha _0}:\mathbb {R}\rightarrow \mathbb {R}$ satisfy

$$\begin{aligned} \int _{\mathbb {R}} |z|^p f_{\alpha _0}(z) \textrm{d}{z} < \infty . \end{aligned}$$

Assumption 2.4

The family $\left\{ {f_{\alpha }}\right\} _{\alpha \in \bar{\Theta }_3}$ satisfies either of the following condtions:

• (i) $f_{\alpha }$, $\alpha \in \bar{\Theta }_3$ are positive and continuous on $\mathbb {R}$.

• (ii) $f_{\alpha }$, $\alpha \in \bar{\Theta }_3$ are positive and continuous on $\mathbb {R}_+(=(0,\infty ))$, and are zero on $(-\infty ,0]$.

Assumption 2.5

The family $\left\{ {b(\cdot ,\sigma )}\right\} _{\sigma \in \bar{\Theta }_2}$ satisfies

$$\begin{aligned} \inf _{(x,\sigma )\in I_{x_0}\times \Theta _2} |b(x_t, \sigma )| > 0. \end{aligned}$$

Assumption 2.6

The familiy $\left\{ {c(\cdot ,\alpha )}\right\} _{\sigma \in \bar{\Theta }_3}$ satisfies

$$\begin{aligned} 0 < c_1 \le |c(x, \alpha )| \le c_2 \quad \text {for } (x,\alpha ) \in I_{x_0}\times \Theta _3 \end{aligned}$$

with some positve constants $c_1$ and $c_2$. In this paper, without loss of generality, we may assume

$$\begin{aligned} c(x_t, \alpha ) > c_1 \quad \text {for } (x,\alpha ) \in I_{x_0}\times \Theta _3. \end{aligned}$$

Assumption 2.7

If $\mu \ne \mu _0$, $\sigma \ne \sigma _0$ or $\alpha \ne \alpha _0$, then

$$\begin{aligned} a(y,\mu )\not \equiv a(y,\mu _0), \quad b(y,\sigma )\not \equiv b(y,\sigma _0) \quad \text {or} \quad \psi (y,c(y,z,\alpha ) \ne \psi (y,z,\alpha _0), \quad \text {respectively} \end{aligned}$$

for some $y\in I_{x_0}^{\delta }$ with some $\delta >0$, and for some $z\in \mathbb {R}$.

Assumption 2.8

$v_{n1},\dots ,v_{nn}$ are random variables such that $v_{nk}$ is $\mathcal {F}_{t_{k-1}}$-measurable (or measurable with respect to $\{X_{t_j};j<k\}$), and they satisfy

$$\begin{aligned} 0 < v_1 \le v_{nk} \le v_2 \end{aligned}$$

for some constants $v_1$ and $v_2$.

Assumption 2.9

There exists $\delta >0$ such that for $(x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}\times \Theta$ with $\psi \ne 0$, $\psi$ is differentiable with respect to $\alpha _i$ ($i=1,\dots ,d_3$). For $\alpha \in \Theta _3$

$$\begin{aligned} x \mapsto \int \psi (x,c(x,\alpha _0)z,\alpha ) f_{\alpha _0}(z) \textrm{d}{z}, \quad x \mapsto \int |\psi (x,c(x,\alpha _0)z,\alpha )|^2 f_{\alpha _0}(z) \textrm{d}{z} \end{aligned}$$

are continuous at every points in $I_{x_0}$, and there exist $\delta >0$ and $C>0$ such that

$$\begin{aligned} \int \left\{ \sup _{(x,\alpha )\in I_{x_0}^\delta \times \Theta _3} \left| \psi (x,c(x,\alpha _0)z,\alpha ) \right| + \sum _{j=1}^{d_3} \sup _{(x,\alpha )\in I_{x_0}^\delta \times \Theta _3} \left| { \frac{\partial \psi }{\partial \alpha _j} \left( x,c(x,\alpha _0)z,\alpha \right) }\right| \right\} f_{\alpha _0} (z) \textrm{d}{z} < \infty . \end{aligned}$$

Assumption 2.10

Relative to the choice (i) or (ii) in Assumption 2.4, we assume either of the following conditions (i) or (ii), respectively:

• (i) Under Assumption 2.4 (i), there exist constants $C>0$, $q\ge 1$ and $\delta >0$ such that

  $$\begin{aligned} \sup _{(x,\alpha ) \in I_{x_0}^\delta \times \Theta _3} \left| \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right| \le C (1+|y|^q) \quad (y \in \mathbb {R}). \end{aligned}$$

• (ii) Under Assumption 2.4 (ii), we assume the following three conditions:

  • (ii.a) There exists $\delta >0$ and $L>0$ such that if $0<y_1\le y \le y_2$, then

    $$\begin{aligned} \left| \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right|\le & {} \left| \frac{\partial \psi }{\partial y} \left( x,y_1,\alpha \right) \right| \\{} & {} + \left| \frac{\partial \psi }{\partial y} \left( x,y_2,\alpha \right) \right| + L \quad \text {for all } (x,\alpha )\in I_{x_0}^\delta \times \Theta _3. \end{aligned}$$

  • (ii.b) There exist constants $q\ge 0$ and $\delta >0$ such that

    $$\begin{aligned} \sup _{(x,\alpha ) \in I_{x_0}^\delta \times \Theta _3} \left| \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right| \le O \left( \frac{1}{|y|^q} \right) \quad \text {as } |y| \rightarrow 0. \end{aligned}$$

  • (ii.c) There exists $\delta >0$ such that for any $C_1>0$ and $C_2\ge 0$ the map

    $$\begin{aligned} x \mapsto \int \sup _{\alpha \in \Theta _3} \left| { \frac{\partial \psi }{\partial y} \left( x, C_1 y + C_2,\alpha \right) }\right| f_{\alpha _0}(y) \textrm{d}{y} \end{aligned}$$

    takes values in $\mathbb {R}$ from $I_{x_0}^\delta$, and is continous on $I_{x_0}^\delta$.

Assumption 2.11

For $(x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}\times \Theta$ with $\psi \ne 0$, $\psi$ is differentiable with respect to $\alpha \in \Theta _3$, and

$$\begin{aligned} x \mapsto \int \frac{\partial \psi }{\partial \alpha _i} \frac{\partial \psi }{\partial \alpha _j} \left( x, c(x,\alpha _0\right) z, \alpha _0 ) f_{\alpha _0}(z) \textrm{d}{z} \quad (i,j=1,\dots ,d_3) \end{aligned}$$

is continuous at every point $x\in I_{x_0}$.

Assumption 2.12

The functions a, b, c are twice differentiable with respect to $\theta$ on $I_{x_0}^\delta \times \Theta$ for some $\delta$, and the families $\left\{ { \frac{\partial ^{2}a}{\partial \theta _{i}\partial \theta _j} \left( \cdot ,\mu \right) }\right\} _{\mu \in \Theta _1}$, $\left\{ { \frac{\partial ^{2}b}{\partial \theta _{i}\partial \theta _j} \left( \cdot ,\sigma \right) }\right\} _{\sigma \in \Theta _2}$, $\left\{ { \frac{\partial ^{2}c}{\partial \theta _{i}\partial \theta _j} \left( \cdot ,\alpha \right) }\right\} _{\alpha \in \Theta _3}$ $(i,j=1,\dots ,d)$ are equi-Lipschitz continuous on $I_{x_0}^\delta$. There exists $\delta >0$ such that for $(x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}\times \Theta$ with $\psi \ne 0$, $\psi$ is twice differentiable with respect to $\alpha _i$ ($i=1,\dots ,d_3$). For $\alpha \in \Theta$, $i=1,\dots ,d_3$

$$\begin{aligned} x \mapsto \int \frac{\partial \psi }{\partial \alpha _i} \big (x,c(x,\alpha _0)z,\alpha \big ) f_{\alpha _0}(z) \textrm{d}{z}, \quad x \mapsto \int \left| { \frac{\partial \psi }{\partial \alpha _i} \big (x,c(x,\alpha _0)z,\alpha \big ) }\right| ^2 f_{\alpha _0}(z) \textrm{d}{z} \end{aligned}$$

are continuous at every points $x\in I_{x_0}$, and there exist $\delta >0$ such that

$$\begin{aligned} \int \sum _{i,j=1}^{d_3} \sup _{(x,\theta )\in I_{x_0}^\delta \times \Theta } \left| { \frac{\partial ^{2}\psi }{\partial \alpha _i}{\alpha _j} \big (x,c(x,\alpha _0)z,\theta \big ) }\right| f_{\alpha _0} (z) \textrm{d}{z} < \infty . \end{aligned}$$

Relative to the choice (i) or (ii) in Assumption 2.4, we assume either of the following conditions (i) or (ii), respectively:

• (i) Under Assumption 2.4 (i), there exist constants $C>0$, $q\ge 1$ and $\delta >0$ such that

  $$\begin{aligned} \sup _{(x,\alpha ) \in I_{x_0}^\delta \times \Theta _3} \left| { \frac{\partial ^{2}\psi }{\partial y}{\alpha _i} \left( x,y,\alpha \right) }\right| \le C (1+|y|^q) \quad (y \in \mathbb {R}). \end{aligned}$$

• (ii) Under Assumption 2.4 (ii), we assume the following three conditions:

  • (ii.a) There exists $\delta >0$ and $L>0$ such that if $0<y_1\le y \le y_2$, then

    $$\begin{aligned} \left| { \frac{\partial ^{2}\psi }{\partial y}{\alpha _i} \left( x,y,\alpha \right) }\right|\le & {} \left| { \frac{\partial ^{2}\psi }{\partial y}{\alpha _i} \left( x,y_1,\alpha \right) }\right| \\{} & {} + \left| { \frac{\partial ^{2}\psi }{\partial y}{\alpha _i} \left( x,y_2,\alpha \right) }\right| + L \quad \text {for all } (x,\alpha )\in I_{x_0}^\delta \times \Theta _3. \end{aligned}$$

  • (ii.b) There exist constants $q\ge 0$ and $\delta >0$ such that

    $$\begin{aligned} \sup _{(x,\alpha ) \in I_{x_0}^\delta \times \Theta _3} \left| { \frac{\partial ^{2}\psi }{\partial y}{\alpha _i} \left( x,y,\alpha \right) }\right| \le O \left( \frac{1}{|y|^q} \right) \quad \text {as } |y| \rightarrow 0. \end{aligned}$$

  • (ii.c) There exists $\delta >0$ such that for any $C_1>0$ and $C_2\ge 0$ the map

    $$\begin{aligned} x \mapsto \int \sup _{\alpha \in \Theta _3} \left| { \frac{\partial ^{2}\psi }{\partial y}{\alpha _i} \left( x, C_1 y + C_2,\alpha \right) }\right| f_{\alpha _0}(y) \textrm{d}{y} \end{aligned}$$

    takes values in $\mathbb {R}$ from $I_{x_0}^\delta$, and is continous on $I_{x_0}^\delta$.

Remark 2.1

Instead of Assumptions 2.5 and 2.6, the following stronger assumptions are often used:

$$\begin{aligned} \inf _{(x,\sigma )\in \mathbb {R}\times \bar{\Theta }_2} |b(x, \sigma )|> 0, \quad \inf _{(x,\alpha )\in \mathbb {R} \times \bar{\Theta }_3} |c(x, \alpha )| > 0. \end{aligned}$$

(see, e.g., Remark 1 in Sørensen and Uchida 2003). However, the ‘classical’ localization argument mentioned in Sørensen and Uchida (2003) is hard to apply for our purpose. Thus, we employ our milder assumptions and show how it works well.

Remark 2.2

Under Assumption 2.9,

$$\begin{aligned} \int \frac{\partial \psi }{\partial \alpha _i} \left( x, c(x,\alpha _0) z, \alpha _0 \right) f_{\alpha _0}(z) \textrm{d}{z} = \frac{\partial }{\partial \alpha _i} \left( \int \psi \big ( x, c(x,\alpha _0) z, \alpha \big ) f_{\alpha _0}(z) \textrm{d}{z} \right) _{\alpha =\alpha _0}, \end{aligned}$$

at every $x\in I_{x_0}^\delta$.

Remark 2.3

Assumption 2.12 is given by replacing $a,b,c,\psi$ with $\frac{\partial a}{\partial \mu _i}$, $\frac{\partial b}{\partial \sigma _i}$, $\frac{\partial c}{\partial \alpha _i}$, $\frac{\partial \psi }{\partial \alpha _i}$, respectively, in Assumptions 2.2, 2.9 and 2.10, and is needed for obtaining the convergence (4.16) of the matrix containing the second derivatives of the contrast function.

Furthermore, we introduce the following notations:

Notation 2.3

Denote

$$\begin{aligned} \Delta X^\varepsilon _t := X^\varepsilon _t - X^\varepsilon _{t-} \quad \text {for}~t>0, \end{aligned}$$

where $\varepsilon >0$.

Notation 2.4

Denote

$$\begin{aligned} \Delta ^n_k X^{\varepsilon } := X^\varepsilon _{t_k} - X^\varepsilon _{t_{k-1}}, \quad \Delta ^n_k N^{\lambda _\varepsilon } := N^{\lambda _\varepsilon }_{t_k} - N^{\lambda _\varepsilon }_{t_{k-1}} \quad \text {for}~k=1,\dots ,n, \end{aligned}$$

where $n\in \mathbb {N}$, $\varepsilon >0$.

Notation 2.5

Define random times

$$\begin{aligned} \tau _k&:=\inf \{t\in [t_{k-1},t_k]\,|\,\Delta X^\varepsilon _t\ne 0~\text {or}~t=t_k\}, \\ \eta _k&:=\sup \{t\in [t_{k-1},t_k]\,|\,\Delta X^\varepsilon _t\ne 0~\text {or}~t=t_{k-1}\}. \end{aligned}$$

Notation 2.6

Define events $J^{n,\varepsilon }_{k,i}$ $(k=1,\dots ,n,~i=0,1,2)$ by

$$\begin{aligned} J^{n,\varepsilon }_{k,0} := \left\{ \Delta ^n_k N^{\lambda _\varepsilon }= 0 \right\} , \quad J^{n,\varepsilon }_{k,1} := \left\{ \Delta ^n_k N^{\lambda _\varepsilon }= 1 \right\} , \quad J^{n,\varepsilon }_{k,2} := \left\{ \Delta ^n_k N^{\lambda _\varepsilon }\ge 2 \right\} \end{aligned}$$

where $n\in \mathbb {N}$, $\varepsilon >0$.

Notation 2.7

Under Assumption 2.8, set events $C^{n,\varepsilon ,\rho }_k$ and $D^{n,\varepsilon ,\rho }_k$ $(k=1,\dots ,n)$ by

$$\begin{aligned} C^{n,\varepsilon ,\rho }_k&:= {\left\{ \begin{array}{ll} \left\{ \left| \Delta ^n_k X^{\varepsilon } \right| \le \frac{v_{nk}}{n^\rho } \right\} &{} \text {under Assumption 2.4 (i),}\\ \left\{ \Delta ^n_k X^{\varepsilon } \le \frac{v_{nk}}{n^\rho } \right\} &{} \text {under Assumption 2.4 (ii),} \end{array}\right. } \\ D^{n,\varepsilon ,\rho }_k&:= {\left\{ \begin{array}{ll} \left\{ \left| \Delta ^n_k X^{\varepsilon } \right|> \frac{v_{nk}}{n^\rho } \right\} &{} \text {under Assumption 2.4 (i),}\\ \left\{ \Delta ^n_k X^{\varepsilon } > \frac{v_{nk}}{n^\rho } \right\} &{} \text {under Assumption 2.4 (ii),} \end{array}\right. } \end{aligned}$$

where $n\in \mathbb {N}$, $\varepsilon >0$, $\rho \in (0,1/2)$. Then, put

$$\begin{aligned} C^{n,\varepsilon ,\rho }_{k,i} := C^{n,\varepsilon ,\rho }_k \cap J^{n,\varepsilon }_{k,i}, \quad D^{n,\varepsilon ,\rho }_{k,i} := D^{n,\varepsilon ,\rho }_k \cap J^{n,\varepsilon }_{k,i} \quad \text {for}~k=1,\dots ,n,~i=0,1,2, \end{aligned}$$

where $n\in \mathbb {N}$, $\varepsilon >0$, $\rho \in (0,1/2)$. Furthermore, for sufficiently small $\delta >0$, we may put

$$\begin{aligned} \begin{aligned} \tilde{C}^{n,\varepsilon ,\rho }_{k,i}&:= C^{n,\varepsilon ,\rho }_{k,i} \cap \{ X^\varepsilon _t \in I_{x_0}^\delta \text { for all } t\in [0,1] \}, \\ \tilde{D}^{n,\varepsilon ,\rho }_{k,i}&:= D^{n,\varepsilon ,\rho }_{k,i} \cap \{ X^\varepsilon _t \in I_{x_0}^\delta \text { for all } t\in [0,1] \} \end{aligned} \end{aligned}$$

for $k=1,\dots ,n$, $i=0,1,2$.

Remark 2.4

We treat $(n,\varepsilon )$ as a directed set with a suitable order according to a convergence. For examples, when we say that $n\rightarrow \infty$, $\varepsilon \rightarrow 0$ and $\lambda _\varepsilon \rightarrow \infty$, we mean that the index set $(n,\varepsilon )$ is a directed set in $\mathbb {N}\times (0,\infty )$ with order $\prec _1$ defined by

$$\begin{aligned} (n_1,\varepsilon _1) \prec _1 (n_2,\varepsilon ) \quad \text {if} ~ n_1< n_2, ~ \varepsilon _1 > \varepsilon _2 ~ \text {and} ~ \lambda _{\varepsilon _1} < \lambda _{\varepsilon _2}, \end{aligned}$$

and when we say that $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\lambda _\varepsilon \int _{|z|\le C/n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$ with some constants $C,\rho >0$, we mean that the index set $(n,\varepsilon )$ is a directed set in $\mathbb {N}\times (0,\infty )$ with order $\prec _2$ defined by

$$\begin{aligned}&(n_1,\varepsilon _1) \prec _2 (n_2,\varepsilon ) \quad \text {if} ~ n_1< n_2, ~ \varepsilon _1> \varepsilon _2, ~ \lambda _{\varepsilon _1} < \lambda _{\varepsilon _2}~\\&\quad \text {and} ~ \lambda _{\varepsilon _1} \int _{|z|\le \frac{C}{n^\rho _1}} f_{\alpha _0}(z) \textrm{d}{z} > \lambda _{\varepsilon _2} \int _{|z|\le \frac{C}{n^\rho _2}} f_{\alpha _0}(z) \textrm{d}{z}. \end{aligned}$$

Needless to say, the identity map ${{\,\textrm{Id}\,}}$ from $(\{(n,\varepsilon )\},\prec _2)$ to $(\{(n,\varepsilon )\},\prec _1)$ is monotone, and ${{\,\textrm{Id}\,}}(\{(n,\varepsilon )\})$ is cofinal in $(\{(n,\varepsilon )\},\prec _1)$.

Remark 2.5

In this paper, we can assume $\lambda _\varepsilon$ does not depend on $\varepsilon$. In this case, we treat $\{(n,\varepsilon ,\lambda )\}$ instead of $\{(n,\varepsilon )\}$ as a driected set, and we must write $X^{\varepsilon ,\lambda }$, $Z^{\lambda }$, $\Psi _{n,\varepsilon ,\lambda }$, etc., instead of $X^\varepsilon$, $Z^{\lambda _\varepsilon }$, $\Psi _{n,\varepsilon }$, etc., respectively. But, for simplicity, we assume $\lambda _\varepsilon$ depends on $\varepsilon$.

3 Main results

We define the following contrast function $\Psi _{n,\varepsilon }(\theta )$ after the quasi-log likelihood proposed in Shimizu (2017):

$$\begin{aligned} \Psi _{n,\varepsilon }(\theta ) := \Psi _{n,\varepsilon }^{(1)}(\mu ,\sigma ) + \Psi _{n,\varepsilon }^{(2)}(\alpha ) \quad \text {for } \theta =(\mu ,\sigma ,\alpha )\in \Theta , \end{aligned}$$

where for $\rho \in (0,1/2)$, $\Psi _{n,\varepsilon }^{(1)}(\mu ,\sigma )$ and $\Psi _{n,\varepsilon }^{(2)}(\alpha )$ are given by using Notations 2.4 and 2.7 as the following:

$$\begin{aligned} \Psi _{n,\varepsilon }^{(1)}(\mu ,\sigma ):= & {} - \frac{1}{n} \sum _{k=1}^{n} \left\{ \frac{ \left| { \Delta ^n_k X^{\varepsilon } - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu ) }\right| ^2 }{2 \frac{1}{n} \left| { \varepsilon b(X^\varepsilon _{t_{k-1}},\sigma ) }\right| ^2} + \frac{1}{2} \log |b(X^\varepsilon _{t_{k-1}},\sigma )|^2 \right\} 1_{C^{n,\varepsilon ,\rho }_k}, \nonumber \\ \Psi _{n,\varepsilon }^{(2)}(\alpha ):= & {} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^{n} \psi \left( X^\varepsilon _{t_{k-1}}, \frac{\Delta ^n_k X^{\varepsilon }}{\varepsilon }, \alpha \right) 1_{D^{n,\varepsilon ,\rho }_k} \end{aligned}$$

(3.1)

with

$$\begin{aligned} \psi (x,y,\alpha ) := \left\{ \begin{array}{ll} \log \left| \frac{1}{c(x,\alpha )} f_\alpha \left( \frac{y}{c(x,\alpha )} \right) \right| &{} \text {if } c(x,\alpha ) \ne 0 \text { and } f_{\alpha }\left( \frac{y}{c(x,\alpha )} \right) >0, \\ 0 &{} \text {otherwise}. \end{array} \right. \end{aligned}$$

Then, the quasi-maximum likelihood estimator is given by

$$\begin{aligned} \hat{\theta }_{n,\varepsilon } := \mathop {\textrm{argmax}}\limits _{\theta \in \Theta } \Psi _{n,\varepsilon }(\theta ). \end{aligned}$$

(3.2)

The goal is to show the asymptotic normality of $\hat{\theta }_{n,\varepsilon }$ when $n\rightarrow \infty$ and $\varepsilon \rightarrow 0$ at the sametime. In the sequel, we will also assume that $\lambda _\varepsilon \rightarrow \infty$ as $\varepsilon \downarrow 0$ for consistency of $\hat{\theta }_{n,\varepsilon }$. Our interest is in a situation where the number of jumps is large and the Lévy noise is small. In practice, $\lambda _\varepsilon$, the intensity of jumps, should be estimated, and it is possible by Lemma 4.8:

$$\begin{aligned} \lambda _\varepsilon \overset{p}{\sim }\ \sum _{k=1}^n 1_{D^{n,\varepsilon ,\rho }_{k}} \quad \text {as}~\varepsilon \downarrow 0. \end{aligned}$$

Theorem 3.1

Under Assumptions 2.1 to 2.10, take $\rho$ as either of the following:

• (i) Under Assumption 2.4 (i), take $\rho \in (0,1/2)$.

• (ii) Under Assumption 2.4 (ii), take $\rho \in (0,\min \{1/2,1/4q\})$, where q is the constant in Assumption 2.10 Assumption (ii.b).

Then,

$$\begin{aligned} \hat{\theta }_{n,\varepsilon } \overset{p}{\longrightarrow }\theta _0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon ^2/n \rightarrow 0$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$ with $\lim (\varepsilon ^2 n)^{-1}<\infty$. Here, the constants $c_1$ and $v_2$ are taken as in Assumptions 2.6 and 2.8, respectively.

Theorem 3.2

Under Assumptions 2.1 to 2.12, take $\rho$ as either of the following:

• (i) Under Assumption 2.4 (i), take $\rho \in (0,1/2)$.

• (ii) Under Assumption 2.4 (ii), take $\rho \in (0,\min \{1/2,1/4q\})$, where q is the constant in Assumptions 2.10 (ii.b) and 2.12 (ii.b).

If $\theta _0\in \Theta$ and $I_{\theta _0}$ is positive definite, then

$$\begin{aligned} \begin{pmatrix} \varepsilon ^{-1} (\hat{\mu }_{n,\varepsilon }-\mu _0) \\ \sqrt{n} (\hat{\sigma }_{n,\varepsilon }-\sigma _0) \\ \sqrt{\lambda _\varepsilon } (\hat{\alpha }_{n,\varepsilon }-\alpha _0) \end{pmatrix} \overset{d}{\longrightarrow }\mathcal {N} \left( 0, I_{\theta _0}^{-1} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon ^2/n \rightarrow 0$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$ with $\lim (\varepsilon ^2 n)^{-1}<\infty$, where

$$\begin{aligned} I_{\theta _0} := \left( \begin{array}{ccc} I_1&{}0&{}0\\ 0&{}I_2&{} 0\\ 0&{}0&{}I_2\\ \end{array}\right) \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} I^{ij}_1&:= \int _0^1 \frac{ \frac{\partial a}{\partial \mu _i} \frac{\partial a}{\partial \mu _j} \left( x_t,\mu _0\right) }{|b(x_t,\mu _0)|^2} \textrm{d}{t}{} & {} (i,j=1,\dots ,d_1), \\ I^{ij}_2&:= 2 \int _0^1 \frac{\frac{\partial b}{\partial \sigma _i} \frac{\partial b}{\partial \sigma _j} \left( x_t,\sigma _0\right) }{|b(x_t,\sigma _0)|^2} \textrm{d}{t}{} & {} (i,j=1,\dots ,d_2), \\ I^{ij}_3&:= \int _0^1 \int \frac{\partial \psi }{\partial \alpha _i} \frac{\partial \psi }{\partial \alpha _j} \big (x_t,c(x_t,\alpha _0\big ) z,\alpha _0) f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{t}{} & {} (i,j=1,\dots ,d_4). \end{aligned} \end{aligned}$$

(3.3)

Remark 3.1

If $\left\{ {f_\alpha }\right\} _{\alpha \in \Theta _3}$ is given by the class of the densities of normal distributions as in Example 5.1, then the range of $\rho$ in Theorems 3.1 and 3.2 is same as in Shimizu and Yoshida (2006) and Ogihara and Yoshida (2011). However, if $\left\{ {f_\alpha }\right\} _{\alpha \in \Theta _3}$ is given by the class of the densities of gamma distributions as in Example 5.2, then the range of $\rho$ is (0, 1/4) which is different from the range $(3/8+b,1/2)$ of $\rho$ in Ogihara and Yoshida (2011), where b is the constant defined in the equation (1) in Ogihara and Yoshida (2011).

4 Proofs

4.1 Inequalities

Lemma 4.1

Under Assumptions 2.1 and 2.3, suppose that $0<\varepsilon \le 1$, $\lambda _\varepsilon \ge 1$, $\varepsilon \lambda _\varepsilon \le 1$ and $0\le s<t\le 1$. Then, for $p\ge 2$,

$$\begin{aligned}{} & {} E \left[ \sup _{u\in [s,t]} \left| X^\varepsilon _u - X^\varepsilon _s \right| ^p \, \Big | \, \mathcal {F}_s \right] \\{} & {} \quad \le C \left\{ (t-s)^p + \varepsilon ^p \left( (t-s)^{p/2} + \lambda _\varepsilon (t-s) + \lambda _\varepsilon ^{p/2}(t-s)^{p/2} + \lambda _\varepsilon ^{p}(t-s)^{p} \right) \right\} \left( 1 + |X^\varepsilon _s|^p \right) , \end{aligned}$$

where C depends only on p, a, b, c and $f_{\alpha _0}$. In particular, when $\lambda _\varepsilon /n\le 1$ and $\lambda _\varepsilon \ge 1$, it holds for $p\ge 2$ and $k=1,\dots ,n$ that

$$\begin{aligned} E \left[ \sup _{t\in [t_{k-1},t_k]} \frac{\left| X^\varepsilon _t - X^\varepsilon _{t_{k-1}} \right| ^p}{\varepsilon ^p} \, \Big | \, \mathcal {F}_{t_{k-1}} \right]&\le C \left\{ \frac{1}{\varepsilon ^p n^p} + \frac{1}{n^{p/2}} + \frac{\lambda _\varepsilon }{n} \right\} \left( 1 + |X^\varepsilon _s|^p \right) , \\ E \left[ \sup _{t\in [0,1]} \left| X^\varepsilon _t - x_0 \right| ^p \, \Big | \, \mathcal {F}_{t_0} \right]&\le C \left\{ 1 + \varepsilon ^p \lambda _\varepsilon ^p \right\} \left( 1 + |x_0|^p \right) , \end{aligned}$$

where C depends only on p, a, b, c and $f_{\alpha _0}$.

Proof

For any $p\ge 2$, we have

$$\begin{aligned}&\left( E \left[ \sup _{u\in [s,t]} \left| X^\varepsilon _u - X^\varepsilon _{s} \right| ^p \, \Big | \, \mathcal {F}_{s} \right] \right) ^{1/p}\nonumber \\&\quad \le \left( E \left[ \left| \int _s^t \left| a(X^\varepsilon _u,\mu _0) - a(X^\varepsilon _s,\mu _0) \right| \textrm{d}{u} \right| ^p \, \Big | \, \mathcal {F}_s \right] \right) ^{1/p} \nonumber \\&\qquad + \varepsilon \left( E \left[ \sup _{u\in [s,t]} \left| \int _s^u \left\{ { b(X^\varepsilon _{v}, \sigma _0) - b(X^\varepsilon _s, \sigma _0) }\right\} \textrm{d}{W_v} \right| ^p \, \Big | \, \mathcal {F}_s \right] \right) ^{1/p} \nonumber \\&\qquad + \varepsilon \left( E \left[ \sup _{u\in [s,t]} \left| { \int _s^u \left\{ { c(X^\varepsilon _v, \alpha _0) - c(X^\varepsilon _s, \alpha _0) }\right\} \textrm{d}{Z_v^{\lambda _\varepsilon }} }\right| ^p \, \Big | \, \mathcal {F}_s \right] \right) ^{1/p} \nonumber \\&\qquad + (t-s) \left| a(X^\varepsilon _s,\mu _0) \right| + C \varepsilon \sqrt{t-s} \left| b(X^\varepsilon _s,\sigma _0) \right| \nonumber \\&\qquad + \varepsilon \left| c(X^\varepsilon _s, \alpha _0) \right| \left( E \left[ \sup _{u\in [s,t]} \left| \int _s^u \textrm{d}{Z_v^{\lambda _\varepsilon }} \right| ^p \right] \right) ^{1/p}, \end{aligned}$$

(4.1)

where C depends only on p. Then, it follows from the Lipschitz continuity of $a(\cdot ,\mu _0)$ that

$$\begin{aligned} E \left[ \left( \int _s^t \left| a(X^\varepsilon _u,\mu _0) - a(X^\varepsilon _s,\mu _0) \right| \textrm{d}{u} \right) ^p \, \Big | \, \mathcal {F}_s \right]&\le C E \left[ \left( \int _s^t |X^\varepsilon _u - X^\varepsilon _s| \textrm{d}{u} \right) ^p \, \Big | \, \mathcal {F}_s \right] \nonumber \\&\le C (t-s)^{p-1} \int _s^t E \left[ \left| X^\varepsilon _u - X^\varepsilon _s \right| ^p \, \Big | \, \mathcal {F}_s \right] \textrm{d}{u}, \end{aligned}$$

(4.2)

where C depends only on a, and it follows from the Lipschitz continuity of $b(\cdot ,\sigma _0)$ and Burkholder’s inequality (see, e.g., Theorem 4.4.21 in Applebaum (2009)) that

$$\begin{aligned}{} & {} E \left[ \sup _{u\in [s,t]} \left| \int _s^u \left\{ { b(X^\varepsilon _v, \sigma _0) - b(X^\varepsilon _s, \sigma _0) }\right\} \textrm{d}{W_v} \right| ^p \, \Big | \, \mathcal {F}_s \right] \nonumber \\{} & {} \quad \le C E \left[ \left| { \int _s^t \left| { X^\varepsilon _u - X^\varepsilon _s }\right| ^2 \textrm{d}{u} }\right| ^{p/2} \, \Big | \, \mathcal {F}_s \right] \nonumber \\{} & {} \quad \le C (t-s)^{p/2-1} \int _s^t E \left[ \left| { X^\varepsilon _u - X^\varepsilon _s }\right| ^p \, \Big | \, \mathcal {F}_s \right] \textrm{d}{u}, \end{aligned}$$

(4.3)

where C depends only on p and b, and from the Lipschitz continuity of $c(\cdot ,\alpha _0)$, it is analogous to the proof of Theorem 4.4.23 in Applebaum (2009) that

$$\begin{aligned}{} & {} E \left[ \sup _{u\in [s,t]} \left| \int _s^u \left\{ { c(X^\varepsilon _v, \alpha _0) - c(X^\varepsilon _s, \alpha _0) }\right\} \textrm{d}{Z_v^{\lambda _\varepsilon }} \right| ^p \, \Big | \, \mathcal {F}_s \right] \\{} & {} \quad \le C \left\{ E \left[ \left( \int _s^t \int _{\mathbb {R}} \left| { X^\varepsilon _u - X^\varepsilon _s }\right| ^2 \left| {z}\right| ^2 \lambda _\varepsilon \, f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{u} \right) ^{p/2} \, \Big | \, \mathcal {F}_s \right] \right. \\{} & {} \qquad + E \left[ \int _s^t \int _{\mathbb {R}} \left| { X^\varepsilon _u - X^\varepsilon _s }\right| ^p \left| {z}\right| ^p \lambda _\varepsilon \, f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{u} \, \Big | \, \mathcal {F}_s \right] \\{} & {} \qquad + \left. E \left[ \left( \int _s^t \int _{\mathbb {R}} \left| X^\varepsilon _u - X^\varepsilon _s \right| \left| z \right| \lambda _\varepsilon \, f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{u} \right) ^{p} \, \Big | \, \mathcal {F}_s \right] \right\} , \end{aligned}$$

where C depends only on p and c. Here, we have

$$\begin{aligned}{} & {} E \left[ \left( \int _s^t \int _{\mathbb {R}} \left| X^\varepsilon _u - X^\varepsilon _s \right| ^2 \left| z \right| ^2 \lambda _\varepsilon \, f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{u} \right) ^{p/2} \, \Big | \, \mathcal {F}_s \right] \\{} & {} \quad \le C \lambda _\varepsilon ^{p/2} \left( \int _s^t \left( E \left[ \left| X^\varepsilon _u - X^\varepsilon _s \right| ^p \, \Big | \, \mathcal {F}_s \right] \right) ^{2/p} \textrm{d}{u} \right) ^{p/2}, \\{} & {} \quad \le C \lambda _\varepsilon ^{p/2} (t-s)^{p/2-1} \int _s^t E \left[ \left| X^\varepsilon _u - X^\varepsilon _s \right| ^p \, \Big | \, \mathcal {F}_s \right] \textrm{d}{u}, \end{aligned}$$

where C depends only on p and $f_{\alpha _0}$, and

$$\begin{aligned}{} & {} E \left[ \left( \int _s^t \int _{\mathbb {R}} \left| X^\varepsilon _u - X^\varepsilon _s \right| \left| z \right| \lambda _\varepsilon \, f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{u} \right) ^{p} \, \Big | \, \mathcal {F}_s \right] \\{} & {} \quad \le C \lambda _\varepsilon ^{p} \left( \int _s^t \left( E \left[ \left| X^\varepsilon _u - X^\varepsilon _s \right| ^p \, \Big | \, \mathcal {F}_s \right] \right) ^{1/p} \textrm{d}{u} \right) ^p \\{} & {} \quad \le C \lambda _\varepsilon ^p (t-s)^{p-1} \int _s^t E \left[ \left| X^\varepsilon _u - X^\varepsilon _s \right| ^p \, \Big | \, \mathcal {F}_s \right] \textrm{d}{u}, \end{aligned}$$

where C depends only on p and $f_{\alpha _0}$. Thus,

$$\begin{aligned}{} & {} E \left[ \sup _{u\in [s,t]} \left| \int _s^u \left( c(X^\varepsilon _v, \alpha _0) - c(X^\varepsilon _s, \alpha _0) \right) \textrm{d}{Z_v^{\lambda _\varepsilon }} \right| ^p \, \Big | \, \mathcal {F}_s \right] \nonumber \\{} & {} \quad \le C \left( \lambda _\varepsilon ^{p/2}(t-s)^{p/2-1} + \lambda _\varepsilon + \lambda _\varepsilon ^p (t-s)^{p-1} \right) \int _s^t E \left[ \left| X^\varepsilon _u - X^\varepsilon _s \right| ^p \, \Big | \, \mathcal {F}_s \right] \textrm{d}{u}, \end{aligned}$$

(4.4)

where C depends only on p, c and $f_{\alpha _0}$. By using the Burkholder-Davis-Gundy inequality,

$$\begin{aligned} E \left[ \sup _{u\in [s,t]} \left| \int _s^u \textrm{d}{Z_v^{\lambda _\varepsilon }} \right| ^p \right] \le C \left( \lambda _\varepsilon ^{p/2} (t-s)^{p/2} + \lambda _\varepsilon (t-s) + \lambda _\varepsilon ^p (t-s)^p \right) , \end{aligned}$$

(4.5)

where C depends only on p and $f_{\alpha _0}$. From (4.1), (4.2), (4.3), (4.4) and (4.5),

$$\begin{aligned}{} & {} E \left[ \sup _{u\in [s,t]} \left| X^\varepsilon _u - X^\varepsilon _s \right| ^p \, \Big | \, \mathcal {F}_s \right] \\{} & {} \quad \le C \left\{ \left( (t-s)^{p-1} + \varepsilon ^p(t-s)^{\frac{p-2}{2}} + \varepsilon ^p \left( \lambda _\varepsilon + \lambda _\varepsilon ^{\frac{p}{2}}(t-s)^{\frac{p-2}{2}} + \lambda _\varepsilon ^{p}(t-s)^{p-1} \right) \right) \right. \\{} & {} \quad \times \int _s^t E \left[ \left| X^\varepsilon _u - X^\varepsilon _s \right| ^p \, \Big | \, \mathcal {F}_s \right] \textrm{d}{u} \\{} & {} \quad + (t-s)^p \left| a(X^\varepsilon _s,\mu _0) \right| ^p + \varepsilon ^p(t-s)^{p/2} \left| b(X^\varepsilon _s,\sigma _0) \right| ^p \\{} & {} \quad \left. + \varepsilon ^p \left( \lambda _\varepsilon (t-s) + \lambda _\varepsilon ^{p/2} (t-s)^{p/2} + \lambda _\varepsilon ^p (t-s)^p \right) \left| c(X^\varepsilon _s, \alpha _0) \right| ^p \right\} , \end{aligned}$$

where C depends only on p, a, b, c and $f_{\alpha _0}$. By Gronwall’s inequality,

$$\begin{aligned}{} & {} E \left[ \sup _{u\in [s,t]} \left| X^\varepsilon _u - X^\varepsilon _s \right| ^p \, \Big | \, \mathcal {F}_s \right] \\{} & {} \quad \le C \left\{ (t-s)^p \left| a(X^\varepsilon _s,\mu _0) \right| ^p + \varepsilon ^p(t-s)^{p/2} \left| b(X^\varepsilon _s,\sigma _0) \right| ^p \right. \\{} & {} \qquad \left. + \varepsilon ^p \left( \lambda _\varepsilon (t-s) + \lambda _\varepsilon ^{p/2}(t-s)^{p/2} + \lambda _\varepsilon ^p (t-s)^p \right) \left| c(X^\varepsilon _s, \alpha _0) \right| ^p \right\} \\{} & {} \qquad \times \exp \left( C \left\{ (t-s)^p + \varepsilon ^p(t-s)^{p/2} + \varepsilon ^p \lambda _\varepsilon (t-s)\right. \right. \\{} & {} \qquad \left. \left. + \varepsilon ^p \lambda _\varepsilon ^{p/2}(t-s)^{p/2} + \varepsilon ^p \lambda _\varepsilon ^{p}(t-s)^{p} \right\} \right) . \end{aligned}$$

This implies the conclusion. $\square$

Lemma 4.2

Under Assumptions 2.1 and 2.3, suppose that $0<\varepsilon \le 1$, $\lambda _\varepsilon \ge 1$, $\varepsilon \lambda _\varepsilon \le 1$ and $0\le s<t\le 1$. Then, for $p\ge 2$

$$\begin{aligned} E \left[ \sup _{u\in [s,t]} \left| X^\varepsilon _u - x_u \right| ^p \, \Big | \, \mathcal {F}_s \right] \le C \varepsilon ^p \left( (t-s)^{p/2} + \lambda _\varepsilon (t-s) + \lambda _\varepsilon ^{p/2}(t-s)^{p/2} + \lambda _\varepsilon ^{p}(t-s)^{p} \right) , \end{aligned}$$

where C depends only on p, a and b.

Proof

Same as the proof of Lemma 4.1, for any $p\ge 2$, we obtain

$$\begin{aligned}{} & {} E \left[ \sup _{u\in [s,t]} \left| X^\varepsilon _u - x_u \right| ^p \, \Big | \, \mathcal {F}_s \right] \\{} & {} \quad \le C \varepsilon ^p \left( (t-s)^{p/2} + \lambda _\varepsilon (t-s) + \lambda _\varepsilon ^{p/2}(t-s)^{p/2} + \lambda _\varepsilon ^{p}(t-s)^{p} \right) \\{} & {} \qquad \times \exp \left( C \left\{ (t-s)^p + \varepsilon ^p \left( (t-s)^{p/2} + \lambda _\varepsilon (t-s) + \lambda _\varepsilon ^{p/2}(t-s)^{p/2} + \lambda _\varepsilon ^{p}(t-s)^{p} \right) \right\} \right) , \end{aligned}$$

where C depends only on p, a, b, c and $f_{\alpha _0}$. $\square$

Lemma 4.3

Under Assumptions 2.1 and 2.3, for $p\ge 1$

$$\begin{aligned} \left\| X^\varepsilon _\cdot - x_\cdot \right\| _{L^p(\Omega ;L^\infty ([0,1]))} = O(\varepsilon \lambda _\varepsilon ) \end{aligned}$$

as $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, and

$$\begin{aligned} \left\| \sup _{\begin{array}{c} 0\le u,s\le 1 \\ |u-s|\le 1/n \end{array}} \left| X^\varepsilon _u - x_s \right| \right\| _{L^p(\Omega )} = O(1/n+\varepsilon \lambda _\varepsilon ) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

Proof

Both rates of convergence are obtained immediately from Lemma 4.2. $\square$

Lemma 4.4

Under Assumptions 2.1 and 2.3, suppose that a family $\left\{ {g(\cdot ,\theta )}\right\} _{\theta \in \Theta }$ of functions from $\mathbb {R}$ to $\mathbb {R}$ is equicontinuous at every points in $I_{x_0}$. Then,

$$\begin{aligned} \frac{1}{n} \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \theta \right) \overset{p}{\longrightarrow }\int _0^1 g(x_t, \theta ) \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$.

Proof

This follows from Lemmas 4.3 and A.2. $\square$

Lemma 4.5

Under Assumptions 2.1 and 2.3 with Notation 2.5, suppose that $0<\varepsilon \le 1$, $\lambda _\varepsilon \ge 1$ and $\varepsilon \lambda _\varepsilon \le 1$. Then, for any $p\in [1,\infty )$,

$$\begin{aligned} E \left[ \sup _{t\in [t_{k-1},\tau _k)} \left| X^\varepsilon _t - X^\varepsilon _{t_{k-1}} \right| ^p \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le C \left( \frac{1}{n^p} + \frac{\varepsilon ^p}{n^{p/2}} \right) \left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) , \end{aligned}$$

where C depends only on p, a and b, and

$$\begin{aligned} E \left[ \sup _{t\in [\eta _k,t_k]} \left| X^\varepsilon _t - X^\varepsilon _{t_k} \right| ^p \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le C \left( \frac{1}{n^p} + \frac{\varepsilon ^p}{n^{p/2}} \right) \left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) , \end{aligned}$$

where C depends only on p, a, b, c and $f_{\alpha _0}$.

Proof

For $t\in [t_{k-1},\tau _{k})$ and $p\ge 2$,

$$\begin{aligned}&\left( E \left[ \sup _{s\in [t_{k-1},t)} \left| X^\varepsilon _s - X^\varepsilon _{t_{k-1}} \right| ^p \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right) ^{1/p} \\&\quad \le C \int _{t_{k-1}}^t \left( E \left[ \left| X^\varepsilon _s - X^\varepsilon _{t_{k-1}} \right| ^p \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right) ^{1/p} \textrm{d}{s} + \frac{1}{n} \left| a(X^\varepsilon _{t_{k-1}},\mu _0) \right| \\&\qquad + C \varepsilon \left( \int _{t_{k-1}}^t \left( E \left[ \left| X^\varepsilon _s - X^\varepsilon _{t_{k-1}} \right| ^p \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right) ^{2/p} \textrm{d}{s} \right) ^{1/2} + \frac{\varepsilon }{\sqrt{n}} \left| b(X^\varepsilon _{t_{k-1}},\sigma _0) \right| , \end{aligned}$$

where C depnds only on p, a and b. By using Gronwall’s inequality, we obtain

$$\begin{aligned} \left( E \left[ \sup _{s\in [t_{k-1},t)} \left| X^\varepsilon _s - X^\varepsilon _{t_{k-1}} \right| ^p \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right) ^{2/p} \le C e^{C(1/n+\varepsilon ^2)t} \left( \frac{1}{n^2} + \frac{\varepsilon ^2}{n} \right) (1+|X^\varepsilon _{t_{k-1}}|^2), \end{aligned}$$

where C depnds only on p, a and b. Similarly,

$$\begin{aligned} \left( E \left[ \sup _{s\in [\eta _k,t_k]} \left| X^\varepsilon _s - X^\varepsilon _{t_k} \right| ^p \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right) ^{2/p} \le C \left( \frac{1}{n^2} + \frac{\varepsilon ^2}{n} \right) \left( 1 + E \left[ |X^\varepsilon _{t_k}|^2 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right) , \end{aligned}$$

where C depnds only on p, a and b. From Lemma 4.1, we have

$$\begin{aligned} E \left[ \sup _{u,s\in [t_{k-1},t_k]} \left| X^\varepsilon _u - X^\varepsilon _s \right| ^p \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le C \left( \frac{1}{n^p} + \varepsilon ^p\frac{\lambda _\varepsilon }{n} \right) \left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) , \end{aligned}$$

where C depnds only on p, a, b, c and $f_{\alpha _0}$. We can easily extend this result to the case $p\in [1,2)$ by using Hölder inequality. $\square$

Lemma 4.6

Under Assumptions 2.1 and 2.3, suppose that $0<\varepsilon \le 1$, $\lambda _\varepsilon \ge 1$, $\varepsilon \lambda _\varepsilon \le 1$. Let

$$\begin{aligned} Y^\varepsilon _k := \sup _{t\in [t_{k-1},\tau _k)} \frac{| X^\varepsilon _t - X^\varepsilon _{t_{k-1}} |}{\varepsilon } + \sup _{t\in [\eta _k,t_k]} \frac{| X^\varepsilon _t - X^\varepsilon _{t_k}|}{\varepsilon }. \end{aligned}$$

Then, for any $p\in (2,\infty )$,

$$\begin{aligned} \sup _{k=1,\dots ,n} Y^\varepsilon _k = O_p \left( \frac{1}{\varepsilon n^{1-1/p}} + \frac{1}{n^{1/2-1/p}} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

Proof

By using Lemmas 4.4 and 4.5, we have

$$\begin{aligned} \sum _{k=1}^n E \left[ \left| Y_k^{\varepsilon } \right| ^p \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le C \left( \frac{n}{(\varepsilon n)^p} + \frac{n}{n^{p/2}} \right) \frac{1}{n} \sum _{k=1}^n \left( 1 + \left| X^\varepsilon _{t_{k-1}} \right| ^p \right) = O_p \left( \frac{n}{(\varepsilon n)^p} + \frac{n}{n^{p/2}} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. It follows from Lemma A.3 that

$$\begin{aligned} \sup _{k=1,\dots ,n} |Y^\varepsilon _k| \le \left( \sum _{k=1}^n \left| Y_k^{\varepsilon } \right| ^p \right) ^{1/p} = O_p \left( \frac{1}{\varepsilon n^{1-1/p}} + \frac{1}{n^{1/2-1/p}} \right) . \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. $\square$

4.2 Limit theorems

We make a version of Lemma 2.2 in Shimizu (2017) in the sequel.

Lemma 4.7

Under Assumptions 2.1, 2.3, 2.6 and 2.8 with Notations 2.3 to 2.5 and 2.7, suppose that $0<\varepsilon \le 1$, $\lambda _\varepsilon \ge 1$ and $\varepsilon \lambda _\varepsilon \le 1$. Then, for $p\ge 2$ and $\rho \in (0,1/2)$

$$\begin{aligned}&P \left[ C^{n,\varepsilon ,\rho }_{k,0} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \ge e^{-\lambda _\varepsilon /n} \left\{ 1 - C \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} \right) \left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) \right\} , \\&P \left[ D^{n,\varepsilon ,\rho }_{k,0} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le C \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} \right) \left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) , \\&P \left[ C^{n,\varepsilon ,\rho }_{k,1} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad \le \frac{\lambda _\varepsilon }{n} \left\{ C \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} + \frac{\varepsilon ^p\lambda _\varepsilon }{n} \right) \left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) + \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \right\} , \\&P \left[ D^{n,\varepsilon ,\rho }_{k,1} \,\Big | \, \mathcal {F}_{t_{k-1}} \right] \le \frac{\lambda _\varepsilon }{n} \left\{ C \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} + \frac{\varepsilon ^p\lambda _\varepsilon }{n} \right) \left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) + 1 \right\} , \\&P \left[ C^{n,\varepsilon ,\rho }_{k,2} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le \frac{\lambda _\varepsilon ^2}{n^2}, \qquad P \left[ D^{n,\varepsilon ,\rho }_{k,2} \,\Big | \, \mathcal {F}_{t_{k-1}} \right] \le \frac{\lambda _\varepsilon ^2}{n^2}, \end{aligned}$$

where $c_1:=\inf _{t\in [0,1]} |c(x_t,\alpha _0)|>0$, $c_2:=\sup _{t\in [0,1]} |c(x,\alpha _0)|$, and C depends only on $p,a,b,c,f_{\alpha _0}$ and $v_1$.

Proof

We only give a proof for the case (i) in Assumption 2.4, because the same argument still works under the case (ii) in Assumption 2.4-. Same as in the proof of Lemma 2.2 in Shimizu and Yoshida (2006), Section 4.2, it follows that

$$\begin{aligned} P \left[ C^{n,\varepsilon ,\rho }_{k,2} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] ,~ P \left[ D^{n,\varepsilon ,\rho }_{k,2} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le \frac{\lambda _\varepsilon ^2}{n^2}. \end{aligned}$$

Also, it follows from

$$\begin{aligned}&P \left[ \left| X^\varepsilon _{t_k} - X^\varepsilon _{\tau _k} + \Delta X^\varepsilon _{\tau _k} + X^\varepsilon _{\tau _k-} - X^\varepsilon _{t_{k-1}} \right| \le \frac{v_{nk}}{n^\rho } \, \Big | \, \mathcal {F}_{t_{k-1}},~\Delta ^n_k N^{\lambda _\varepsilon }=1 \right] \\&\quad \le P \left[ \left| X^\varepsilon _{t_k} - X^\varepsilon _{\tau _k} \right| + \sup _{t\in [t_{k-1},\tau _k)} \left| X^\varepsilon _t - X^\varepsilon _{t_{k-1}} \right|> \frac{v_{nk}}{n^\rho } \, \Big | \, \mathcal {F}_{t_{k-1}},~\Delta ^n_k N^{\lambda _\varepsilon }=1 \right] \\&\qquad + P \left[ \left| \Delta Z^{\lambda _\varepsilon }_{\tau _k} \right| \le \frac{4v_{nk}}{c_1n^\rho } ~\text {or}~ \sup _{t\in [t_k,t_{k-1}]} \left| c(X^\varepsilon _t,\alpha _0) - c(x_t, \alpha _0) \right|> \frac{c_1}{2} \, \Big | \, \mathcal {F}_{t_{k-1}},~\Delta ^n_k N^{\lambda _\varepsilon }=1 \right] , \\&P \left[ \left| X^\varepsilon _{t_k} - X^\varepsilon _{\tau _k} + \Delta X^\varepsilon _{\tau _k} + X^\varepsilon _{\tau _k-} - X^\varepsilon _{t_{k-1}} \right|> \frac{v_{nk}}{n^\rho }\, \Big | \, \mathcal {F}_{t_{k-1}},~\Delta ^n_k N^{\lambda _\varepsilon }=1 \right] \\&\quad \le P \left[ \left| X^\varepsilon _{t_k} - X^\varepsilon _{\tau _k} \right| + \sup _{t\in [t_{k-1},\tau _k)} \left| X^\varepsilon _t - X^\varepsilon _{t_{k-1}} \right|> \frac{v_{nk}}{2n^\rho } \, \Big | \, \mathcal {F}_{t_{k-1}},~\Delta ^n_k N^{\lambda _\varepsilon }=1 \right] \\&\qquad + P \left[ \left| \Delta Z^{\lambda _\varepsilon }_{\tau _k} \right|> \frac{v_{nk}}{4c_2n^\rho } ~\text {or}~ \sup _{t\in [t_k,t_{k-1}]} \left| c(X^\varepsilon _t,\alpha _0) - c(x_t, \alpha _0) \right| > c_2 \, \Big | \, \mathcal {F}_{t_{k-1}},~\Delta ^n_k N^{\lambda _\varepsilon }=1 \right] \end{aligned}$$

and Lemmas 4.2, 4.5 and A.2 that

$$\begin{aligned}{} & {} P \left[ C^{n,\varepsilon ,\rho }_{k,1} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le \frac{\lambda _\varepsilon }{n} e^{-\lambda _\varepsilon /n} \left\{ C \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} \right) \left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) \right. \\{} & {} \qquad \left. + \int _{|z|\le 4v_{nk}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} + C \frac{\varepsilon ^p \lambda _\varepsilon }{n} \right\} , \\{} & {} P \left[ D^{n,\varepsilon ,\rho }_{k,1} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le \frac{\lambda _\varepsilon }{n} e^{-\lambda _\varepsilon /n} \left\{ C \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} \right) \left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) \right. \\{} & {} \qquad \left. + \int _{|z|>v_{nk}/4c_2n^\rho } f_{\alpha _0}(z) \textrm{d}{z} + C \frac{\varepsilon ^p \lambda _\varepsilon }{n} \right\} , \end{aligned}$$

where C depends only on $p,a,b,c,f_{\alpha _0}$ and $v_1$. The other inequalities follow from Lemma 4.5. $\square$

In the proof of Proposition 3.3 (ii) in Shimizu (2017), the intensity of the Poisson process driving on the background is constant, although we assume the intensity $\lambda _\varepsilon$ goes to infinity. So, we prepare the following lemma.

Lemma 4.8

Under Assumptions 2.1, 2.3, 2.4, 2.6 and 2.8, for $\rho \in (0,1/2)$

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n 1_{D^{n,\varepsilon ,\rho }_{k}} \overset{p}{\longrightarrow }1, \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon /n \rightarrow 0$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. More precisely, for $\rho \in (0,1/2)$ and $p\in [2/(1-2\rho ),\infty )$

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n 1_{D^{n,\varepsilon ,\rho }_{k,0}}&= O_p \left( \frac{1}{\lambda _\varepsilon n^{p(1-\rho )-1}} + \frac{\varepsilon ^p}{\lambda _\varepsilon n^{p(1/2-\rho )-1}} \right) , \\ \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n 1_{D^{n,\varepsilon ,\rho }_{k,1}}&= 1 + O_p \left( \frac{\lambda _\varepsilon }{n} + \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} + \int _{|z|\le 4v_{nk}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \right) , \\ \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n 1_{D^{n,\varepsilon ,\rho }_{k,2}}&= O_p \left( \frac{\lambda _\varepsilon }{n} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

Proof

Since

$$\begin{aligned} \left| \frac{\lambda _\varepsilon }{n} - P \left[ D^{n,\varepsilon ,\rho }_{k,1} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right|&\le \frac{\lambda _\varepsilon }{n} - \frac{\lambda _\varepsilon }{n} e^{-\lambda _\varepsilon /n} + \left| \frac{\lambda _\varepsilon }{n} e^{-\lambda _\varepsilon /n} - P \left[ D^{n,\varepsilon ,\rho }_{k,1} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right| \\&\le \left( \frac{\lambda _\varepsilon }{n} \right) ^2 + P \left[ C^{n,\varepsilon ,\rho }_{k,1} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] , \end{aligned}$$

it follows from Lemmas 4.4 and 4.7 that for $p\ge 2$ and $\rho \in (0,1/2)$

$$\begin{aligned}{} & {} \sum _{k=1}^n E \left[ \left| \frac{1}{\lambda _\varepsilon } 1_{D^{n,\varepsilon ,\rho }_{k,1}} - \frac{1}{n} \right| \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le \frac{\lambda _\varepsilon }{n} + \frac{1}{\lambda _\varepsilon }\sum _{k=1}^n P \left[ C^{n,\varepsilon ,\rho }_{k,1} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\{} & {} \quad \le \frac{\lambda _\varepsilon }{n} + C \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} + \frac{\varepsilon ^p\lambda _\varepsilon }{n} \right) \frac{1}{n} \sum _{k=1}^n\left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) \\{} & {} \qquad +\int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \overset{p}{\longrightarrow }0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon /n \rightarrow 0$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. Similarly, we obtain

$$\begin{aligned} \sum _{k=1}^n E \left[ \frac{1}{\lambda _\varepsilon } 1_{D^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right]&\le C \frac{1}{\lambda _\varepsilon } \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} \right) \sum _{k=1}^n \left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) , \\ \sum _{k=1}^n E \left[ \frac{1}{\lambda _\varepsilon } 1_{D^{n,\varepsilon ,\rho }_{k,2}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right]&\le \frac{\lambda _\varepsilon }{n}. \end{aligned}$$

Hence, the conclusion follows from Lemma A.3. $\square$

Lemma 4.9

Under Assumptions 2.1, 2.3, 2.4, 2.6 and 2.8, for $\rho \in (0,1/2)$

$$\begin{aligned} \frac{1}{n} \sum _{k=1}^n 1_{C^{n,\varepsilon ,\rho }_{k}} \overset{p}{\longrightarrow }1, \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon /n \rightarrow 0$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. More precisely, for $\rho \in (0,1/2)$ and $p\in [2,\infty )$

$$\begin{aligned} \frac{1}{n} \sum _{k=1}^n 1_{C^{n,\varepsilon ,\rho }_{k,0}}&= 1 + O_p \left( \frac{\lambda _\varepsilon }{n} \right) , \\ \frac{1}{n} \sum _{k=1}^n 1_{C^{n,\varepsilon ,\rho }_{k,1}}&= O_p \left( \frac{\lambda _\varepsilon }{n^{p(1-\rho )+1}} + \frac{\varepsilon ^p \lambda _\varepsilon }{n^{p(1/2-\rho )+1}} + \frac{\lambda _\varepsilon }{n} \int _{|z|\le 4v_{nk}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \right) , \\ \frac{1}{n} \sum _{k=1}^n 1_{C^{n,\varepsilon ,\rho }_{k,2}}&= O_p \left( \frac{\lambda _\varepsilon ^2}{n^2} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon /n \rightarrow 0$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

Proof

From Lemma 4.8 we have

$$\begin{aligned} \frac{1}{n} \sum _{k=1}^n 1_{C^{n,\varepsilon ,\rho }_{k}} - 1 = \frac{\lambda _\varepsilon }{n} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n 1_{D^{n,\varepsilon ,\rho }_{k}} = O_p \left( \frac{\lambda _\varepsilon }{n} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon /n \rightarrow 0$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. It follows from Lemmas 4.4 and 4.7 that for any $p\in [0,\infty )$

$$\begin{aligned}&\sum _{k=1}^n E \left[ \left| \frac{1}{n} 1_{C^{n,\varepsilon ,\rho }_{k,1}} \right| \, \Big | \, \mathcal {F}_{t_{k-1}} \right] = \frac{1}{n} \sum _{k=1}^n P \left[ C^{n,\varepsilon ,\rho }_{k,1} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad \le C \frac{\lambda _\varepsilon }{n} \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} + \frac{\varepsilon ^p\lambda _\varepsilon }{n} \right) \\&\qquad \times \frac{1}{n} \sum _{k=1}^n \left( 1+|X^\varepsilon _{t_{k-1}}|^p \right) + \frac{\lambda _\varepsilon }{n} \int _{|z|\le 4v_{nk}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z}, \\&\sum _{k=1}^n E \left[ \left| \frac{1}{n} 1_{C^{n,\varepsilon ,\rho }_{k,2}} \right| \, \Big | \, \mathcal {F}_{t_{k-1}} \right] = \frac{1}{n} \sum _{k=1}^n P \left[ C^{n,\varepsilon ,\rho }_{k,2} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le \frac{\lambda _\varepsilon ^2}{n^2}. \end{aligned}$$

The conclusion follows from Lemma A.3. $\square$

Remark 4.1

From this lemma, under Assumptions 2.1, 2.3, 2.4, 2.6 and 2.8, for $\rho \in (0,1/2)$ and for any random variables $\xi _{k,\theta }^{n,\varepsilon }$ $(k=1,\dots ,n,~n\in \mathbb {N},~\varepsilon >0,~\theta \in \bar{\Theta })$, when

$$\begin{aligned} \lambda _\varepsilon \rightarrow \infty , \quad \varepsilon \lambda _\varepsilon \rightarrow 0, \quad \frac{\lambda _\varepsilon ^2}{n} \rightarrow 0, \quad \lambda _\varepsilon \int _{|z|\le 4v_{2}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0 \end{aligned}$$

as $\varepsilon \rightarrow 0$,

$$\begin{aligned} \sum _{k=1}^n \xi _{k,\theta }^{n,\varepsilon } \left\{ 1_{C^{n,\varepsilon ,\rho }_{k}} - 1_{C^{n,\varepsilon ,\rho }_{k,0}} \right\} = o_p(1) \end{aligned}$$

as $n\rightarrow \infty$ and $\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$, since for any $\eta >0$

$$\begin{aligned} P \left( \sup _{\theta \in \Theta } \left| \sum _{k=1}^n \xi _{k,\theta }^{n,\varepsilon } 1_{C^{n,\varepsilon ,\rho }_{k,j}} \right|> \eta \right) \le P \left( \left| \sum _{k=1}^n 1_{C^{n,\varepsilon ,\rho }_{k,j}} \right| > 1/2 \right) \quad \text {for } j=1,2. \end{aligned}$$

Similarly, from Lemma 4.8, when

$$\begin{aligned} \lambda _\varepsilon \rightarrow \infty , \quad \varepsilon \lambda _\varepsilon \rightarrow 0, \quad \frac{\lambda _\varepsilon ^2}{n} \rightarrow 0 \end{aligned}$$

as $\varepsilon \rightarrow 0$,

$$\begin{aligned} \sum _{k=1}^n \xi _{k,\theta }^{n,\varepsilon } \left\{ 1_{D^{n,\varepsilon ,\rho }_{k}} - 1_{D^{n,\varepsilon ,\rho }_{k,1}} \right\} = o_p(1) \end{aligned}$$

as $n\rightarrow \infty$ and $\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$,

Lemma 4.10

Under Assumptions 2.1, 2.3, 2.4, 2.6 and 2.8, let $\rho \in (0,1/2)$, $\delta >0$ and $\tilde{D}^{n,\varepsilon ,\rho }_{k,1}$ be an event defined by

$$\begin{aligned} \tilde{D}^{n,\varepsilon ,\rho }_{k,1} := D^{n,\varepsilon ,\rho }_{k,1} \cap \{ X^\varepsilon _t \in I_{x_0}^\delta \text { for all } t\in [0,1] \}, \end{aligned}$$

and let $\xi _{k,\theta }^{n,\varepsilon }$ $(k=1,\dots ,n,~n\in \mathbb {N},~\varepsilon >0,~\theta \in \bar{\Theta })$ be random variables. If

$$\begin{aligned} \lambda _\varepsilon \rightarrow \infty , \quad \varepsilon \lambda _\varepsilon \rightarrow 0, \quad \frac{\lambda _\varepsilon ^2}{n} \rightarrow 0, \quad \lambda _\varepsilon \int _{|z|\le 4v_{2}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0 \end{aligned}$$

(4.6)

as $\varepsilon \rightarrow 0$, then

$$\begin{aligned} \sum _{k=1}^n \xi _{k,\theta }^{n,\varepsilon } \left\{ 1_{D^{n,\varepsilon ,\rho }_{k}} - 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} \right\} = o_p(1), \quad \sum _{k=1}^n \xi _{k,\theta }^{n,\varepsilon } \left\{ 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} - 1_{J^{n,\varepsilon }_{k,1}} \right\} = o_p(1) \end{aligned}$$

as $n\rightarrow \infty$ and $\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$.

Proof

Since from Lemma 4.3

$$\begin{aligned} P \left( X^\varepsilon _t \in I_{x_0}^\delta \text { for all } t\in [0,1] \right) \ge P \left( \sup _{t\in [0,1]} | X^\varepsilon _t - x_t | \le \delta \right) \rightarrow 1 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, for any $\eta >0$

$$\begin{aligned}&P \left( \left| \sum _{k=1}^n \xi _{k,\theta }^{n,\varepsilon } \left\{ 1_{D^{n,\varepsilon ,\rho }_{k}} - 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} \right\} \right|> \eta \right) \\&\quad \le P \left( \left| \sum _{k=1}^n \xi _{k,\theta }^{n,\varepsilon } \left\{ 1_{D^{n,\varepsilon ,\rho }_{k}} - 1_{D^{n,\varepsilon ,\rho }_{k,1}} \right\} \right|> \eta /2 \right) + P \left( \{ X^\varepsilon _t \not \in I_{x_0}^\delta , ~ \exists \, t\in [0,1] \} \right) , \\&P \left( \left| \sum _{k=1}^n \xi _{k,\theta }^{n,\varepsilon } \left\{ 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} - 1_{J^{n,\varepsilon }_{k,1}} \right\} \right|> \eta \right) \\&\quad \le P \left( \{ X^\varepsilon _t \not \in I_{x_0}^\delta , ~ \exists \, t\in [0,1] \} \right) + P \left( \left| \sum _{k=1}^n \xi _{k,\theta }^{n,\varepsilon } 1_{C^{n,\varepsilon ,\rho }_{k,1}} \right| > \eta /2 \right) . \end{aligned}$$

Take sufficiently large $p\in [2/(1-2\rho ),\infty )$. Thus, we obtain from Remark 4.1 the conclusion. $\square$

Remark 4.2

In this lemma, if $\left\{ {\xi _{k,\theta }^{n,\varepsilon }}\right\} _{n,\varepsilon ,k,\theta }$ is bounded in probability, we can replace the condition (4.6) with a milder condition

$$\begin{aligned} \lambda _\varepsilon \rightarrow \infty , \quad \varepsilon \lambda _\varepsilon \rightarrow 0, \quad \frac{\lambda _\varepsilon }{n} \rightarrow 0. \end{aligned}$$

But, we will never use this fact in this paper.

Lemma 4.11

Under Assumptions 2.1, 2.3, 2.4, 2.6 and 2.8, let $\rho \in (0,1/2)$, and suppose that a family $\left\{ {g(\cdot ,\theta )}\right\} _{\theta \in \Theta }$ of functions from $\mathbb {R}$ to $\mathbb {R}$ is equicontinuous at every points in $I_{x_0}$. Then,

$$\begin{aligned} \frac{1}{n} \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) 1_{C^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }\int _0^1 g(x_t, \theta ) \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon /n\rightarrow 0$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$. Also, for $p\in [2,\infty )$

$$\begin{aligned}&\frac{1}{n} \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \theta \right) 1_{C^{n,\varepsilon ,\rho }_{k,0}} \overset{p}{\longrightarrow }\int _0^1 g(x_t,\theta ) \textrm{d}{t}, \\&\frac{1}{n} \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \theta \right) 1_{C^{n,\varepsilon ,\rho }_{k,1}} = O_p \left( \frac{\lambda _\varepsilon }{n^{p(1-\rho )+1}} + \frac{\varepsilon ^p \lambda _\varepsilon }{n^{p(1/2-\rho )+1}} + \frac{\lambda _\varepsilon }{n} \int _{|z|\le 4v_{nk}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \right) , \\&\frac{1}{n} \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \theta \right) 1_{C^{n,\varepsilon ,\rho }_{k,2}} = O_p \left( \frac{\lambda _\varepsilon ^2}{n^2} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon /n\rightarrow 0$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$.

Proof of Lemma 4.11

Since $\left\{ {g(\cdot ,\theta )}\right\} _{\theta \in \Theta }$ is equicontinuous at every points in $I_{x_0}$, there exists $\delta >0$ such that

$$\begin{aligned} \sup _{(x,\theta )\in I_{x_0}^\delta \times \Theta } |g(x,\theta )| < \infty . \end{aligned}$$

For any $\eta >0$

$$\begin{aligned}&P \left( \left| \frac{1}{n} \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \theta \right) 1_{D^{n,\varepsilon ,\rho }_{k}} \right|> \eta \right) \\&\quad \le P \left( \sup _{k=0,\dots ,n-1} |X^\varepsilon _{t_k} - x_{t_k}| \ge \delta \right) + P \left( \sup _{(x,\theta )\in I_{x_0}^\delta \times \Theta } |g(x,\theta )| \frac{\lambda _\varepsilon }{n} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n 1_{D^{n,\varepsilon ,\rho }_k}> \eta \right) , \\&P \left( \left| \frac{1}{n} \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \theta \right) 1_{C^{n,\varepsilon ,\rho }_{k,j}} \right|> \eta \right) \\&\quad \le P \left( \sup _{k=0,\dots ,n-1} |X^\varepsilon _{t_k} - x_{t_k}| \ge \delta \right) \\&\qquad + P \left( \sup _{(x,\theta )\in I_{x_0}^\delta \times \Theta } |g(x,\theta )| \frac{1}{n} \sum _{k=1}^n 1_{C^{n,\varepsilon ,\rho }_{k,j}} > \eta \right) \quad \text {for } j=1,2. \end{aligned}$$

It follows from Lemmas 4.3, 4.4 and 4.9 that

$$\begin{aligned}&\left| { \frac{1}{n} \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \theta \right) 1_{C^{n,\varepsilon ,\rho }_k} -\int _0^1 g(x_t,\theta ) \textrm{d}{t} }\right| \\&\quad \le \left| { \frac{1}{n} \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \theta \right) 1_{D^{n,\varepsilon ,\rho }_k} }\right| + \left| { \frac{1}{n} \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \theta \right) - \int _0^1 g(x_t,\theta ) \textrm{d}{t} }\right| \overset{p}{\longrightarrow }0, \\&\frac{1}{n} \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \theta \right) 1_{C^{n,\varepsilon ,\rho }_{k,1}} = O_p \left( \frac{\lambda _\varepsilon }{n^{p(1-\rho )+1}} + \frac{\varepsilon ^p \lambda _\varepsilon }{n^{p(1/2-\rho )+1}} + \frac{\lambda _\varepsilon }{n} \int _{|z|\le 4v_{nk}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \right) , \\&\frac{1}{n} \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \theta \right) 1_{C^{n,\varepsilon ,\rho }_{k,2}} = O_p \left( \frac{\lambda _\varepsilon ^2}{n^2} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon /n\rightarrow 0$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$. $\square$

Lemma 4.12

Under Assumptions 2.1, 2.3, 2.4, 2.6 and 2.8, let $\rho \in (0,1/2)$. We assume either of the following conditions (i) or (ii):

• (i) Under Assumption 2.4 (i), we assume the following four conditions:

  • (i.a) There exists $\delta >0$ such that for every $(x,\theta )\in I_{x_0}^\delta \times \bar{\Theta }$, $g(x,y,\theta )$ is continuously differentiable with respect to $y \in \mathbb {R}$.

  • (i.b) There exist constants $C>0$, $q\ge 1$ and $\delta >0$ such that

    $$\begin{aligned} \sup _{(x,\theta ) \in I_{x_0}^\delta \times \bar{\Theta }} \left| { \frac{\partial g}{\partial y} \left( x,y,\theta \right) }\right| \le C (1+|y|^q) \quad (y \in \mathbb {R}). \end{aligned}$$

  • (i.c) There exists a sufficiently large $p\ge 2$ such that

    $$\begin{aligned} \lambda _\varepsilon \rightarrow \infty , \quad \frac{\lambda _\varepsilon ^2}{n} \rightarrow 0, \quad \varepsilon \lambda _\varepsilon \rightarrow 0, \quad \varepsilon n^{1-1/p}\rightarrow \infty , \quad \lambda _\varepsilon \int _{|z|\le 4v_{2}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0 \end{aligned}$$

    as $n\rightarrow \infty$ and $\varepsilon \rightarrow 0$.

  • (i.d) Let p be taken as in the condition (i.c). Put $r_{n,\varepsilon }$ by

    $$\begin{aligned} r_{n,\varepsilon } := \frac{1}{\varepsilon n^{1-1/p}} + \frac{1}{n^{1/2-1/p}}. \end{aligned}$$

• (ii) Under Assumption 2.4 (ii), we assume the following six conditions:

  • (ii.a) There exists $\delta >0$ such that for every $(x,\theta )\in I_{x_0}^\delta \times \bar{\Theta }$, $g(x,y,\theta )$ is continuously differentiable with respect to $y \in (0,\infty )$.

  • (ii.b) There exists $\delta >0$ and $L>0$ such that if $0<y_1\le y \le y_2$, then

    $$\begin{aligned} \left| \frac{\partial g}{\partial y} \left( x,y,\theta \right) \right| \le \left| { \frac{g}{y} \left( x,y_1,\theta \right) }\right| + \left| { \frac{\partial g}{\partial y} \left( x,y_2,\theta \right) }\right| + L \quad \text {for all } (x,\theta )\in I_{x_0}^\delta \times \bar{\Theta }. \end{aligned}$$

  • (ii.c) There exist $q\ge 0$ and $\delta >0$ such that

    $$\begin{aligned} \sup _{(x,\theta ) \in I_{x_0}^\delta \times \bar{\Theta }} \left| \frac{\partial g}{\partial y} \left( x,y,\theta \right) \right| \le O \left( \frac{1}{|y|^q} \right) \quad \text {as } |y| \rightarrow 0. \end{aligned}$$

  • (ii.d) There exists $\delta >0$ such that for any $C_1>0$ and $C_2\ge 0$ the map

    $$\begin{aligned} x \mapsto \int \sup _\theta \left| { \frac{\partial g}{\partial y} \left( x, C_1 y + C_2,\theta \right) }\right| f_{\alpha _0}(y) \textrm{d}{y} \end{aligned}$$

    takes values in $\mathbb {R}$ from $I_{x_0}^\delta$, and is continous on $I_{x_0}^\delta$.

  • (ii.e) Let q be the constant in the condition (ii.c), and let $\rho <1/4q$. For any large $p\ge 2/(1-2q\rho )$,

    $$\begin{aligned}{} & {} \lambda _\varepsilon \rightarrow \infty , \quad \frac{\lambda _\varepsilon ^2}{n} \rightarrow 0, \quad \varepsilon \lambda _\varepsilon \rightarrow 0, \quad \varepsilon n^{1-q\rho -1/p}\rightarrow \infty , \\{} & {} \lambda _\varepsilon \int _{|z|\le 4v_{2}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0 \end{aligned}$$

    as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$.

  • (ii.f) Let p and q be the constants in the condition (ii.e). Put $r_{n,\varepsilon }$ by

    $$\begin{aligned} r_{n,\varepsilon } := \frac{1}{\varepsilon n^{1-1/p-q\rho }} + \frac{1}{n^{1/2-1/p-q\rho }}. \end{aligned}$$

Then,

$$\begin{aligned}{} & {} \left| \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, \frac{\Delta ^n_k X^{\varepsilon }}{\varepsilon }, \theta \right) 1_{D^{n,\varepsilon ,\rho }_{k}} \right. \\{} & {} \qquad \left. - \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) 1_{J^{n,\varepsilon }_{k,1}} \right| = O_p(r_{n,\varepsilon }) \end{aligned}$$

as $n\rightarrow \infty$ and $\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$.

Remark 4.3

Assumption 2.4 is used only for defining $D^{n,\varepsilon ,\rho }_{k}$ in Lemmas 4.7, 4.8, 4.10 and 4.11, while it is essentially used in Lemma 4.12.

Remark 4.4

The assumptions (i.c) and (ii.e) in Lemma 4.12 are ensured if

$$\begin{aligned} \lambda _\varepsilon \rightarrow \infty , \quad \varepsilon \lambda _\varepsilon \rightarrow 0, \quad (\varepsilon \sqrt{n})^{-1} <\infty \quad \text {and} \quad \lambda _\varepsilon \int _{|z|\le 4v_{2}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0 \end{aligned}$$

as $n\rightarrow \infty$ and $\varepsilon \rightarrow 0$. This condition seems to be natural when we consider the asymptotic normality for our estimator (see, e.g., the condition (B2) in Sørensen and Uchida (2003)).

Proof of Lemma 4.12

Let $\delta >0$ be a sufficiently small number satisfying the conditions of the statement and

$$\begin{aligned} \frac{c_1}{2} \le c(x,\alpha _0) \le 2 c_2 \quad \text {for } x\in I_{x_0}^\delta , \end{aligned}$$

where $c_1$ and $c_2$ are the constants from Assumption 2.6. In this proof, we may simply write the maps

$$\begin{aligned} (y,\theta ) \mapsto g(X^\varepsilon _{t_{k-1}},y,\theta ) =: g_k(y,\theta ) \quad \text {and} \quad (y,\theta ) \mapsto \frac{\partial g}{\partial y} \left( x,y,\theta \right) \Big |_{x=X^\varepsilon _{t_{k-1}}} =: \frac{\partial g_k}{\partial y} \left( y, \theta \right) , \end{aligned}$$

and we denote the following event by $\tilde{D}^{n,\varepsilon ,\rho }_{k,1}$

$$\begin{aligned} \tilde{D}^{n,\varepsilon ,\rho }_{k,1} := D^{n,\varepsilon ,\rho }_{k,1} \cap \left\{ X^\varepsilon _t \in I_{x_0}^\delta \text { for all } t\in [0,1] \right\} . \end{aligned}$$

Since

$$\begin{aligned} \frac{\lambda _\varepsilon ^2}{n} \rightarrow 0, \quad \lambda _\varepsilon \int _{|z|\le 4v_{2}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0, \end{aligned}$$

under either of the assumptions (i.c) or (ii.e), we obtain from Lemma 4.10 that for any non-random $r_{n,\varepsilon }'>0$ ($n\in \mathbb {N}, \varepsilon >0$),

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n g_k \left( \frac{\Delta ^n_k X^{\varepsilon }}{\varepsilon }, \theta \right) \left\{ 1_{D^{n,\varepsilon ,\rho }_{k}} - 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} \right\}&= o_p(r_{n,\varepsilon }'), \\ \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n g_k \left( c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) \left\{ 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} - 1_{J^{n,\varepsilon }_{k,1}} \right\}&= o_p(r_{n,\varepsilon }') \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$ $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$. Thus, it is sufficient to show that

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n \left\{ g_k \left( \frac{\Delta ^n_k X^{\varepsilon }}{\varepsilon }, \theta \right) - g_k \left( \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon }, \theta \right) \right\} 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}}&= O_p(r_{n,\varepsilon }), \end{aligned}$$

(4.7)

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n \left\{ g_k \left( \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon }, \theta \right) - g_k \left( c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) \right\} 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}}&= O_p \left( \frac{1}{\varepsilon n^{1-1/p}} + \frac{1}{n^{1/2-1/p}} \right) \end{aligned}$$

(4.8)

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$.

Put

$$\begin{aligned} Y^\varepsilon _k := \frac{X^\varepsilon _{t_k} - X^\varepsilon _{\eta _k}}{\varepsilon } + \frac{X^\varepsilon _{\tau _k-} - X^\varepsilon _{t_{k-1}}}{\varepsilon } \left( = \frac{\Delta ^n_k X^{\varepsilon }}{\varepsilon } - \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon } \quad \text {on}~D^{n,\varepsilon ,\rho }_{k,1}\right) . \end{aligned}$$

By using Taylor’s theorem under either of the assumptions (i.a) or (ii.a), we have

$$\begin{aligned} g_k \left( \frac{\Delta ^n_k X^{\varepsilon }}{\varepsilon }, \theta \right) - g_k \left( \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon }, \theta \right) = \int _0^1 \frac{\partial g_k}{\partial y} \left( \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon } +\zeta Y^\varepsilon _k, \theta \right) Y^\varepsilon _k \textrm{d}{\zeta } \quad \text {on } \tilde{D}^{n,\varepsilon ,\rho }_{k,1}. \end{aligned}$$

Here, we remark that $\Delta ^n_k X^{\varepsilon }$ and $\Delta X^\varepsilon _{\tau _k}$ are almost surely positive on $\tilde{D}^{n,\varepsilon ,\rho }_{k,1}$ under Assumption 2.4 (ii). To see (4.7), it is sufficient to show that

$$\begin{aligned} \sup _{\theta \in \Theta } \left| \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n \int _0^1 \frac{\partial g_k}{\partial y} \left( \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon } +\zeta Y^\varepsilon _k, \theta \right) Y^\varepsilon _k \textrm{d}{\zeta } 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1} \cap \{|Y^\varepsilon _k|\le 1\}} \right| = O_p(r_{n,\varepsilon }) \end{aligned}$$

(4.9)

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. Indeed, for any $M>0$

$$\begin{aligned}&P \left( \sup _{\theta \in \Theta } \left| \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n \left\{ g_k \left( \frac{\Delta ^n_k X^{\varepsilon }}{\varepsilon }, \theta \right) - g_k \left( \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon }, \theta \right) \right\} 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} \right|> M r_{n,\varepsilon } \right) \\&\quad \le P \left( \sup _{k=1,\dots ,n}|Y^\varepsilon _k|> 1 \right) \\&\qquad + P \left( \sup _{\theta \in \Theta } \left| \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n \int _0^1 \frac{\partial g_k}{\partial y} \left( \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon } +\zeta Y^\varepsilon _k, \theta \right) Y^\varepsilon _k \textrm{d}{\zeta } \, 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1} \cap \{|Y^\varepsilon _k|\le 1\}} \right| > M r_{n,\varepsilon } \right) , \end{aligned}$$

and from Lemma 4.6 the first term converges to zero as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, since from either of the assumptions (i.c) or (ii.e) we have $\varepsilon n^{1-1/p}\rightarrow \infty$ or $\varepsilon n^{1-q\rho -1/p}\rightarrow \infty$, respectively.

We first consider the case (ii) in Assumption 2.4. Since for $\zeta \in [0,1]$ we have

$$\begin{aligned} \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon } + \zeta Y^\varepsilon _k \ge (1-\zeta ) \, c(X^\varepsilon _{\tau _k-},\alpha _0) \, V_{N^{\lambda _\varepsilon }_{\tau _k}} + \zeta \frac{v_{nk}}{n^\rho } \ge \min \left\{ \frac{c_1}{2} V_{N^{\lambda _\varepsilon }_{\tau _k}}, \frac{v_1}{n^\rho } \right\} \quad \text {on } \tilde{D}^{n,\varepsilon ,\rho }_{k,1}, \end{aligned}$$

we obtain from the assumption (ii.b) that

$$\begin{aligned}&\int _0^1 \left| { \frac{\partial g_k}{\partial y} \left( \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon } +\zeta Y^\varepsilon _k, \theta \right) }\right| \textrm{d}{\zeta } 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1} \cap \{|Y^\varepsilon _k|\le 1\}} \\&\quad \le \left\{ { \left| { \frac{\partial g_k}{\partial y} \left( \frac{c_1}{2} V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) }\right| + \left| { \frac{\partial g_k}{\partial y} \left( \frac{v_1}{n^\rho }, \theta \right) }\right| + \left| { \frac{\partial g_k}{\partial y} \left( 2c_2 V_{N^{\lambda _\varepsilon }_{\tau _k}} + 1, \theta \right) }\right| + L }\right\} 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1} \cap \{|Y^\varepsilon _k|\le 1\}} \\&\quad \le \left\{ \left| \frac{\partial g_k}{\partial y} \left( \frac{c_1}{2} V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) \right| + \left| \frac{\partial g_k}{\partial y} \left( \frac{v_1}{n^\rho }, \theta \right) \right| + \left| \frac{\partial g_k}{\partial y} \left( 2c_2 V_{N^{\lambda _\varepsilon }_{\tau _k}} + 1, \theta \right) \right| + L \right\} 1_{J^{n,\varepsilon }_{k,1}} \end{aligned}$$

Since

$$\begin{aligned}{} & {} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n E \left[ \sup _\theta \left| \frac{\partial g_k}{\partial y} \left( \frac{c_1}{2} V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) \right| 1_{J^{n,\varepsilon }_{k,1}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\{} & {} \quad \le \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n \int \sup _\theta \left| \frac{\partial g_k}{\partial y} \left( \frac{c_1}{2} z, \theta \right) \right| f_{\alpha _0}(z) \textrm{d}{z} \cdot P \left( J^{n,\varepsilon }_{k,1} \right) \\{} & {} \quad \le \frac{1}{n} \sum _{k=1}^n \int \sup _\theta \left| \frac{\partial g_k}{\partial y} \left( \frac{c_1}{2} z, \theta \right) \right| f_{\alpha _0}(z) \textrm{d}{z}, \end{aligned}$$

it follows from Lemma A.3 (ii), Lemmas 4.4 and 4.6 and the assumption (ii.d) that

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n \left| { \frac{\partial g_k}{\partial y} \left( \frac{c_1}{2} V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) }\right| 1_{J^{n,\varepsilon }_{k,1}} \sup _{k=1,\dots ,n} \left| Y^\varepsilon _k \right| = O_p \left( \frac{1}{\varepsilon n^{1-1/p}} + \frac{1}{n^{1/2-1/p}} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$, where p is given in the assumption (ii.e). Similarly, it follows from Lemma A.3 (ii), Lemmas 4.4 and 4.6 and the assumption (ii.d) that

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n \left| \frac{\partial g_k}{\partial y} \left( 2c_2 V_{N^{\lambda _\varepsilon }_{\tau _k}} + 1, \theta \right) \right| 1_{J^{n,\varepsilon }_{k,1}} \sup _{k=1,\dots ,n} \left| Y^\varepsilon _k \right| = O_p \left( \frac{1}{\varepsilon n^{1-1/p}} + \frac{1}{n^{1/2-1/p}} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$, and it follows from Lemma A.3 (ii), Lemma 4.6 and the assumption (ii.c) that

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n \left| \frac{\partial g_k}{\partial y} \left( \frac{v_1}{n^\rho }, \theta \right) \right| 1_{J^{n,\varepsilon }_{k,1}} \sup _{k=1,\dots ,n} \left| Y^\varepsilon _k \right| = O_p \left( \frac{1}{\varepsilon n^{1-1/p-q\rho }} + \frac{1}{n^{1/2-1/p-q\rho }} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$. Thus, we obtain (4.9).

Under the case (i) in Assumption 2.4, as in the same argument above, we have

$$\begin{aligned}&\frac{1}{\lambda _\varepsilon } \sum _{k=1}^n \sup _{\theta \in \bar{\Theta }} \left| \int _0^1 \frac{\partial g_k}{\partial y} \left( \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon } +\zeta Y^\varepsilon _k, \theta \right) Y^\varepsilon _k \textrm{d}{\zeta } \right| 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}\cap \{|Y^\varepsilon _k|\le 1\}} \\&\quad \le \frac{C}{\lambda _\varepsilon } \sum _{k=1}^n \left( 2 + \left| 2c_2 V_{N^{\lambda _\varepsilon }_{\tau _k}} \right| ^p \right) 1_{J^{n,\varepsilon }_{k,1}} \sup _{k=1,\dots ,n} \left| Y^\varepsilon _k \right| = O_p \left( \frac{1}{\varepsilon n^{1-1/p}} + \frac{1}{n^{1/2-1/p}} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. Thus, we obtain (4.9).

Analogously, it follows that for $\zeta \in [0,1]$

$$\begin{aligned} (1-\zeta ) \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon } + \zeta c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}} \ge \frac{c_1}{2} V_{N^{\lambda _\varepsilon }_{\tau _k}} \quad \text {on } \tilde{D}^{n,\varepsilon ,\rho }_{k,1}, \end{aligned}$$

and that on $\tilde{D}^{n,\varepsilon ,\rho }_{k,1}$

$$\begin{aligned}&\int _0^1 \left| \frac{\partial g_k}{\partial y} \left( (1-\zeta ) \frac{\Delta X^\varepsilon _{\tau _k}}{\varepsilon } + \zeta c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) \right| \textrm{d}{\zeta } \\&\quad \le \left\{ \begin{array}{ll} C \left( 1 + \left| 2 c_2 V_{N^{\lambda _\varepsilon }_{\tau _k}} \right| ^p \right) &{} \text {in the case (i)}, \\ \left| \frac{\partial g_k}{\partial y} \left( \frac{c_1}{2} V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) \right| + \left| \frac{\partial g_k}{\partial y} \left( 2c_2 V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) \right| &{} \text {in the case (ii)}, \end{array} \right. \end{aligned}$$

so that, (4.8) holds. $\square$

Lemma 4.13

Let $\rho \in (0,1/2)$. Under Assumptions 2.1, 2.3, 2.4, 2.6 and 2.8, suppose that for $\theta \in \Theta$

$$\begin{aligned} x \mapsto \int g(x,c(x,\alpha _0)z,\theta ) f_{\alpha _0}(z) \textrm{d}{z}, \quad x \mapsto \int |g(x,c(x,\alpha _0)z,\theta )|^2 f_{\alpha _0}(z) \textrm{d}{z} \end{aligned}$$

(4.10)

are continuous at every points in $I_{x_0}$, and that there exist $\delta >0$, $C>0$ and $q\ge 0$ such that

$$\begin{aligned}{} & {} \int \left\{ \sup _{(x,\theta )\in I_{x_0}^\delta \times \Theta } \left| g (x,c(x,\alpha _0)z,\theta ) \right| \right. \nonumber \\{} & {} \qquad \left. + \sum _{j=1}^d \sup _{(x,\theta )\in I_{x_0}^\delta \times \Theta } \left| \frac{\partial g}{\partial \theta _j} \left( x,c(x,\alpha _0)z,\theta \right) \right| \right\} f_{\alpha _0} (z) \textrm{d}{z} < \infty . \end{aligned}$$

(11)

Then,

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) 1_{D^{n,\varepsilon ,\rho }_{k,1}} \overset{p}{\longrightarrow }\int _0^1 \int g(x_t,c(x_t,\alpha _0)z,\theta ) f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$.

Proof

It follows from Lemma 4.4 and the assumption (4.10) that for each $\theta \in \Theta$

$$\begin{aligned}&\sum _{k=1}^n E \left[ \frac{1}{\lambda _\varepsilon } g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) 1_{J^{n,\varepsilon }_{k,1}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = \frac{1}{n} \sum _{k=1}^n \int g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) z, \theta \right) f_{\alpha _0}(z) \textrm{d}{z} \\&\quad \overset{p}{\longrightarrow }\int _0^1 \int g(x_t,c(x_t,\alpha _0)z,\theta ) f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, and that

$$\begin{aligned} \sum _{k=1}^n E \left[ \frac{1}{\lambda _\varepsilon ^2} \left| g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) \right| ^2 1_{J^{n,\varepsilon }_{k,1}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \overset{p}{\longrightarrow }0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. Thus, Lemma 9 in Genon-Catalot and Jacod (1993) shows us that for each $\theta \in \Theta$

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) 1_{J^{n,\varepsilon }_{k,1}} \overset{p}{\longrightarrow }\int _0^1 \int g(x_t,c(x_t,\alpha _0)z,\theta ) f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. Put

$$\begin{aligned} \tilde{J}^{n,\varepsilon }_{k,1} := J^{n,\varepsilon }_{k,1} \cap \{ X^\varepsilon _t \in I_{x_0}^\delta \text { for all } t\in [0,1] \}. \end{aligned}$$

Then, by the same argument in the proof of Lemma 4.10, it follows from Lemma 4.3 that

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) \left\{ 1_{J^{n,\varepsilon }_{k,1}} - 1_{\tilde{J}^{n,\varepsilon }_{k,1}} \right\} \overset{p}{\longrightarrow }0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$.

Now, we have for each $\theta \in \Theta$

$$\begin{aligned} \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) 1_{\tilde{J}^{n,\varepsilon }_{k,1}} \overset{p}{\longrightarrow }\int _0^1 \int g(x_t,c(x_t,\alpha _0)z,\theta ) f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. To say the uniformity of this convergence in $\theta \in \Theta$, put

$$\begin{aligned} \chi ^{n,\varepsilon }(\theta ) := \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) 1_{\tilde{J}^{n,\varepsilon }_{k,1}} - \int _0^1 \int g(x_t,c(x_t,\alpha _0)z,\theta ) f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{t} \end{aligned}$$

and we shall use Theorem 5.1 in Billingsley (1999) with the state space $C(\Theta )$, same as in the proofs of Propositions 3.3 and 3.6 in Shimizu and Yoshida (2006)¹. From the assumption (114.11), we obtain

$$\begin{aligned}&E \left[ \sup _{\theta \in \Theta } \left| \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n g\left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) 1_{\tilde{J}^{n,\varepsilon }_{k,1}} \right| \right] \\&\quad \le \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n E \left[ \sup _{(x,\theta )\in I_{x_0}^\delta \times \Theta } \left| g(x,c(x,\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}},\theta ) \right| 1_{J^{n,\varepsilon }_{k,1}} \right] \\&\quad = \int \sup _{(x,\theta )\in I_{x_0}^\delta \times \Theta } \left| g(x,c(x,\alpha _0)z,\theta ) \right| f_{\alpha _0} (z) \textrm{d}{z} \, (< \infty ) \end{aligned}$$

and

$$\begin{aligned}{} & {} E \left[ \sup _{\theta \in \Theta } \left| \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n \frac{\partial g}{\partial \theta _j} \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \theta \right) 1_{\tilde{J}^{n,\varepsilon }_{k,1}} \right| \right] \\{} & {} \quad \le \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n E \left[ \sup _{(x,\theta )\in I_{x_0}^\delta \times \Theta } \left| { \frac{\partial g}{\partial \theta _j} \left( x,c(x,\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}},\theta \right) }\right| 1_{J^{n,\varepsilon }_{k,1}}\right] \\{} & {} \quad = \int \sup _{(x,\theta )\in I_{x_0}^\delta \times \Theta } \left| \frac{\partial g}{\partial \theta _j} \left( x,c(x,\alpha _0)z,\theta \right) \right| f_{\alpha _0} (z) \textrm{d}{z} \, (< \infty ) \qquad \text {for } j = 1,\dots ,d. \end{aligned}$$

The above equalities hold from the fact that $V_{N^{\lambda _\varepsilon }_{\tau _k}}$ and $1_{J^{n,\varepsilon }_{k,1}}$ are independent. Hence, for any closed ball $B_M$ of radius $M>0$ centered at zero in the Sobolev space $W^{1,\infty }(\Theta )$, we obtain from Markov’s inequality that

$$\begin{aligned} \sup _{n,\varepsilon } P \left( \chi ^{n,\varepsilon } \not \in B_M \right) = P \left( \Vert \chi ^{n,\varepsilon } \Vert _{W^{1,\infty }(\Theta )} \ge M \right) \le \frac{2C}{M}, \end{aligned}$$

where C is defined as (114.11) and for $q\ge 1$

$$\begin{aligned} \Vert u \Vert _{W^{1,q}(\Theta )} := \Vert u\Vert _{L^q(\Theta )} + \sum _{j=1}^{d} \left\| {\frac{\partial u}{\partial \theta _j}}\right\| _{L^q(\Theta )} \quad \text {for } u\in W^{1,q}(\Theta ). \end{aligned}$$

From Rellich-Kondrachov’s theorem (see, e.g., Theorem 9.16 in Brezis (2011)), it follows that the balls $B_M$, $M>0$ are compact in $C(\Theta )$, and so from Theorem 5.1 in Billingsley (1999) that $\{\chi ^{n,\varepsilon }\}$ is relatively compact in distribution sense as in the Billingsley’s book. Since for each $\theta \in \Theta$ $\{\chi ^{n,\varepsilon }(\theta )\}$ converges to zero in probability, all convergent subsequences of $\{\chi ^{n,\varepsilon }\}$ converges to zero in probability. Analogously, all subnet of $\{\chi ^{n,\varepsilon }\}$ has a subsequence convergent in probability to zero, and so $\{\chi ^{n,\varepsilon }\}$ converges to zero in probability as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. $\square$

Lemma 4.14

Under Assumptions 2.1, 2.3, 2.4, 2.6 and 2.8, let $\rho \in (0,1/2)$, and let $g:\mathbb {R}\times \Theta \rightarrow \mathbb {R}$ satisfy that $\left\{ {\frac{\partial g}{\partial \theta _j}\left( \cdot ,\theta \right) }\right\} _{\theta \in \Theta }$, $j=1,\dots ,d$ are equi-Lipschitz continuous on $I_{x_0}^\delta$ for some small $\delta >0$. Then,

$$\begin{aligned} \frac{1}{\varepsilon } \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \left\{ \Delta ^n_k X^{\varepsilon } - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu _0) \right\} 1_{C^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }\int _0^1 g(x_t, \theta ) b(x_t,\theta ) \textrm{d}{W_t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon ^2/n \rightarrow 0$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$, uniformly in $\theta \in \Theta$.

Proof

At first, we can easily check that

$$\begin{aligned} \frac{1}{\varepsilon } \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \left\{ \int _{t_{k-1}}^{t_k} a(X^\varepsilon _t,\mu _0) \textrm{d}{t} - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu _0) \right\} 1_{C^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }0 \end{aligned}$$

(4.11)

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon n\rightarrow \infty$, and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$. Indeed, this follows from Lemmas 4.3, A.2 and A.3 with the equicontinuity of g on $I_{x_0}$ and the following estimate:

$$\begin{aligned}&\frac{1}{\varepsilon } \sum _{k=1}^n E \left[ \sup _{\theta \in \Theta } \left| g ( X^\varepsilon _{t_{k-1}}, \theta ) \left\{ \int _{t_{k-1}}^{t_k} a(X^\varepsilon _t,\mu _0) \textrm{d}{t} - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu _0) \right\} 1_{C^{n,\varepsilon ,\rho }_{k}} \right| \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad \le C \left( \frac{1}{n} \sum _{k=1}^n E \left[ \sup _{\theta \in \Theta } \left| g ( X^\varepsilon _{t_{k-1}}, \theta ) \sup _{t\in [t_{k-1},t_k]} \frac{|X^\varepsilon _t -X^\varepsilon _{t_{k-1}}|}{\varepsilon } \right| ^{2} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right) ^{1/2} \\&\qquad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (\because \text {Schwartz's inequality and 2.1}) \\&\quad = O_p \left( \frac{1}{\varepsilon n} + \frac{1}{\sqrt{n}} + \frac{\lambda _\varepsilon }{n} \right) \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (\because \text {Lemma 4.1 and 4.4}) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

At second, we show that

$$\begin{aligned}{} & {} \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \int _{t_{k-1}}^{t_k} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} 1_{C^{n,\varepsilon ,\rho }_k} - \int _0^1 g(X^\varepsilon _t,\theta ) b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \nonumber \\{} & {} \quad = \sum _{k=1}^n \int _{t_{k-1}}^{t_k} \left\{ g ( X^\varepsilon _{t_{k-1}}, \theta ) - g ( X^\varepsilon _{t}, \theta ) \right\} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} 1_{C^{n,\varepsilon ,\rho }_k} \nonumber \\{} & {} \qquad - \sum _{k=1}^n \int _{t_{k-1}}^{t_k} g ( X^\varepsilon _{t}, \theta ) b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} 1_{D^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }0 \end{aligned}$$

(4.12)

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon ^2/n \rightarrow 0$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$, uniformly in $\theta \in \Theta$. When we put

$$\begin{aligned} \tilde{C}^{n,\varepsilon ,\rho }_{k} := C^{n,\varepsilon ,\rho }_{k} \cap \left\{ \sup _{t\in [0,1]} |X^\varepsilon _{t} - x_{t}| < \delta \right\} , \end{aligned}$$

it holds from Morrey’s inequality (see, e.g., Theorem 5 in Evans (2010), Section 5.6) that for $q\in (d,\infty )$

$$\begin{aligned}&\sum _{k=1}^n E \left[ \sup _{\theta \in \Theta } \left| \int _{t_{k-1}}^{t_k} \left\{ g ( X^\varepsilon _{t_{k-1}}, \theta ) - g ( X^\varepsilon _{t}, \theta ) \right\} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} 1_{\tilde{C}^{n,\varepsilon ,\rho }_{k}} \right| \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad \le C_1 \sum _{k=1}^n E \left[ \left\| \int _{t_{k-1}}^{t_k} \left\{ g ( X^\varepsilon _{t_{k-1}}, \theta ) - g ( X^\varepsilon _{t}, \theta ) \right\} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} 1_{\tilde{C}^{n,\varepsilon ,\rho }_{k}} \right\| _{W^{1,q}(\Theta )} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] , \end{aligned}$$

where the constant $C_1$ depends only on d, q and $\Theta$. Then, it follows that

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, where $C_2$ depends only on q, and $C_3$ depends only on q, b, g and $\Theta$. By the same argument with Theorem B.4 in Bhagavatula (1999), it follows that

$$\begin{aligned}&\sum _{k=1}^n E \left[ \left\| { \int _{t_{k-1}}^{t_k} \left\{ { \frac{\partial g}{\partial \theta _j} \left( X^\varepsilon _{t_{k-1}}, \theta \right) - \frac{\partial g}{\partial \theta _j} \left( X^\varepsilon _{t}, \theta \right) }\right\} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} 1_{\tilde{C}^{n,\varepsilon ,\rho }_{k}} }\right\| _{L^{q}(\Theta )} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = O_p \left( \frac{1}{\sqrt{n}} + \varepsilon + \varepsilon \sqrt{\lambda _\varepsilon } \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. Thus, it follows from Lemma A.3 that

$$\begin{aligned} \sum _{k=1}^n \int _{t_{k-1}}^{t_k} \left\{ g ( X^\varepsilon _{t_{k-1}}, \theta ) - g ( X^\varepsilon _{t}, \theta ) \right\} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} 1_{\tilde{C}^{n,\varepsilon ,\rho }_{k}} = O_p \left( \frac{1}{\sqrt{n}} + \varepsilon + \varepsilon \sqrt{\lambda _\varepsilon } \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$, and therefore, from Lemma 4.3 we obtain the convergence of the first term in the left-hand side of (4.13). To obtain (4.13), we remain to prove

$$\begin{aligned} \sum _{k=1}^n \int _{t_{k-1}}^{t_k} g ( X^\varepsilon _{t}, \theta ) b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} 1_{D^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }0 \end{aligned}$$

(4.13)

as $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$, $\frac{\lambda _\varepsilon ^2}{n} \rightarrow 0$, $\lambda _\varepsilon \int _{|z|\le 4v_{2}/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$, uniformly in $\theta \in \Theta$. Put $\tilde{D}^{n,\varepsilon ,\rho }_{k,1} := D^{n,\varepsilon ,\rho }_{k,1} \cap \{ X^\varepsilon _t \in I_{x_0}^\delta \text { for all } t\in [0,1] \}$. We begin with showing that for any $p\in (2,\infty )$ and $q'\in (1,d/(d-1))$

$$\begin{aligned}&\sum _{k=1}^n \int _{t_{k-1}}^{t_k} g ( X^\varepsilon _{t}, \theta ) b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} = O_p \left( \frac{1}{\sqrt{n}} \left( \frac{\varepsilon ^p\lambda _\varepsilon ^2}{n} + \lambda _\varepsilon \right) ^{1/2+1/q'} \right) \end{aligned}$$

(4.14)

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$. It follows from Morrey’s inequality (see, e.g., Theorem 5 in Evans (2010), Section 5.6) that for $q\in (d,\infty )$

$$\begin{aligned}&\sum _{k=1}^n E \left[ \sup _{\theta \in \Theta } \left| \int _{t_{k-1}}^{t_k} g ( X^\varepsilon _{t}, \theta ) b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} \right| \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad \le C_1 \sum _{k=1}^n E \left[ \left\| \int _{t_{k-1}}^{t_k} g ( X^\varepsilon _{t}, \theta ) b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \right\| _{W^{1,q}(\Theta )} 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] , \end{aligned}$$

where the constant $C_1$ depends only on d, q and $\Theta$. If we put $q'=q/(q-1)$, then it follows from Hölder’s inequality, Burkholder’s inequality (see, e.g., Theorem 4.4.21 in Applebaum (2009)), the equicontinuity of g and Assumption 2.1 that

$$\begin{aligned}&\sum _{k=1}^n E \left[ \left\| \int _{t_{k-1}}^{t_k} g ( X^\varepsilon _{t}, \theta ) b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \right\| _{L^{q}(\Theta )} 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad \le \sum _{k=1}^n \left( \int _\Theta E \left[ \left| \int _{t_{k-1}}^{t_k} g ( X^\varepsilon _{t}, \theta ) b(X^\varepsilon _t,\sigma _0) 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} \textrm{d}{W_t} \right| ^{q} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \textrm{d}{\theta } \right) ^{1/q} \\&\qquad \times P \left( \tilde{D}^{n,\varepsilon ,\rho }_{k,1} \, \Big | \, \mathcal {F}_{t_{k-1}} \right) ^{1/q'} \\&\quad \le C_2 \sum _{k=1}^n \left( \int _\Theta E \left[ \int _{t_{k-1}}^{t_k} \left| g ( X^\varepsilon _{t}, \theta ) b(X^\varepsilon _t,\sigma _0) 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} \right| ^{2} \textrm{d}{t} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] ^{q/2} \textrm{d}{\theta } \right) ^{1/q} \\&\qquad \times P \left( \tilde{D}^{n,\varepsilon ,\rho }_{k,1} \, \Big | \, \mathcal {F}_{t_{k-1}} \right) ^{1/q'} \\&\quad \le C_2 \sup _{(x,\theta )\in I_{x_0}^\delta \times \Theta } |g(x,\theta )b(x,\sigma _0)| \frac{|\Theta |^{1/q}}{n^{1/2}} \sum _{k=1}^n P \left( \tilde{D}^{n,\varepsilon ,\rho }_{k,1} \, \Big | \, \mathcal {F}_{t_{k-1}} \right) ^{1/2+1/q'}, \end{aligned}$$

where $C_2$ depends only on q. By using Lemmas 4.4 and 4.7, for any $p>2$ we obtain

$$\begin{aligned}&\sum _{k=1}^n E \left[ \left\| \int _{t_{k-1}}^{t_k} g ( X^\varepsilon _{t}, \theta ) b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \right\| _{L^{q}(\Theta )} 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = O_p \left( \sqrt{n} \left\{ \frac{\lambda _\varepsilon }{n} \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} + \frac{\varepsilon ^p\lambda _\varepsilon }{n} \right) + \frac{\lambda _\varepsilon }{n} \right\} ^{1/2+1/q'} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. Similarly, by using Theorem B.4 in Bhagavatula (1999), we obtain for $j=1,\dots ,d$

$$\begin{aligned}&\sum _{k=1}^n E \left[ \left\| \int _{t_{k-1}}^{t_k} \frac{\partial g}{\partial \theta _j} \left( X^\varepsilon _{t}, \theta \right) b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \right\| _{L^{q}(\Theta )} 1_{\tilde{D}^{n,\varepsilon ,\rho }_{k,1}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = O_p \left( \sqrt{n} \left\{ \frac{\lambda _\varepsilon }{n} \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} + \frac{\varepsilon ^p\lambda _\varepsilon }{n} \right) + \frac{\lambda _\varepsilon }{n} \right\} ^{1/2+1/q'} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. Since we can take $q'<2$ small enough, we obtain (4.15) from Remark A.3. Hence, (4.14) holds from (4.15) and Lemma 4.10.

At last, it is an immediate consequence from Lemma 4.9 that

$$\begin{aligned} \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \int _{t_{k-1}}^{t_k} c(X^\varepsilon _{t-},\alpha _0) \textrm{d}{Z^{\lambda _\varepsilon }_t} 1_{C^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon ^2/n \rightarrow 0$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$, uniformly in $\theta \in \Theta$. $\square$

Lemma 4.15

Under Assumptions 2.1 2.3, 2.4, 2.6 and 2.8, let $\rho \in (0,1/2)$. and let $g:\mathbb {R}\times \Theta \rightarrow \mathbb {R}$ satisfy that $\left\{ {\frac{\partial g}{\partial \theta _i} \left( \cdot ,\theta \right) }\right\} _{\theta \in \Theta }$ ($i=1,\dots ,d$) are equicontinuous on $I_{x_0}^\delta$ for some small $\delta >0$. Then,

$$\begin{aligned} \frac{1}{\varepsilon ^2} \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \left| \Delta ^n_k X^{\varepsilon } - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu _0) \right| ^2 1_{C^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }\int _0^1 g(x_t, \theta ) |b(x_t,\sigma _0)|^2 \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$, $\lambda _\varepsilon ^2/n\rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$, uniformly in $\theta \in \Theta$.

Proof

From Lemma 4.9, it is sufficient to show that

$$\begin{aligned} \frac{1}{\varepsilon ^2} \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \left| \Delta ^n_k X^{\varepsilon } - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu _0) \right| ^2 1_{C^{n,\varepsilon ,\rho }_{k,0}} \overset{p}{\longrightarrow }\int _0^1 g(x_t, \theta ) |b(x_t,\sigma _0)|^2 \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$, $\lambda _\varepsilon ^2/n\rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$, uniformly in $\theta \in \Theta$, and we note that

$$\begin{aligned}&\left| \Delta ^n_k X^{\varepsilon } - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu _0) \right| ^2 1_{J^{n,\varepsilon }_{k,0}} \\&\quad = \left\{ \left| \int _{t_{k-1}}^{t_k} \left\{ a(X^\varepsilon _t,\mu _0) - a(X^\varepsilon _{t_{k-1}}, \mu _0) \right\} \textrm{d}{t} \right| ^2 + \left| \int _{t_{k-1}}^{t_k} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \right| ^2 \right. \\&\qquad \quad \left. + 2 \int _{t_{k-1}}^{t_k} \left\{ a(X^\varepsilon _t,\mu _0) - a(X^\varepsilon _{t_{k-1}}, \mu _0) \right\} \textrm{d}{t} \int _{t_{k-1}}^{t_k} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \right\} 1_{J^{n,\varepsilon }_{k,0}}. \end{aligned}$$

Similarly to the proof of (4.12), it follows that

$$\begin{aligned} \sup _{\theta \in \Theta } \left| \frac{1}{\varepsilon ^2} \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \left| \int _{t_{k-1}}^{t_k} \left\{ a(X^\varepsilon _t,\mu _0) - a(X^\varepsilon _{t_{k-1}}, \mu _0) \right\} \textrm{d}{t} \right| ^2 1_{C^{n,\varepsilon ,\rho }_{k,0}} \right| = O_p \left( \frac{1}{\varepsilon ^2 n^3} + \frac{1}{n^2} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon n\rightarrow \infty$, and $\varepsilon \lambda _\varepsilon \rightarrow 0$. Also, it holds that

$$\begin{aligned}&\sup _{\theta \in \Theta } \left| \frac{2}{\varepsilon } \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \int _{t_{k-1}}^{t_k} \left\{ a(X^\varepsilon _t,\mu _0) - a(X^\varepsilon _{t_{k-1}}, \mu _0) \right\} \textrm{d}{t}\right. \\&\quad \left. \times \int _{t_{k-1}}^{t_k} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \, 1_{C^{n,\varepsilon ,\rho }_{k,0}} \right| = O_p \left( \frac{1}{\varepsilon n^{3/2}} + \frac{1}{n} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$. Indeed, by using Assumption 2.1, Hölder’s inequality and Burkholder’s inequality, we obtain

$$\begin{aligned}&\frac{2}{\varepsilon } \sum _{k=1}^n E \left[ \sup _{\theta \in \Theta } \left| g ( X^\varepsilon _{t_{k-1}}, \theta ) \int _{t_{k-1}}^{t_k} \left\{ a(X^\varepsilon _t,\mu _0) - a(X^\varepsilon _{t_{k-1}}, \mu _0) \right\} \textrm{d}{t}\right. \right. \\&\qquad \left. \left. + \int _{t_{k-1}}^{t_k} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \, 1_{C^{n,\varepsilon ,\rho }_{k,0}} \right| \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad \le \frac{2C}{n} \sum _{k=1}^n \sup _{\theta \in \Theta } |g ( X^\varepsilon _{t_{k-1}}, \theta )| \left( E \left[ \frac{1}{\varepsilon ^2} \sup _{t\in [t_{k-1},\tau _k]} |X^\varepsilon _t - X^\varepsilon _{t_{k-1}}|^2 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right) ^{1/2} \\&\qquad \times \left( \frac{1}{n} E \left[ \sup _{t\in [t_{k-1},\tau _k]} |X^\varepsilon _t - X^\varepsilon _{t_{k-1}}|^2 + |b(X^\varepsilon _{t_{k-1}},\sigma _0)|^2 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right) ^{1/2} \end{aligned}$$

where C depends only on a, b. By applying Lemmas 4.3 to 4.5 and A.3 and the boundedness of g on $I_{x_0}^\delta \times \Theta$ for some small $\delta >0$, we obtain the above convergence.

From Lemma 4.11, we remain to prove that

$$\begin{aligned} \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \left| \int _{t_{k-1}}^{t_k} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \right| ^2 1_{C^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }\int _0^1 g(x_t, \theta ) |b(x_t,\sigma _0)|^2 \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, uniformly in $\theta \in \Theta$. At first, by using Lemma 4.4, we have

$$\begin{aligned}&\sum _{k=1}^n E \left[ g ( X^\varepsilon _{t_{k-1}}, \theta ) \left| \int _{t_{k-1}}^{t_k} b(X^\varepsilon _{t_{k-1}},\sigma _0) \textrm{d}{W_t} \right| ^2 \, \Big | \, \mathcal {F}_{t_{k-1}} \right]&= \frac{1}{n} \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) | b(X^\varepsilon _{t_{k-1}},\sigma _0) |^2 \\&\quad \overset{p}{\longrightarrow }\int _0^1 g(x_t, \theta ) |b(x_t,\sigma _0)|^2 \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, and

$$\begin{aligned} \sum _{k=1}^n E \left[ \left| g ( X^\varepsilon _{t_{k-1}}, \theta ) \left| \int _{t_{k-1}}^{t_k} b(X^\varepsilon _{t_{k-1}},\sigma _0) \textrm{d}{W_t} \right| ^2 \right| ^2 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \overset{p}{\longrightarrow }0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. Thus, by Lemma 9 in Genon-Catalot and Jacod (1993), we obtain

$$\begin{aligned} \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \left| \int _{t_{k-1}}^{t_k} b(X^\varepsilon _{t_{k-1}},\sigma _0) \textrm{d}{W_t} \right| ^2 \overset{p}{\longrightarrow }\int _0^1 g(x_t, \theta ) |b(x_t,\theta )|^2 \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. From the equidifferentiablities of g on $I_{x_0}^\delta$ for some $\delta >0$, the uniform tightness is shown by the same argument in the proof of Lemma 4.13. At second, we shall see

$$\begin{aligned} \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \left\{ \left| \int _{t_{k-1}}^{t_k} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \right| ^2 - \left| \int _{t_{k-1}}^{t_k} b(X^\varepsilon _{t_{k-1}},\sigma _0) \textrm{d}{W_t} \right| ^2 \right\} 1_{C^{n,\varepsilon ,\rho }_{k,0}} \overset{p}{\longrightarrow }0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon /n\rightarrow 0$, uniformly in $\theta \in \Theta$. This convergence is obtained from Lemma A.3 and the following estimate:

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

At last, since

$$\begin{aligned} \sup _{k=1,\dots ,n} n \left| { \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} }\right| ^2 \end{aligned}$$

is bounded in probability, it follows from Lemmas 4.1,  4.8 and 4.9 and the linearity of b that

$$\begin{aligned} \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}}, \theta ) \left| \int _{t_{k-1}}^{t_k} b(X^\varepsilon _{t_{k-1}},\sigma _0) \textrm{d}{W_t} \right| ^2 1_{D^{n,\varepsilon ,\rho }_k\cup C^{n,\varepsilon ,\rho }_{k,1} \cup C^{n,\varepsilon ,\rho }_{k,2}} \overset{p}{\longrightarrow }0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon /n\rightarrow 0$, uniformly in $\theta \in \Theta$. $\square$

4.3 Proof of main results

4.3.1 Proof of Theorem 3.1

Proof of Theorem 3.1

It follows from Lemmas 4.11 and 4.14 that

$$\begin{aligned} \Phi _{n,\varepsilon }^{(1)}(\mu ,\sigma )&:= n \varepsilon ^2 \left( \Psi _{n,\varepsilon }^{(1)}(\mu ,\sigma ) - \Psi _{n,\varepsilon }^{(1)}(\mu _0,\sigma )\right) \\&= \sum _{k=1}^{n} \frac{ \left( \Delta ^n_k X^{\varepsilon } - a(X^\varepsilon _{t_{k-1}}, \mu _0) / n \right) \left( a(X^\varepsilon _{t_{k-1}}, \mu ) - a(X^\varepsilon _{t_{k-1}}, \mu _0) \right) }{\left| b(X^\varepsilon _{t_{k-1}},\sigma ) \right| ^2} 1_{C^{n,\varepsilon ,\rho }_k}\\&\qquad - \frac{1}{2n} \sum _{k=1}^{n} \frac{ \left| a(X^\varepsilon _{t_{k-1}}, \mu ) - a(X^\varepsilon _{t_{k-1}}, \mu _0) \right| ^2}{\left| b(X^\varepsilon _{t_{k-1}},\sigma ) \right| ^2} 1_{C^{n,\varepsilon ,\rho }_k}\\&\quad \overset{p}{\longrightarrow }- \frac{1}{2} \int _0^1 \frac{ \left| a(x_t, \mu ) - a(x_t, \mu _0) \right| ^2}{\left| b(x_{t},\sigma ) \right| ^2} \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon ^2/n \rightarrow 0$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$, uniformly in $(\mu ,\sigma )\in \bar{\Theta }_1\times \bar{\Theta }_2$, and from Lemmas 4.11, 4.14 and 4.15 that

$$\begin{aligned}&\Psi _{n,\varepsilon }^{(1)}(\mu ,\sigma ) \\&\quad = \frac{1}{\varepsilon ^2 n} \Phi _{n,\varepsilon }^{(1)}(\mu ,\sigma ) + \Psi _{n,\varepsilon }^{(1)}(\mu _0,\sigma ) \\&\quad = \frac{1}{\varepsilon ^2 n} \Phi _{n,\varepsilon }^{(1)}(\mu ,\sigma ) - \frac{1}{n} \sum _{k=1}^{n} \left\{ \frac{ \left| \Delta ^n_k X^{\varepsilon } - a(X^\varepsilon _{t_{k-1}}, \mu _0) / n \right| ^2 }{2 \frac{1}{n} \left| \varepsilon b(X^\varepsilon _{t_{k-1}},\sigma ) \right| ^2} + \frac{1}{2} \log |b(X^\varepsilon _{t_{k-1}},\sigma )|^2 \right\} 1_{C^{n,\varepsilon ,\rho }_k} \\&\quad \overset{p}{\longrightarrow }- \left( \lim _{\begin{array}{c} n\rightarrow \infty \\ \varepsilon \rightarrow 0 \end{array}} \frac{1}{\varepsilon ^2 n} \right) \int _0^1 \frac{ \left| a(x_t, \mu ) - a(x_t, \mu _0) \right| ^2}{2\left| b(x_{t},\sigma ) \right| ^2} \textrm{d}{t} \\&\qquad - \frac{1}{2} \int _0^1 \left| \frac{b(x_t,\sigma _0)}{b(x_t,\sigma )} \right| ^2 \textrm{d}{t} - \frac{1}{2} \int _0^1 \log |b(x_t,\sigma )|^2 \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon ^2/n \rightarrow 0$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$, uniformly in $(\mu ,\sigma )\in \bar{\Theta }_1\times \bar{\Theta }_2$, Also, it follows from Lemmas 4.12 and 4.13 that

$$\begin{aligned} \Psi _{n,\varepsilon }^{(2)}(\alpha )&= \frac{1}{\lambda _\varepsilon } \sum _{k=1}^{n} \psi \left( X^\varepsilon _{t_{k-1}}, \frac{\Delta ^n_k X^{\varepsilon }}{\varepsilon }, \alpha \right) 1_{D^{n,\varepsilon ,\rho }_k} \\&\overset{p}{\longrightarrow }\int _0^1 \int _{-\infty }^\infty \frac{1}{c(x_t,\alpha _0)} f_{\alpha _0}\left( \frac{y}{c(x_t,\alpha _0)} \right) \log \left\{ \frac{1}{c(x_t,\alpha )} f_{\alpha }\left( \frac{y}{c(x_t,\alpha )} \right) \right\} \textrm{d}{y} \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\lambda _\varepsilon ^2/n \rightarrow 0$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$, uniformly in $\alpha \in \bar{\Theta }_3$. Thus, by using usual argument (see, e.g., the proof of Theorem 1 in Sørensen and Uchida (2003)), the consistency of $\hat{\theta }_{n,\varepsilon }$ holds under Assumption 2.7. $\square$

4.3.2 Proof of Theorem 3.2

To establish the proof of this theorem, we set up random variables $\xi ^i_{\ell k}$, $\tilde{\xi }^i_{\ell k}$ ($\ell =1,\dots ,3$, $i=1,\dots ,d_\ell$, $k=1,\dots ,n$) as the followings:

$$\begin{aligned} {\sqrt{\varepsilon ^{-2}} \frac{\partial \Phi _{n,\varepsilon }^{(1)}}{\partial \mu _i} \left( \mu ,\sigma \right) }\Big |_{\theta =\theta _0}&= - { \frac{1}{\varepsilon }\sum _{k=1}^{n} \frac{ \left\{ { \Delta ^n_k X^{\varepsilon } - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu ) }\right\} \frac{\partial a}{\partial \mu _i} \left( X^\varepsilon _{t_{k-1}}, \mu \right) }{\left| b(X^\varepsilon _{t_{k-1}},\sigma ) \right| ^2} 1_{C^{n,\varepsilon ,\rho }_k} }\Big |_{\theta =\theta _0} \\&=: \sum _{k=1}^n \xi ^i_{1,k} \left( \overset{p}{\longrightarrow }\int _0^1 \frac{ \frac{\partial a}{\partial \mu _i} \left( x_{t},\mu _0\right) }{b(x_{t},\sigma _0)} \textrm{d}{W_t} \right) , \quad (\because \text {Lemma 4.14})\\ \sqrt{n} { \frac{\partial \Psi _{n,\varepsilon }^{(1)}}{\partial \sigma _i} \left( \mu ,\sigma \right) }\Big |_{\theta =\theta _0}&= - \frac{1}{\sqrt{n}} \sum _{k=1}^{n} \left\{ { - \frac{ \left| \Delta ^n_k X^{\varepsilon } - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu ) \right| ^2 }{\frac{1}{n} \left| \varepsilon b(X^\varepsilon _{t_{k-1}},\sigma ) \right| ^2} + 1 }\right\} \\&\qquad \times \frac{ \frac{\partial b}{\partial \sigma _i} \left( X^\varepsilon _{t_{k-1}},\sigma \right) }{b(X^\varepsilon _{t_{k-1}},\sigma )} 1_{C^{n,\varepsilon ,\rho }_k} \Big |_{\theta =\theta _0} \\&=: \sum _{k=1}^n \xi ^i_{2,k}, \\ { \sqrt{\lambda _\varepsilon } \frac{\partial \Psi _{n,\varepsilon }^{(2)}}{\partial \alpha _i} \left( \alpha \right) }\Big |_{\alpha =\alpha _0}&= \frac{1}{\sqrt{\lambda _\varepsilon }} \sum _{k=1}^{n} \frac{\partial \psi }{\partial \alpha _i} \left( X^\varepsilon _{t_{k-1}}, \frac{\Delta ^n_k X^{\varepsilon }}{\varepsilon }, \alpha _0 \right) 1_{D^{n,\varepsilon ,\rho }_k} =: \sum _{k=1}^n \xi ^i_{3,k}, \end{aligned}$$

and

$$\begin{aligned} \sum _{k=1}^n \tilde{\xi }^i_{1,k}&:= \sum _{k=1}^{n} \frac{ \frac{\partial a}{\partial \mu _i} \left( X^\varepsilon _{t_{k-1}}, \mu _0\right) }{b(X^\varepsilon _{t_{k-1}},\sigma _0)} \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} 1_{C^{n,\varepsilon ,\rho }_{k,0}}, \\ \sum _{k=1}^n \tilde{\xi }^i_{2,k}&:= - \sqrt{n} \sum _{k=1}^{n} \left\{ { - \left| { \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} }\right| ^2 + \frac{1}{n} }\right\} \frac{ \frac{\partial b}{\partial \sigma _i} \left( X^\varepsilon _{t_{k-1}},\sigma _0\right) }{b(X^\varepsilon _{t_{k-1}},\sigma _0)} 1_{C^{n,\varepsilon ,\rho }_{k,0}}, \\ \sum _{k=1}^n \tilde{\xi }^i_{3,k}&:= \sum _{k=1}^{n} \frac{1}{\sqrt{\lambda _\varepsilon }} \frac{\partial \psi }{\partial \alpha _i} \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}}, \alpha _0 \right) 1_{J^{n,\varepsilon }_{k,1}}. \end{aligned}$$

Lemma 4.16

Under Assumptions 2.1 to 2.6, 2.8 and 2.10, the following convergences are holds.

For $\ell =1,2$

$$\begin{aligned} \sum _{k=1}^n \xi ^i_{\ell k} - \sum _{k=1}^{n} \tilde{\xi }^i_{\ell k} \overset{p}{\longrightarrow }0 \quad (i=1,\dots ,d_\ell ) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon n\rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$, $\lambda _\varepsilon ^2/n\rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$.

For $\ell =3$, take $\rho$ as either of the following:

• (i) Under Assumption 2.4 (i), take $\rho \in (0,1/2)$.

• (ii) Under Assumption 2.4 (ii), take $\rho \in (0,\min \{1/2,1/4q\})$, where q is the constant given in Assumption 2.10 Assumption (ii.b).

Then,

$$\begin{aligned} \sum _{k=1}^n \xi ^i_{\ell k} - \sum _{k=1}^{n} \tilde{\xi }^i_{\ell k} \overset{p}{\longrightarrow }0 \quad (i=1,\dots ,d_\ell ) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$, $\lambda _\varepsilon ^2/n\rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$ with $\lim (\varepsilon ^2 n)^{-1}<\infty$.

Proof

For $\ell =1,2$, from Lemmas 4.9 and A.3, it is suffcient to show that for $\rho \in (0,1/2)$

$$\begin{aligned}&\sum _{k=1}^n E \left[ \left| \xi ^i_{\ell k} 1_{J^{n,\varepsilon }_{k,0}} - \tilde{\xi }^i_{\ell k} \right| \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \overset{p}{\longrightarrow }0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon n\rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$, $\lambda _\varepsilon ^2/n\rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$.

For $\ell =1$, let $i\in \{1,\dots ,d_1\}$, and put $g(x)=\frac{\partial a}{\partial \mu _i}\left( x,\mu _0\right) /|b(x,\sigma _0)|^2$. Then,

$$\begin{aligned}&\xi ^i_{1, k} 1_{J^{n,\varepsilon }_{k,0}} - \tilde{\xi }^i_{1, k} = g \left( X^\varepsilon _{t_{k-1}}\right) \left\{ \int _{t_{k-1}}^{t_k} a(X^\varepsilon _t,\mu _0) \textrm{d}{t} - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu _0) \right. \\&\qquad \qquad \qquad \qquad \qquad \left. + \, \varepsilon \int _{t_{k-1}}^{t_k} \{ b(X^\varepsilon _{t},\sigma _0) - b(X^\varepsilon _{t_{k-1}},\sigma _0) \} \textrm{d}{W_t} \right\} 1_{C^{n,\varepsilon ,\rho }_{k,0}}. \end{aligned}$$

As in the same argument in Lemma 4.14, it holds from Assumptions 2.1 to 2.3 and Lemmas 4.4 and 4.5 that

$$\begin{aligned} \frac{1}{\varepsilon } \sum _{k=1}^n E \left[ |g ( X^\varepsilon _{t_{k-1}} )| \left| \int _{t_{k-1}}^{t_k} a(X^\varepsilon _t,\mu _0) \textrm{d}{t} - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu _0) \right| 1_{C^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] = O_p \left( \frac{1}{\varepsilon n} + \frac{1}{\sqrt{n}} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, and from Assumption 2.1, Burkholder’s inequality, Lemmas 4.4 and 4.5 that

$$\begin{aligned}&\sum _{k=1}^n E \left[ |g ( X^\varepsilon _{t_{k-1}} )| \left| \int _{t_{k-1}}^{t_k} \{ b(X^\varepsilon _{t},\sigma _0) - b(X^\varepsilon _{t_{k-1}},\sigma _0) \} \textrm{d}{W_t} \right| 1_{C^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad \le \frac{C}{\sqrt{n}} \sum _{k=1}^n |g ( X^\varepsilon _{t_{k-1}} )| \left( E \left[ \sup _{t\in [t_{k-1},\tau _k]} |X^\varepsilon _{t} - X^\varepsilon _{t_{k-1}} |^2 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right) ^{1/2} = O_p \left( \frac{1}{\sqrt{n}} + \varepsilon \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

For $\ell =2$, let $i\in \{1,\dots ,d_2\}$, and put $g(x)=-\frac{1}{|b|^3}\frac{\partial b}{\partial \sigma _i}\left( x,\sigma _0\right)$. Then, we have

$$\begin{aligned}&\xi ^i_{2, k} 1_{J^{n,\varepsilon }_{k,0}} - \tilde{\xi }^i_{2, k} = g \left( X^\varepsilon _{t_{k-1}}\right) \left\{ \left| { \int _{t_{k-1}}^{t_k} \left\{ { a(X^\varepsilon _t,\mu _0) - a(X^\varepsilon _{t_{k-1}}, \mu _0) }\right\} \textrm{d}{t} }\right| ^2 \right. \\&\qquad \qquad + 2 \varepsilon \int _{t_{k-1}}^{t_k} \left\{ { a(X^\varepsilon _t,\mu _0) - a(X^\varepsilon _{t_{k-1}}, \mu _0) }\right\} \textrm{d}{t} \int _{t_{k-1}}^{t_k} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \\&\qquad \qquad \left. + \left| { \varepsilon \int _{t_{k-1}}^{t_k} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} }\right| ^2 - \left| { \varepsilon \int _{t_{k-1}}^{t_k} b(X^\varepsilon _{t_{k-1}},\sigma _0) \textrm{d}{W_t} }\right| ^2 \right\} 1_{C^{n,\varepsilon ,\rho }_{k,0}}, \end{aligned}$$

and by the same argument as in the proof of Lemma 4.15, we obtain

$$\begin{aligned}&\frac{\sqrt{n}}{\varepsilon ^2} \sum _{k=1}^n E \left[ |g ( X^\varepsilon _{t_{k-1}})| \left| \int _{t_{k-1}}^{t_k} \left\{ a(X^\varepsilon _t,\mu _0) - a(X^\varepsilon _{t_{k-1}}, \mu _0) \right\} \textrm{d}{t} \right| ^2 1_{C^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = O_p \left( \frac{1}{\varepsilon ^2 n^{5/2}} + \frac{1}{n^{3/2}} \right) , \\&\frac{\sqrt{n}}{\varepsilon } \sum _{k=1}^{n} E \left[ \left| g(X^\varepsilon _{t_{k-1}}) \int _{t_{k-1}}^{t_k} \left\{ a(X^\varepsilon _t,\mu _0) - a(X^\varepsilon _{t_{k-1}}, \mu _0) \right\} \textrm{d}{t} \right. \right. \\&\qquad \quad \left. \left. +\int _{t_{k-1}}^{t_k} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \right| 1_{C^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = O_p \left( \frac{1}{\varepsilon n} + \frac{1}{\sqrt{n}} \right) , \\&\sqrt{n} \sum _{k=1}^{n} E \left[ \left| g(X^\varepsilon _{t_{k-1}}) \left| \int _{t_{k-1}}^{t_k} b(X^\varepsilon _t,\sigma _0) \textrm{d}{W_t} \right| ^2 \right. \right. \\&\qquad \quad \left. \left. - \left| \int _{t_{k-1}}^{t_k} b(X^\varepsilon _{t_{k-1}},\sigma _0) \textrm{d}{W_t} \right| ^2 \right| 1_{C^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = O_p \left( \frac{1}{\sqrt{n}} + \varepsilon \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon n\rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

For $\ell =3$, let $r_{n,\varepsilon }$ be defined as either of the following:

• (i) Under Assumption 2.4 (i), $r_{n,\varepsilon }=\frac{1}{\varepsilon n^{1-1/p}} + \frac{1}{n^{1/2-1/p}}$ with sufficiently large $p>1$.

• (ii) Under Assumption 2.4 (ii), $r_{n,\varepsilon }=\frac{1}{\varepsilon n^{1-1/p-q\rho }} + \frac{1}{n^{1/2-1/p-q\rho }}$ with sufficiently large $p>1$.

Then, it follows from Lemmas 4.10 4.12 and A.3 that

$$\begin{aligned}&\sum _{k=1}^n E \left[ \left| \xi ^i_{3,k} 1_{J^{n,\varepsilon }_{k,1}} - \tilde{\xi }^i_{3,k} \right| \, \Big | \, \mathcal {F}_{t_{k-1}} \right] = O_p \left( \sqrt{\lambda _\varepsilon } r_{n,\varepsilon } \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon n\rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$, $\lambda _\varepsilon ^2/n\rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$. $\square$

Lemma 4.17

Under Assumptions 2.1 to 2.3, 2.5, 2.6, 2.8 and 2.9,

$$\begin{aligned} \sum _{k=1}^n E \left[ \tilde{\xi }^i_{\ell k} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \overset{p}{\longrightarrow }0 \quad (\ell =1,2,3) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon /n\rightarrow 0$.

Proof

For $\ell =1$, let $i\in \{1,\dots ,d_1\}$, and put $g(x)=\frac{\partial a}{\partial \mu _i} \left( x,\mu _0\right) /b(x,\sigma _0)$. Since

$$\begin{aligned} E \left[ \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] = 0, \quad \text {and } \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} \text { and } 1_{J^{n,\varepsilon }_{k,i}}~(i=1,2) \text { are independent}, \end{aligned}$$

it holds from Lemmas 4.4 and 4.7 that for any $p\ge 1$

$$\begin{aligned} \left| \sum _{k=1}^n E \left[ \tilde{\xi }^i_{1,k} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right|&= \left| \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}} ) E \left[ \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} 1_{C^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right| \\&= \left| \sum _{k=1}^n g ( X^\varepsilon _{t_{k-1}} ) E \left[ \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} 1_{D^{n,\varepsilon ,\rho }_{k,0}\cup J^{n,\varepsilon }_{k,1}\cup J^{n,\varepsilon }_{k,2}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right| \\&= O_p \left( \frac{1}{n^{p(1-\rho )-1/2}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )-1/2}} \right) + O_p \left( \frac{\lambda _\varepsilon }{n} \right) + O_p \left( \frac{\lambda _\varepsilon ^2}{n^2} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

For $\ell =2$, let $i\in \{1,\dots ,d_2\}$, and put $g(x)=-\frac{1}{b}\frac{\partial b}{\partial \sigma _i} \left( x,\sigma _0\right)$. Since

$$\begin{aligned} E \left[ \left| { \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} }\right| ^2 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] = \frac{1}{n}, \quad \text {and} ~\left| { \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} }\right| ^2 \text { and } 1_{J^{n,\varepsilon }_{k,i}}~(i=1,2) \text { are independent}, \end{aligned}$$

it follows from Lemmas 4.4 and 4.7 that for any $p\ge 1$

$$\begin{aligned} \left| \sum _{k=1}^n E \left[ \tilde{\xi }^i_{2,k} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right|&=\left| \sqrt{n} \sum _{k=1}^{n} g(X^\varepsilon _{t_{k-1}}) E \left[ \left\{ \left| { \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} }\right| ^2 - \frac{1}{n} \right\} 1_{C^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right| \\&= \left| \sqrt{n} \sum _{k=1}^{n} g(X^\varepsilon _{t_{k-1}}) E \left[ \left\{ \left| \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} \right| ^2 - \frac{1}{n} \right\} 1_{D^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right| \\&= O_p \left( \frac{1}{n^{p(1-\rho )-1/2}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )-1/2}} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

For $\ell =3$, we may assume $\sup _t |X^\varepsilon _{t} - x_t |<\delta$ for some enough small $\delta >0$. From Assumption 2.9, we obtain

$$\begin{aligned} \sum _{k=1}^n E \left[ \tilde{\xi }^i_{3, k} \, \Big | \, \mathcal {F}_{t_{k-1}} \right]&= \frac{\sqrt{\lambda _\varepsilon }}{n} \sum _{k=1}^n \int \frac{\partial \psi }{\partial \alpha _i} \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) z, \alpha _0 \right) f_{\alpha _0}(z) \textrm{d}{z} \\&= \frac{\sqrt{\lambda _\varepsilon }}{n} \sum _{k=1}^n \frac{\partial }{\partial \alpha _i} \left( \int \psi \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) z, \alpha \right) f_{\alpha _0}(z) \textrm{d}{z} \right) _{\alpha =\alpha _0} = 0. \end{aligned}$$

The last equality holds from the fact that

$$\begin{aligned} \alpha \mapsto \int \psi \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) z, \alpha \right) f_{\alpha _0}(z) \textrm{d}{z} \end{aligned}$$

behaves like the Kullback Leibler divergence from $p_{\alpha ,x}$ to $p_{\alpha _0,x}$ at $x=X^\varepsilon _{t_{k-1}}$, where $p_{\alpha ,x}(y)=f_{\alpha }(y/c(x,\alpha ))/c(x,\alpha )$. $\square$

Lemma 4.18

Under Assumptions 2.1 to 2.6, 2.8, 2.9 and 2.11,

$$\begin{aligned} \sum _{k=1}^n E \left[ \tilde{\xi }^{i_1}_{\ell k} \tilde{\xi }^{i_2}_{\ell k} \, \Big | \, \mathcal {F}_{t_{k-1}} \right]&\overset{p}{\longrightarrow }I^{i_1i_2}_{\ell }{} & {} (\ell =1,2,3,~i_1,i_2=1,\dots ,d_\ell ),\\ \sum _{k=1}^n E \left[ \tilde{\xi }^{i_1}_{\ell _1k} \tilde{\xi }^{i_2}_{\ell _2k} \, \Big | \, \mathcal {F}_{t_{k-1}} \right]&\overset{p}{\longrightarrow }0{} & {} (\ell _1,\ell _2=1,2,3,~\ell _1\ne \ell _2,~i_j=1,\dots ,d_{\ell _j},~j=1,2) \end{aligned}$$

as $n\!\rightarrow \!\infty$, $\varepsilon \!\rightarrow \!0$, $\lambda _\varepsilon \!\rightarrow \!\infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, where $I_1,\dots ,I_3$ are the matrices defined as (3.3).

Proof

For $\ell =1$, $i,j\in \{1,\dots ,d_1\}$, put $g(x)=\frac{\partial a}{\partial \mu _i}\frac{\partial a}{\partial \mu _j}\left( x,\mu _0\right) /b(x,\sigma _0)^2$. Since from Lemmas 4.4 and 4.7 for any $p>1$ we have

$$\begin{aligned}&\sum _{k=1}^n g(X^\varepsilon _{t_{k-1}}) E \left[ \left| \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} \right| ^2 1_{D^{n,\varepsilon ,\rho }_{k,0}\cup J^{n,\varepsilon }_{k,1} \cup J^{n,\varepsilon }_{k,2}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] = O_p \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} + \frac{\lambda _\varepsilon }{n} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, we obtain

$$\begin{aligned} \sum _{k=1}^n E \left[ \tilde{\xi }^i_{1,k} \tilde{\xi }^j_{1,k} \, \Big | \, \mathcal {F}_{t_{k-1}} \right]&=\sum _{k=1}^n g(X^\varepsilon _{t_{k-1}}) E \left[ \left| \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} \right| ^2 1_{C^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&= \frac{1}{n} \sum _{k=1}^n g(X^\varepsilon _{t_{k-1}}) + O_p \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} + \frac{\lambda _\varepsilon }{n} \right) \\&\overset{p}{\longrightarrow }\int _0^1 g(x_t) \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$, and $\lambda _\varepsilon /n\rightarrow 0$.

For $\ell =2$, $i,j\in \{1,\dots ,d_2\}$, put $g(x)=\frac{1}{b^2}\frac{\partial b}{\partial \sigma _i}\frac{\partial b}{\partial \sigma _j} \left( x,\sigma _0\right)$. Since similarly to the proof of Lemma 4.17, it follows from Lemmas 4.4 and 4.7 that for any $p>1$

$$\begin{aligned}&n \sum _{k=1}^{n} g(X^\varepsilon _{t_{k-1}}) E \left[ \left| \left| { \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} }\right| ^2 - \frac{1}{n} \right| ^2 1_{D^{n,\varepsilon ,\rho }_{k,0}\cup J^{n,\varepsilon }_{k,1} \cup J^{n,\varepsilon }_{k,2}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = O_p \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} \right) + O_p \left( \frac{\lambda _\varepsilon }{n} \right) + O_p \left( \frac{\lambda _\varepsilon ^2}{n^2} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$, we obtain from Lemma 4.4 that

$$\begin{aligned}&\sum _{k=1}^n E \left[ \tilde{\xi }^i_{2,k} \tilde{\xi }^j_{2,k} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] = n \sum _{k=1}^{n} g(X^\varepsilon _{t_{k-1}}) E \left[ \left| \left| { \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} }\right| ^2 - \frac{1}{n} \right| ^2 1_{C^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = n \sum _{k=1}^{n} g(X^\varepsilon _{t_{k-1}}) E \left[ \left| \left| { \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} }\right| ^2 - \frac{1}{n} \right| ^2 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] + O_p \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} + \frac{\lambda _\varepsilon }{n} \right) \\&\quad \overset{p}{\longrightarrow }2\int _0^1 g(x_t) \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon /n\rightarrow 0$.

For $\ell =3$, $i,j\in \{1,\dots ,d_3\}$, put $g(x,y)=\frac{\partial \psi }{\partial \alpha _i} \frac{\partial \psi }{\partial \alpha _j}\left( x,y,\alpha _0\right)$. Then, it follows from Lemma 4.4 and Assumption 2.11 that

$$\begin{aligned} \sum _{k=1}^n E \left[ \tilde{\xi }^i_{3,k} \tilde{\xi }^j_{3,k} \, \Big | \, \mathcal {F}_{t_{k-1}} \right]&= \frac{1}{\lambda _\varepsilon } \sum _{k=1}^n E \left[ g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}} \right) 1_{J^{n,\varepsilon }_{k,1}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&= \frac{1}{n} \sum _{k=1}^n \int g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) z \right) f_{\alpha _0}(z) \textrm{d}{z} \\&\overset{p}{\longrightarrow }\int _0^1 \int g \left( x_t, c(x_t,\alpha _0) z \right) f_{\alpha _0}(z) \textrm{d}{z} \textrm{d}{t} \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. The second equality holds from the fact that $V_{N^{\lambda _\varepsilon }_{\tau _k}}$ and $1_{J^{n,\varepsilon }_{k,1}}$ are independent.

For $\ell _j=j$, $i_j=1,\dots ,d_j$ ($j=1,2$), put $g(x)=-\frac{\partial a}{\partial \mu _{i_1}} \left( x,\mu _0\right) \frac{1}{b^2}\frac{\partial b}{\partial \sigma _{i_2}} \left( x,\sigma _0\right)$. Since

$$\begin{aligned}&E \left[ \left( \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} \right) ^i \, \Big | \, \mathcal {F}_{t_{k-1}} \right] = 0, \text { and } \\&\left( \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} \right) ^i \text { and } 1_{J^{n,\varepsilon }_{k,j}} \text { are independent } (i=1,3,~j=1,2), \end{aligned}$$

it follows from Lemmas 4.4 and 4.7 that for any $p\ge 1$

$$\begin{aligned}&\sum _{k=1}^n E \left[ \tilde{\xi }^{i_1}_{1,k} \tilde{\xi }^{i_2}_{2,k} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = \sqrt{n} \sum _{k=1}^n g( X^\varepsilon _{t_{k-1}}) E \left[ \left\{ - \left| \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} \right| ^2 + \frac{1}{n} \right\} \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} 1_{C^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = \sqrt{n} \sum _{k=1}^n g( X^\varepsilon _{t_{k-1}}) E \left[ \left\{ - \left| \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} \right| ^2 + \frac{1}{n} \right\} \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} 1_{D^{n,\varepsilon ,\rho }_{k,0}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \\&\quad = O_p \left( \frac{1}{n^{p(1-\rho )}} + \frac{\varepsilon ^p}{n^{p(1/2-\rho )}} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. $\square$

Lemma 4.19

Under Assumptions 2.1 to 2.3, 2.5, 2.6, 2.8 and 2.9,

$$\begin{aligned} \sum _{k=1}^n \left| E \left[ \tilde{\xi }^i_{\ell k} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \right| ^2 \overset{p}{\longrightarrow }0 \quad (\ell =1,2,3) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$ and $\lambda _\varepsilon /n\rightarrow 0$.

Proof

This follows from the same argument as in the proof of Lemma 4.17. $\square$

Lemma 4.20

Under Assumptions 2.1 to 2.3, 2.5, 2.6, 2.8 and 2.11,

$$\begin{aligned} \sum _{k=1}^n E \left[ \left| {\tilde{\xi }^i_{\ell k}}\right| ^4 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \overset{p}{\longrightarrow }0 \quad (\ell =1,2,3) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

Proof

For $\ell =1$, let $i\in \{1,\dots ,d_1\}$, and put $g(x)=|\frac{\partial a}{\partial \mu _i} \left( x,\mu _0\right) /b(x,\sigma _0)|^4$. Then, it holds from Lemma 4.4 that

$$\begin{aligned} \sum _{k=1}^n E \left[ \left| {\tilde{\xi }^i_{1,k}}\right| ^4 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le \sum _{k=1}^n g \left( X^\varepsilon _{t_{k-1}} \right) E \left[ \left| { \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} }\right| ^4 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \overset{p}{\longrightarrow }0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

For $\ell =2$, let $i\in \{1,\dots ,d_2\}$, and put $g(x)=\left| {\frac{1}{b} \frac{\partial b}{\partial \sigma _i} \left( x,\sigma _0\right) }\right| ^4$. Then, it follows from Lemma 4.4 that

$$\begin{aligned} \sum _{k=1}^n E \left[ \left| { \tilde{\xi }^i_{2,k} }\right| ^4 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \le n^2 \sum _{k=1}^{n} g(X^\varepsilon _{t_{k-1}}) E \left[ \left| { \left| { \int _{t_{k-1}}^{t_k} \textrm{d}{W_t} }\right| ^2 - \frac{1}{n} }\right| ^4 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \overset{p}{\longrightarrow }0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$.

For $\ell =3$, $i\in \{1,\dots ,d_3\}$, put $g(x,y)=\left| {\frac{\partial \psi }{\partial \alpha _{i}}\left( x,y,\alpha _0\right) }\right| ^4$. Then, similarly to the proof of Lemma 4.18, it follows from Lemma 4.4 and Assumption 2.11 that

$$\begin{aligned} \sum _{k=1}^n E \left[ \left| { \tilde{\xi }^i_{3,k} }\right| ^4 \, \Big | \, \mathcal {F}_{t_{k-1}} \right] = \frac{1}{\lambda _\varepsilon ^2} \sum _{k=1}^n E \left[ g \left( X^\varepsilon _{t_{k-1}}, c(X^\varepsilon _{t_{k-1}},\alpha _0) V_{N^{\lambda _\varepsilon }_{\tau _k}} \right) 1_{J^{n,\varepsilon }_{k,1}} \, \Big | \, \mathcal {F}_{t_{k-1}} \right] \overset{p}{\longrightarrow }0 \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$ and $\varepsilon \lambda _\varepsilon \rightarrow 0$. $\square$

Proof of Theorem 3.2

From Theorem A.3 in Shimizu (2007) and Lemmas 4.16 to 4.20,

$$\begin{aligned} \Lambda _{n,\varepsilon } := \sum _{k=1}^n \left( \xi ^1_{1,k}, \dots , \xi ^{d_1}_{1,k}, \xi ^1_{2,k}, \dots , \xi ^{d_2}_{2,k}, \xi ^1_{3,k}, \dots , \xi ^{d_3}_{3,k} \right) ^T \overset{d}{\longrightarrow }\mathcal {N} \left( 0, I_{\theta _0} \right) \end{aligned}$$

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$, $\lambda _\varepsilon ^2/n\rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$ with $\lim (\varepsilon ^2 n)^{-1}<\infty$. Also, it follows from Lemmas 4.11 to 4.15 under Assumption 2.12 that

$$\begin{aligned} C_{\varepsilon ,n} (\theta ) := \begin{pmatrix} \varepsilon ^2n \left( \frac{\partial ^{2}\Psi _{n,\varepsilon }}{\partial \mu _i\partial \mu _j} \left( \theta \right) \right) _{i,j}&{} \varepsilon ^2n \left( \frac{\partial ^{2}\Psi _{n,\varepsilon }}{\partial \mu _i\partial \sigma _j} \left( \theta \right) \right) _{i,j} &{}0 \\ \left( \frac{\partial ^{2}\Psi _{n,\varepsilon }}{\partial \sigma _i\partial \mu _j} \left( \theta \right) \right) _{i,j}&{} \left( \frac{\partial ^{2}\Psi _{n,\varepsilon }}{\partial \sigma _i\partial \sigma _j} \left( \theta \right) \right) _{i,j} &{} 0 \\ 0 &{} 0 &{} \left( \frac{\partial ^{2}\Psi _{n,\varepsilon }}{\partial \alpha _i\partial \alpha _j} \left( \theta \right) \right) _{i,j} \\ \end{pmatrix} \overset{p}{\longrightarrow }- I_{\theta _0} \end{aligned}$$

(4.15)

as $n\rightarrow \infty$, $\varepsilon \rightarrow 0$, $\lambda _\varepsilon \rightarrow \infty$, $\varepsilon \lambda _\varepsilon \rightarrow 0$, $\lambda _\varepsilon ^2/n\rightarrow 0$ and $\lambda _\varepsilon \int _{|z|\le 4v_2/c_1n^\rho } f_{\alpha _0}(z) \textrm{d}{z} \rightarrow 0$ with $\lim (\varepsilon ^2 n)^{-1}<\infty$, uniformly in $\theta \in \Theta$. Indeed,

$$\begin{aligned} \varepsilon ^2n \frac{\partial ^{2}\Psi _{n,\varepsilon }}{\partial \mu _i\partial \mu _j} \left( \theta \right)&= \sum _{k=1}^{n} \left\{ \Delta ^n_k X^{\varepsilon } - \frac{1}{n}a(X^\varepsilon _{t_{k-1}}, \mu ) \right\} \frac{ \frac{\partial ^{2}a}{\partial \mu _i\partial \mu _j} \left( X^\varepsilon _{t_{k-1}}, \mu \right) }{\left| b(X^\varepsilon _{t_{k-1}},\sigma ) \right| ^2} 1_{C^{n,\varepsilon ,\rho }_k} \\&\quad - \frac{1}{n} \sum _{k=1}^{n} \frac{ \frac{\partial a}{\partial \mu _i} \frac{\partial a}{\partial \mu _j} \left( X^\varepsilon _{t_{k-1}}, \mu \right) }{\left| b(X^\varepsilon _{t_{k-1}},\sigma ) \right| ^2} 1_{C^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }- I^{ij}_1, \\ \varepsilon ^2n \frac{\partial ^{2}\Psi _{n,\varepsilon }}{\partial \mu _i\partial \sigma _j} \left( \theta \right)&= - 2 \sum _{k=1}^{n} \left\{ {\Delta ^n_k X^{\varepsilon } - \frac{1}{n}a(X^\varepsilon _{t_{k-1}}, \mu ) }\right\} \\&\qquad \times \frac{ \frac{\partial a}{\partial \mu _i} \left( X^\varepsilon _{t_{k-1}}, \mu \right) \frac{\partial b}{\partial \sigma _j} \left( X^\varepsilon _{t_{k-1}},\sigma \right) }{\left| { b(X^\varepsilon _{t_{k-1}},\sigma ) }\right| ^3} 1_{C^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }0, \\ \frac{\partial ^{2}\Psi _{n,\varepsilon }}{\partial \sigma _i\partial \sigma _j} \left( \theta \right)&= - \frac{1}{n} \sum _{k=1}^{n} \left\{ { - \frac{ \left| { \Delta ^n_k X^{\varepsilon } - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu ) }\right| ^2 }{\frac{1}{n} \left| { \varepsilon b(X^\varepsilon _{t_{k-1}},\sigma ) }\right| ^2} + 1 }\right\} \frac{\partial \left( \frac{1}{b} \frac{\partial b}{\partial \sigma _i}\right) }{\partial \sigma _j} \left( X^\varepsilon _{t_{k-1}},\sigma \right) 1_{C^{n,\varepsilon ,\rho }_k} \\&\quad - \frac{2}{\varepsilon ^2} \sum _{k=1}^{n} \left| { \Delta ^n_k X^{\varepsilon } - \frac{1}{n} a(X^\varepsilon _{t_{k-1}}, \mu ) }\right| ^2 \frac{ \frac{\partial b}{\partial \sigma _i} \frac{\partial b}{\partial \sigma _j} \left( X^\varepsilon _{t_{k-1}},\sigma \right) }{|b(X^\varepsilon _{t_{k-1}},\sigma )|^4} 1_{C^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }- I^{i_1i_2}_2, \\ \frac{\partial ^{2}\Psi _{n,\varepsilon }}{\partial \alpha _i\partial \alpha _j} \left( \theta \right)&= \frac{1}{\lambda _\varepsilon } \sum _{k=1}^{n} \frac{1}{\varphi } \frac{\partial ^{2}\varphi }{\partial \alpha _i\partial \alpha _j} \left( X^\varepsilon _{t_{k-1}}, \frac{\Delta ^n_k X^{\varepsilon }}{\varepsilon }, \alpha \right) 1_{D^{n,\varepsilon ,\rho }_k} \\&\quad - \frac{1}{\lambda _\varepsilon } \sum _{k=1}^{n} \frac{1}{|\varphi |^2} \frac{\partial \varphi }{\partial \alpha _i} \frac{\partial \varphi }{\partial \alpha _j} \left( X^\varepsilon _{t_{k-1}}, \frac{\Delta ^n_k X^{\varepsilon }}{\varepsilon }, \alpha \right) 1_{D^{n,\varepsilon ,\rho }_k} \overset{p}{\longrightarrow }- I^{i_1i_2}_3, \end{aligned}$$

where $\varphi (x,y,\alpha ) := \exp \psi (x,y,\alpha )$. Since

$$\begin{aligned} D_{n,\varepsilon } \begin{pmatrix} \varepsilon ^{-1} (\hat{\mu }_{n,\varepsilon }-\mu _0) \\ \sqrt{n} (\hat{\sigma }_{n,\varepsilon }-\sigma _0) \\ \sqrt{\lambda _\varepsilon } (\hat{\alpha }_{n,\varepsilon }-\alpha _0) \end{pmatrix} = \Lambda _{n,\varepsilon }, \end{aligned}$$

where

$$\begin{aligned} D_{n,\varepsilon } := \int _0^1 C_{n,\varepsilon }(\theta _0+u(\hat{\theta }_{n,\varepsilon }-\theta _0)), \end{aligned}$$

the conclusion follows by the same argument in the proof of Theorem 1 in Sørensen and Uchida (2003). $\square$

5 Examples

This section is devoted to give some examples of densities which satisfy Assumptions 2.9 to 2.12. For simplicity, suppose that $c(x,\alpha )$ is an enough smooth postive function on $I_{x_0}^\delta \times \Theta _3$, and derivatives of c are uniformly continuous. Let $D_+$ is the interior of the common support of $\left\{ {f_\alpha }\right\} _{\alpha \in \Theta _3}$, i.e.,

$$\begin{aligned} f_\alpha (z) \left\{ \begin{aligned}&> 0{} & {} \text {for } z \in D_+, \\&= 0{} & {} \text {otherwise}. \end{aligned} \right. \end{aligned}$$

Note that $y\in D_+(=\mathbb {R}\text { or }\mathbb {R}_+)$ if and only if $y/c(x,\alpha )\in D_+$ for $(x,\alpha )\in I_{x_0}^\delta \times \Theta _3$ owing to Assumption 2.4. If $(x,y,\alpha )\in I_{x_0}^\delta \times D_+\times \Theta _3$,

$$\begin{aligned} \frac{\partial \psi }{\partial y} \left( x, y, \alpha \right)&= \frac{1}{c(x,\alpha )} \frac{ f_\alpha ' \left( \frac{y}{c(x,\alpha )}\right) }{ f_\alpha \left( \frac{y}{c(x,\alpha )} \right) }, \\ \frac{\partial \psi }{\partial \alpha _j} \left( x, y, \alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _j} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) + \frac{\frac{\partial f_\alpha }{\partial \alpha _j} \left( \frac{y}{c(x,\alpha )} \right) }{c(x,\alpha ) f_\alpha \left( \frac{y}{c(x,\alpha )} \right) } \end{aligned}$$

for $(x,\alpha )\in I_{x_0}^\delta \times \Theta _3$. The values of these functions may be undefined if $(x,y,\alpha )\in I_{x_0}^\delta \times \partial D_+ \times \Theta _3$. Otherwise their values are equal to zero.

First, we show an example such that the class of jump size densities satisfies Assumption 2.4 (i).

Example 5.1

(Normal distribution) Let $\Theta _3$ be a smooth open convex set which is compactly contained in $\mathbb {R}\times \mathbb {R}_+\times \mathbb {R}^{d_3-2}$, and let $f_\alpha$ be of the form

$$\begin{aligned} f_\alpha (z) = \frac{1}{\sqrt{2\pi \alpha _2^2}} \exp \left( - \frac{|z-\alpha _1|^2}{2\alpha _2^2} \right) \quad \text {for } \alpha =(\alpha _1,\alpha _2)\in \Theta _3. \end{aligned}$$

Then,

$$\begin{aligned} \psi (x,y,\alpha ) = - \log c(x,\alpha ) - \frac{1}{2} \log (2\pi \alpha _2^2) - \frac{|\frac{y}{c(x,\alpha )}-\alpha _1|^2}{2\alpha _2^2} \quad \text {on } I_{x_0}^\delta \times \mathbb {R}\times \Theta _3. \end{aligned}$$

Since

$$\begin{aligned} f_\alpha '(z) = - \frac{z-\alpha _1}{\alpha _2^2} f_\alpha (z) , \quad \frac{\partial f_{\alpha }}{\partial \alpha _1} (z) = \frac{z - \alpha _1}{\alpha _2} f_{\alpha }(z), \quad \frac{\partial f_{\alpha }}{\partial \alpha _2} (z) = \left\{ - \frac{1}{\alpha _2} + \frac{(z-\alpha _1)^2}{\alpha _2^3} \right\} f_\alpha (z), \end{aligned}$$

we have

$$\begin{aligned} \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right)&= - \frac{1}{c(x,\alpha )} \frac{1}{\alpha _2^2} \left( \frac{y}{c(x,\alpha )} - \alpha _1 \right) , \\ \frac{\partial \psi }{\partial \alpha _1} \left( x,y,\alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _1} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) - \frac{\frac{y}{c(x,\alpha )} - \alpha _1}{\alpha _2 c(x,\alpha )}, \\ \frac{\partial \psi }{\partial \alpha _2} \left( x, y, \alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _2} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) \\&\qquad + \frac{1}{c(x,\alpha )} \left\{ - \frac{1}{\alpha _2} + \frac{|\frac{y}{c(x,\alpha )}-\alpha _1|^2}{\alpha _2^3} \right\} , \\ \frac{\partial \psi }{\partial \alpha _2} \left( x, y, \alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _j} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) \end{aligned}$$

for $(x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}_+\times \Theta _3$ and $j=3,\dots ,d_3$. Furthermore, the derivatives of c and $\log c$ with respect to $\alpha$ are bounded on $I_{x_0}^\delta \times \Theta _3$, and so

$$\begin{aligned}&\left| { \frac{\partial ^{2}\psi }{\partial \alpha _j\partial y} \left( x,y,\alpha \right) }\right| \le C (1 + |y|), \\&\left| { \frac{\partial ^{2}\psi }{\partial \alpha _i\partial \alpha _j} \left( x,y,\alpha \right) }\right| \le C (1 + |y|^2) \quad \text {for } (x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}\times \Theta _3, \end{aligned}$$

where C is a constant not depending on $(x,y,\alpha )$. Thus, Assumptions 2.9 to 2.12 are satisfied.

Next, we show examples such that the class of jump size densities satisfies Assumption 2.4 (ii).

Example 5.2

(Gamma distribution) Let $\Theta _3$ be an open interval compactly contained in $\mathbb {R}_+\times (1,\infty )\times \mathbb {R}^{d_3-2}$, and let $f_\alpha$ be of the form

$$\begin{aligned} f_\alpha (z) = \left\{ \begin{aligned}&\frac{1}{\Gamma (\alpha _2)\alpha _1^{\alpha _2}} z^{\alpha _2-1} e^{-z/{\alpha _1}}{} & {} (z>0), \\&0{} & {} (z\le 0) \end{aligned} \right. \end{aligned}$$

for $\alpha \in \Theta _3$. Then,

$$\begin{aligned} \psi (x,y,\alpha ) = - \log c(x,\alpha ) - \log \Gamma (\alpha _2) - \alpha _2 \log \alpha _1 + (\alpha _2-1) \log z - \frac{z}{\alpha _1} \quad \text {on } I_{x_0}^\delta \times \mathbb {R}_+\times \Theta _3. \end{aligned}$$

Since

$$\begin{aligned} f_\alpha ' (z)= & {} \left( \frac{\alpha _2-1}{z} - \frac{1}{\alpha _1} \right) f_\alpha (z), \\ \frac{\partial f_{\alpha }}{\partial \alpha _1} (z)= & {} \left( -\frac{\alpha _2}{\alpha _1} + \frac{z}{\alpha _1^2} \right) f_\alpha (z), \quad \frac{\partial f_{\alpha }}{\partial \alpha _2} (z) = \left\{ - \frac{\Gamma '(\alpha _2)}{\Gamma (\alpha _2)} - \log \alpha _1 + \log z \right\} f_\alpha (z), \end{aligned}$$

for $z>0$ and $\alpha \in \Theta _3$, we have

$$\begin{aligned} \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right)&= \frac{\alpha _2-1}{y} - \frac{1}{\alpha _1 c(x,\alpha )}, \\ \frac{\partial \psi }{\partial \alpha _1} \left( x,y,\alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _1} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) + \frac{1}{c(x,\alpha )} \left\{ -\frac{\alpha _2}{\alpha _1} + \frac{y}{\alpha _1^2 c(x,\alpha )} \right\} , \\ \frac{\partial \psi }{\partial \alpha _2} \left( x, y, \alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _2} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) \\&\qquad + \frac{1}{c(x,\alpha )} \left\{ - \frac{\Gamma '(\alpha _2)}{\Gamma (\alpha _2)} - \log \alpha _1 + \log \frac{y}{c(x,\alpha )} \right\} , \\ \frac{\partial \psi }{\partial \alpha _j} \left( x,y,\alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _j} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) \end{aligned}$$

for $(x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}_+\times \Theta _3$ and $j=3,\dots ,d_3$. Furthermore, the derivatives of c and $\log c$ with respect to $\alpha$ are bounded on $I_{x_0}^\delta \times \Theta _3$, and so

$$\begin{aligned} \left| { \frac{\partial ^{2}\psi }{\partial \alpha _j\partial y} \left( x,y,\alpha \right) }\right| \le C, \quad \left| { \frac{\partial ^{2}\psi }{\partial \alpha _i\partial \alpha _j} \left( x,y,\alpha \right) }\right| \le C (1 + |y|) \quad \text {for } (x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}\times \Theta _3, \end{aligned}$$

where C is a constant not depending on $(x,y,\alpha )$. Thus, Assumptions 2.9 to 2.12 are satisfied, and $\rho$ in Theorems 3.1 and 3.2 can be taken as $\rho \in (0,1/4)$. Here, we remark that

$$\begin{aligned} \int \frac{1}{z} f_{\alpha }(z) \textrm{d}{z} < \infty \quad \text {if and only if} \quad \alpha _2>1. \end{aligned}$$

Example 5.3

(Inverse Gaussian distribution) Let $\Theta _3$ be smooth, open, convex and compactly contained in $\mathbb {R}_+^2\times \mathbb {R}^{d_3-2}$, and let $f_\alpha$ be of the form

$$\begin{aligned} f_\alpha (z) = \left\{ \begin{array}{ll} \sqrt{\frac{\alpha _2}{2\pi z^3}} e^{-\alpha _2(z - \alpha _1)^2/2\alpha _1^2z}&{} (z>0), \\ 0 &{} (z\le 0) \end{array}\right. \end{aligned}$$

for $\alpha \in \Theta _3$. Then,

$$\begin{aligned} \psi (x,y,\alpha ) = \frac{1}{2c(x,\alpha )} \left\{ \log \frac{\alpha _2}{2\pi } - 3 \log \frac{y}{c(x,\alpha )} \right\} - \frac{\alpha _2 \left| \frac{y}{c(x,\alpha )} - \alpha _1 \right| ^2}{2 \alpha _1^2 y} \quad \text {on } I_{x_0}^\delta \times \mathbb {R}_+\times \Theta _3. \end{aligned}$$

Since

$$\begin{aligned} f_\alpha ' (z)= & {} \left\{ - \frac{3}{2z} - \frac{\alpha _2(z-\alpha _1)}{\alpha _1^2z} - \frac{\alpha _2(z-\alpha _1)^2}{2\alpha _1z^2} \right\} f_\alpha (z),\\ \frac{\partial f_{\alpha }}{\partial \alpha _1} (z)= & {} \frac{\alpha _2(z-\alpha _1)}{\alpha _1^2} f_\alpha (z), \quad \frac{\partial f_{\alpha }}{\partial \alpha _2} (z) = \left\{ \frac{1}{2\alpha _2} - \frac{|z-\alpha _1|^2}{2\alpha _1^2z} \right\} f_\alpha (z) \end{aligned}$$

for $z>0$ and $\alpha \in \Theta _3$, we have

$$\begin{aligned} \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right)&= - \frac{3}{2y} - \frac{ \alpha _2 \left\{ {\frac{y}{c(x,\alpha )}-\alpha _1}\right\} }{\alpha _1^2y} - \frac{\alpha _2 \left| {\frac{y}{c(x,\alpha )}-\alpha _1}\right| ^2}{2\alpha _1\frac{y^2}{c(x,\alpha )}}, \\ \frac{\partial \psi }{\partial \alpha _1} \left( x, y, \alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _1} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) + \frac{\alpha _2\left( \frac{y}{c(x,\alpha )}-\alpha _1\right) }{\alpha _1^2 c(x,\alpha )}, \\ \frac{\partial \psi }{\partial \alpha _2} \left( x, y, \alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _2} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) \\&\qquad + \frac{1}{2\alpha _2 c(x,\alpha )} - \frac{|\frac{y}{c(x,\alpha )}-\alpha _1|^2}{2\alpha _1^2y}, \\ \frac{\partial \psi }{\partial \alpha _j} \left( x,y,\alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _j} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) \end{aligned}$$

for $(x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}_+\times \Theta _3$ and $j=3,\dots ,d_3$. Furthermore, the derivatives of c and $\log c$ with respect to $\alpha$ are bounded on $I_{x_0}^\delta \times \Theta _3$, and so

$$\begin{aligned} \sup _{(x,\alpha ) \in I_{x_0}^\delta \times \Theta _3} \left| { \frac{\partial ^{2}\psi }{\partial \alpha _j\partial y} \left( x,y,\alpha \right) }\right| \le O \left( \frac{1}{|y|^2} \right) \quad \text {as} ~ y\rightarrow 0, \end{aligned}$$

for $(x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}_+\times \Theta _3$ with $y/c(x,\alpha )\ne \alpha _1$. Thus, Assumptions 2.9 to 2.12 are satisfied, and $\rho$ in Theorems 3.1 and 3.2 can be taken as $\rho \in (0,1/8)$.

Example 5.4

(Weibull distribution) Let $\Theta _3$ be smooth, open, convex and compactly contained in $\mathbb {R}_+\times (1,\infty )\times \mathbb {R}^{d_3-2}$, and let $f_\alpha$ be of the form

$$\begin{aligned} f_\alpha (z) = \left\{ \begin{aligned}&\frac{\alpha _2}{\alpha _1} \left( \frac{z}{\alpha _1} \right) ^{\alpha _2-1} e^{-(z/\alpha _1)^{\alpha _2}}{} & {} (z>0), \\&0{} & {} (z\le 0) \end{aligned} \right. \end{aligned}$$

for $\alpha \in \Theta _3$. Then,

$$\begin{aligned} \psi (x,y,\alpha ) = \frac{1}{c(x,\alpha )} \left\{ \log \alpha _2 - \alpha _2 \log \alpha _1 - (\alpha _2-1) \log \frac{y}{c(x,\alpha )} \right\} \quad \text {on } I_{x_0}^\delta \times \mathbb {R}_+\times \Theta _3. \end{aligned}$$

Since

$$\begin{aligned} f_\alpha ' (z)= & {} \left( \frac{\alpha _2-1}{z} - \alpha _2 \left( \frac{z}{\alpha _1} \right) ^{\alpha _2-1} \right) f_\alpha (z) \quad (z\ne 0), \\ \frac{\partial f_\alpha }{\partial \alpha _1} (z)= & {} - \frac{\alpha _2}{\alpha _1} \left\{ 1 + \left( \frac{z}{\alpha _1} \right) ^{\alpha _2} \right\} f_\alpha (z), \quad \frac{\partial f_\alpha }{\partial \alpha _2} (z) = \left\{ \frac{1}{\alpha _2} + \log \frac{z}{\alpha _1} - \left( \frac{z}{\alpha _1} \right) ^{\alpha _2} \log \frac{z}{\alpha _1} \right\} f_\alpha (z) \end{aligned}$$

for $z>0$ and $\alpha \in \Theta _3$, we have

$$\begin{aligned} \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right)&= \frac{(\alpha _2-1)}{y} - \frac{\alpha _2}{c(x,\alpha )} \left( \frac{y}{\alpha _1 c(x,\alpha )} \right) ^{\alpha _2-1} \\ \frac{\partial \psi }{\partial \alpha _1} \left( x, y, \alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _1} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) - \frac{\alpha _2}{\alpha _1 c(x,\alpha )} \left\{ 1 + \left( \frac{y}{\alpha _1 c(x,\alpha )} \right) ^{\alpha _2} \right\} , \\ \frac{\partial \psi }{\partial \alpha _2} \left( x, y, \alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _2} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) \\&\quad + \frac{1}{c(x,\alpha )} \left\{ \frac{1}{\alpha _2} + \log \frac{y}{\alpha _1c(x,\alpha )} - \left( \frac{y}{\alpha _1 c(x,\alpha )} \right) ^{\alpha _2} \log \frac{y}{\alpha _1 c(x,\alpha )} \right\} , \\ \frac{\partial \psi }{\partial \alpha _j} \left( x,y,\alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _j} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) \end{aligned}$$

for $(x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}_+\times \Theta _3$ and $j=3,\dots ,d_3$. Furthermore, the derivatives of c and $\log c$ with respect to $\alpha$ are bounded on $I_{x_0}^\delta \times \Theta _3$, and so

$$\begin{aligned} \sup _{(x,\alpha ) \in I_{x_0}^\delta \times \Theta _3} \left| { \frac{\partial ^{2}\psi }{\partial \alpha _j\partial y} \left( x,y,\alpha \right) }\right| \le O \left( \frac{1}{y} \right) \quad \text {as} ~ y\rightarrow 0, \end{aligned}$$

for $(x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}\times \Theta _3$ with $y/c(x,\alpha )\ne \alpha _1$, where C is a constant not depending on $(x,y,\alpha )$. Here, we remark that

$$\begin{aligned} \int \frac{1}{y} f_\alpha (y) \textrm{d}{y} < \infty \quad \text {if and only if} \quad \alpha _2 > 1 \end{aligned}$$

and that there exists a constant $C>0$ such that

$$\begin{aligned} | y^{\alpha _2-1} \log y | \le |y_1^{\alpha _2-1} \log y_1| + | y_2^{\alpha _2-1} \log y_2| + C \quad \text {for } y_1 \le y \le y_2. \end{aligned}$$

Thus, Assumptions 2.9 to 2.12 are satisfied, and $\rho$ in Theorems 3.1 and 3.2 can be taken as $\rho \in (0,1/4)$.

Example 5.5

(Log-normal distribution) Let $\Theta _3$ be smooth, open, convex and compactly contained in $\mathbb {R}\times [0,\infty )$, and let $f_\alpha$ be of the form

$$\begin{aligned} f_\alpha (z) = \left\{ \begin{aligned}&\frac{1}{\sqrt{2\pi }\alpha _2 z} e^{-(\log z - \alpha _1)^2/2\alpha _2^2}{} & {} (z>0), \\&0{} & {} (z\le 0) \end{aligned} \right. \end{aligned}$$

for $\alpha \in \Theta _3$. Then,

$$\begin{aligned} \psi (x,y,\alpha ) = \frac{1}{c(x,\alpha )} \left\{ - \log \frac{\sqrt{2\pi }\alpha _2 y}{c(x,\alpha )}- \frac{1}{2\alpha _2} \left| \log \frac{y}{c(x,\alpha )} - \alpha _1 \right| ^2 \right\} \quad \text {on } I_{x_0}^\delta \times \mathbb {R}_+\times \Theta _3. \end{aligned}$$

Since

$$\begin{aligned} f_\alpha ' (z)= & {} \left\{ - \frac{1}{z} - \frac{\log z - \alpha _1}{\alpha _2^2 z} \right\} f_\alpha (z), \\ \frac{\partial f_\alpha }{\partial \alpha _1} (z)= & {} \frac{\log z - \alpha _1}{\alpha _2^2} f_\alpha (z), \quad \frac{\partial f_\alpha }{\partial \alpha _2} (z) = \left\{ - \frac{1}{\alpha _2} + \frac{|\log z - \alpha _1|^2}{\alpha _2^3} \right\} f_\alpha (z) \end{aligned}$$

for $z>0$ and $\alpha \in \Theta _3$, we have

$$\begin{aligned} \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right)&= - \frac{1}{\alpha _2^2 y} \left( \alpha _1 + \alpha _2^2 + \log \frac{y}{c(x,\alpha )} \right) \\ \frac{\partial \psi }{\partial \alpha _1} \left( x, y, \alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _1} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) + \frac{\log \frac{y}{c(x,\alpha )} - \alpha _1}{\alpha _2^2 c(x,\alpha )}, \\ \frac{\partial \psi }{\partial \alpha _2} \left( x, y, \alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _2} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) \\&\qquad + \frac{1}{c(x,\alpha )} \left\{ - \frac{1}{\alpha _2} + \frac{|\log \frac{y}{c(x,\alpha )} - \alpha _1|^2}{\alpha _2^3} \right\} , \\ \frac{\partial \psi }{\partial \alpha _j} \left( x,y,\alpha \right)&= - \frac{\partial (\log c)}{\partial \alpha _j} \left( x,\alpha \right) \left( 1 + y \frac{\partial \psi }{\partial y} \left( x,y,\alpha \right) \right) \end{aligned}$$

for $(x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}_+\times \Theta _3$ and $j=3,\dots ,d_3$. Furthermore, the derivatives of c and $\log c$ with respect to $\alpha$ are bounded on $I_{x_0}^\delta \times \Theta _3$, and so

$$\begin{aligned} \sup _{(x,\alpha ) \in I_{x_0}^\delta \times \Theta _3} \left| { \frac{\partial ^{2}\psi }{\partial \alpha _j\partial y} \left( x,y,\alpha \right) }\right| \le O \left( \frac{1}{y} + \frac{1}{y} \log y \right) \quad \text {as} ~ y\rightarrow 0, \end{aligned}$$

for $(x,y,\alpha )\in I_{x_0}^\delta \times \mathbb {R}\times \Theta _3$ with $y/c(x,\alpha )\ne \alpha _1$, where C is a constant not depending on $(x,y,\alpha )$. Here, we remark that

$$\begin{aligned} \int \left( \frac{1}{y} + \frac{\log y}{y} \right) f_\alpha (y) \textrm{d}{y} < \infty \end{aligned}$$

and that there exists a constant $C>0$ such that

$$\begin{aligned} \left| \frac{1}{y} \log y \right| \le \left| \frac{1}{y_1} \log y_1\right| + \left| \frac{1}{y_2} \log y_2\right| + C \quad \text {for } y_1 \le y \le y_2. \end{aligned}$$

Thus, Assumptions 2.9 to 2.12 are satisfied, and $\rho$ in Theorems 3.1 and 3.2 can be taken as $\rho \in (0,1/4)$.

Remark 5.1

As in the assumptions of Theorems 3.1 and 3.2, the range of $\rho$ depends on q in Assumption 2.10 (ii.b) and Assumption 2.12 (ii.b). So, the differences of the ranges of $\rho$ in the examples above are caused by the differences of q: $q=2$ in Example 5.3, $q=1$ in Examples 5.2 and 5.4, and any $q\in [0,1)$ in Example 5.5.

6 Numerical experiments

In this section, we show some numerical results of our estimator for the Ornstein-Uhlenbeck processes given by

$$\begin{aligned} \textrm{d}{X^\varepsilon _t} = - \mu _0 X^\varepsilon _t \textrm{d}{t} + \varepsilon \sqrt{\sigma _0} \textrm{d}{W_t} + \varepsilon \textrm{d}{Z_t^{\lambda _\varepsilon }}, \quad X_0^{\varepsilon } = x_0 \in \mathbb {R}, \end{aligned}$$

(4.16)

where $Z_t^{\lambda _\varepsilon }$ is a compound Poisson process with the Lévy density $f_{\alpha _0}$ and with the intensity $\lambda _\varepsilon$. In particular, we fix $x_0=0.8$ and $\lambda _\varepsilon =100$, and we employ the inverse Gaussian densities $f_\alpha$’s as in Example 5.3.

To avoid the discussion about how we find some ’appropriate’ $v_{nk}$ and $\rho$, we suppose that the intensity $\lambda _\varepsilon =100$ is known, and we set

$$\begin{aligned} {\hat{C}^{N_D}_k}&:= \left\{ { \Delta ^n_k X^{\varepsilon } \text { is not contained in the}\, \lceil N_D\rceil \,\text { largest positive numbers of}\, \left\{ {\Delta ^n_j X^{\varepsilon }}\right\} _{j=1,\dots ,n}}\right\} , \\ {\hat{D}^{N_D}_k}&:= \left\{ { \Delta ^n_k X^{\varepsilon } \text { is one of the}\, \lceil N_D\rceil \,\text { largest positive values of}\, \left\{ {\Delta ^n_j X^{\varepsilon }}\right\} _{j=1,\dots ,n}}\right\} , \end{aligned}$$

where $N_D>0$ and $\lceil \cdot \rceil$ is the ceil function (we take $N_D=\lambda _\varepsilon$ in Table 1, and $N_D=50,100,150$ in Table 2). Then we replace $1_{C^{n,\varepsilon ,\rho }_k}$ and $1_{D^{n,\varepsilon ,\rho }_k}$ in (3.1) with

$$\begin{aligned} 1_{\hat{C}_k^{N_D}} ~\text {and}~ 1_{\hat{D}_k^{N_D}}, ~\text {respectively,} \end{aligned}$$

and we calculate our estimator $\hat{\theta }_{n,\varepsilon }=(\hat{\mu }_{n,\varepsilon }, \hat{\sigma }_{n,\varepsilon },\hat{\alpha }_{n,\varepsilon ,1},\hat{\alpha }_{n,\varepsilon ,2})$ as in (3.2) from a sample path of (6.1) under the true parameter $(\mu _0,\sigma _0,\alpha _{01},\alpha _{02})$. We iterate this calculation 1000 times with $n=200,500,1500,5000$ and $\varepsilon =1,0.1,0.01$. and we summarize the averages and the standard deviations of $\hat{\theta }_{n,\varepsilon }$’s in Tables 1 and 2.

Table 1 Sample means (with standard deviations in parentheses) of $\hat{\theta }_{n,\varepsilon }$’s, based on 1000 sample paths from the OU process (6.1) with inverse Gaussian $f_\alpha$ as in Example 5.3 with $(\mu _0,\sigma _0,\alpha _{01},\alpha _{02}) = (1.0,2.0,1.2,0.5)$ and with $N_D=\lambda _\varepsilon (=100)$

Table 2 Sample means (with standard deviations in parentheses) of $\hat{\theta }_{n,\varepsilon }$’s, based on 1000 sample paths from the OU process (6.1) with inverse Gaussian $f_\alpha$ as in Example 5.3 with $(\mu _0,\sigma _0,\alpha _{01},\alpha _{02}) = (1.0,2.0,1.2,0.5)$ and with $(n,\varepsilon ,\lambda _\varepsilon )=(5000,0.01,100)$

Remark 6.1

Note that $\hat{D}_k^{N_D}$ (and $\hat{C}_k^{N_D}$) are defined by using the whole data $\{X_{t_j}^\varepsilon \}_{j=1,\dots ,n}$, which conflicts Assumption 2.8, however, for simplicity of our numerical experiment we replace $D^{n,\varepsilon ,\rho }_k$ with $\hat{D}_k^{\lambda _\varepsilon }$ above. We give an intuitive explanation of the reason why we use this setting as follows: The continuous increments go to zero and the jumps are remained as $n\rightarrow \infty$ with $\varepsilon$ fixed (recall that in our asymptotics n increases much faster than $1/\varepsilon$ and $\lambda _\varepsilon$ as in Theorems 3.1 and 3.2), and in this case, from Lemma 4.8, $\{\Delta ^n_k X^{\varepsilon }\,|\,\Delta ^n_k X^{\varepsilon }>v_{nk}/n^\rho \}$ with ‘appropriate’ $v_{nk}$ and $\rho$ would be the $\lambda _\varepsilon$ largest numbers of $\{X_{t_j}^\varepsilon \}_{j}$ in probability. Hence, we replace $D_k^{n,\varepsilon ,\rho }$ with $\hat{D}_k^{\lambda _\varepsilon }$ .

In Table 1, the averages of $(\mu ,\sigma ,\alpha _1,\alpha _2)$ becomes close to the true paramter as n grows and $\varepsilon$ goes to zero. However, the standard deviation of $\alpha _2$ for each fixed $\varepsilon$ increases as n grows. The reason why it happens is expected as follows: If n is not enough large with fixed $\varepsilon$, then the continuous increments in $\Delta ^n_k X^{\varepsilon }$ is too larger than the jumps. In this case, some of $\Delta ^n_k X^{\varepsilon }$’s including positive jumps may be negative, and furthermore even positive $\Delta ^n_k X^{\varepsilon }$’s may be closer to zero than the jumps included in them. This implies that $\Delta ^n_k X^{\varepsilon }$ with small jumps are ignored and the remained $\Delta ^n_k X^{\varepsilon }$ regared as jumps are underestimated, and therefore, the mean and standard deviations of $\alpha _2$ are near zero when n is few with fixed $\varepsilon$.

In Table 2, we consider the following two cases: One is $C^{n,\varepsilon ,\rho }_k$ is too loose, i.e., the case $N_D=50$, and the other is $C^{n,\varepsilon ,\rho }_k$ is too tight, i.e., the case $N_D=150$. In the former case, some small jumps are not removed for the estimation of $(\mu ,\sigma )$ and are in short supply for the estimation of $\alpha$. Thus, it is natural that $\sigma ,\alpha _1,\alpha _2$ take bigger values than true values. In the latter case, some Brownian increments are mistakenly regarded as jumps, and so $\alpha _1$ is closer to zero than the true value.

A Appendix

In this section, we state and prove some slightly different versions of well-known results. More precisely, we prepare Lemma A.2 as localization of the continuous mapping theorem. Lemma A.3 is a slightly different version of Lemma 9 in Genon-Catalot and Jacod (1993).

Lemma A.1

Let $\mathcal {X}$ be a Banach space, and let $\left\{ {g_\theta }\right\} _{\theta \in \Theta }$ be a family of functions from $\mathcal {X}$ to $\mathbb {R}$, and let $T_{g_\theta }$ be the composition operator on $L^\infty ([0,1];\mathcal {X})$ generated by $g_\theta$, i.e.,

$$\begin{aligned} T_{g_\theta }(\tilde{y}_\cdot ):=g_\theta (\tilde{y}_\cdot ) \quad \text {for}~\tilde{y}_\cdot \in L^\infty ([0,1];\mathcal {X}). \end{aligned}$$

Suppose that $y_\cdot$ is a version of a function of $C([0,1];\mathcal {X})$ in $L^\infty ([0,1];\mathcal {X})$, and that $\left\{ {g_\theta }\right\} _{\theta \in \Theta }$ is equicontinuous at every points in ${{\,\textrm{Image}\,}}(y_\cdot ):=\{y_t\,|\,t\in [0,1]\}$. Then, there is a neighborhood $\mathscr {N}_{y_\cdot }$ of $y_\cdot$ in $L^\infty ([0,1];\mathcal {X})$ such that $\left\{ {T_{g_\theta }}\right\} _{\theta \in \Theta }$ is a family of operators from $\mathscr {N}_{y_\cdot }$ to $L^\infty ([0,1])$, and is equicontinuous at $y_\cdot$.

Proof

Fix an arbirary $\eta >0$. For each $x\in {{\,\textrm{Image}\,}}(y_\cdot )$, there exists $\delta _x>0$ such that if $\left\| x-x' \right\| _{\mathcal {X}}<\delta _x$, $x,x'\in \mathcal {X}$, then

$$\begin{aligned} \sup _{\theta \in \Theta } \left| g_\theta (x) - g_\theta (x') \right| < \frac{\eta }{2}. \end{aligned}$$

Since ${{\,\textrm{Image}\,}}(y_\cdot )$ is compact in $\mathcal {X}$, there are finite points $x_1,\dots ,x_k\in {{\,\textrm{Image}\,}}(y_\cdot )$ such that

$$\begin{aligned} {{\,\textrm{Image}\,}}(y_\cdot ) \subset \bigcup _{i=1}^k B(x_i,\delta _{x_i}/2), \end{aligned}$$

where $B(x_i,\delta _{x_i}/2)$ is the ball in $\mathcal {X}$ centered at $x_i$ with radius $\delta _{x_i}/2$. If $\left\| { \tilde{y}_\cdot - y_\cdot }\right\| _{L^\infty ([0,1];\mathcal {X})}<\delta$ with $\delta :=\min \{\delta _{x_1}/2,\dots ,\delta _{x_k}/2\}$, then for a.e. $t\in [0,1]$ there is $i_t\in \{1,\dots ,k\}$ such that $y_t,\tilde{y}_t\in B(x_{i_t},\delta _{x_{i_t}})$. Thus, we obtain

$$\begin{aligned} \sup _{\theta \in \Theta } \left| g_\theta (\tilde{y}_t) - g_\theta (y_t) \right| \le \sup _{\theta \in \Theta } \left| g_\theta (\tilde{y}_t) - g_\theta (x_{i_t}) \right| + \sup _{\theta \in \Theta } \left| g_\theta (x_{i_t}) - g_\theta (y_t) \right| < \eta , \end{aligned}$$

that is,

$$\begin{aligned} \sup _{\theta \in \Theta } \left\| g_\theta (\tilde{y}_\cdot ) - g_\theta (y_\cdot ) \right\| _{L^\infty ([0,1])} < \eta . \end{aligned}$$

This implies the conclusion. $\square$

We prepare the following lemma as localization of the continuous mapping theorem.

Lemma A.2

Under the same assumptions as in Lemma A.1, suppose that $\left\{ {g(\cdot ,\theta )}\right\} _{\theta \in \Theta }$ is equicontinuous at every points in ${{\,\textrm{Image}\,}}(y_\cdot ):=\{y_t\,|\,t\in [0,1]\}$, and that $(Y^\iota _\cdot )_{\iota \in I}$ is a net of $\mathcal {X}$-valued bounded random processes on [0, 1] with a directed set I. If the net $(Y^\iota _\cdot )_{\iota \in I}$ converges in probability to $y_\cdot$ in $L^\infty ([0,1;\mathcal {X}])$, i.e.,

$$\begin{aligned} \left\| Y^\iota _\cdot - y_\cdot \right\| _{L^\infty ([0,1];\mathcal {X})} \overset{p}{\longrightarrow }0, \end{aligned}$$

then

$$\begin{aligned} \sup _{\theta \in \Theta } \left\| g(Y^\iota _\cdot ,\theta ) - g(y_\cdot ,\theta ) \right\| _{L^\infty ([0,1])} \overset{p}{\longrightarrow }0. \end{aligned}$$

Proof

Take an arbitrary $\eta >0$. It follows from Lemma A.1 that there exists a sufficiently small $\delta >0$ such that if $\left\| { \tilde{y}_\cdot - y_\cdot }\right\| _{L^\infty ([0,1];\mathcal {X})}<\delta$, then $\left\{ {g(\tilde{y}_\cdot ,\theta )}\right\} _{\theta \in \Theta }\subset L^\infty ([0,1])$ and

$$\begin{aligned} \sup _{\theta \in \Theta } \left\| g(\tilde{y}_\cdot ,\theta ) - g(y_\cdot ,\theta ) \right\| _{L^\infty ([0,1])} < \eta , \end{aligned}$$

and therefore,

$$\begin{aligned} P \left( \sup _{\theta \in \Theta } \left\| g(Y^\iota _\cdot ,\theta ) - g(y_\cdot ,\theta ) \right\| _{L^\infty ([0,1])}> \eta \right) \le P \left( \left\| Y^\iota _\cdot - y_\cdot \right\| _{L^\infty ([0,1];\mathcal {X})} > \delta \right) . \end{aligned}$$

This implies the conclusion. $\square$

Remark A.1

By the proof of Lemma A.2, it also follows that for any $C_1>0$,

$$\begin{aligned} P \left( \sup _{\theta \in \Theta } \left\| g(Y^\iota _\cdot ,\theta ) - g(y_\cdot ,\theta ) \right\| _{L^\infty ([0,1])}> C_2 \right) \le P \left( \left\| Y^\iota _\cdot - y_\cdot \right\| _{L^\infty ([0,1];\mathcal {X})} > C_1 \right) , \end{aligned}$$

where $C_2$ depends only on $C_1$, g and ${{\,\textrm{Image}\,}}(y_\cdot )$.

Lemma A.3

Suppose that $(\mathcal {X},\Vert \cdot \Vert )$ is a Banach space, $\{(n,\varepsilon )\}$ is a directed set and $\left\{ {\mathcal {G}^{n,\varepsilon }_i}\right\} _i$ is a filtration for each $n,\varepsilon$. Let $\chi ^{n,\varepsilon }_i$, U be $\mathcal {X}$-valued $\mathcal {G}^{n,\varepsilon }_i$-measurable random variables.

• (i) If for any $\eta >0$

  $$\begin{aligned} \lim _{n,\varepsilon } P \left( \sum _{i=1}^n E \left[ \Vert \chi ^{n,\varepsilon }_i \Vert \, \Big | \, \mathcal {G}^{n,\varepsilon }_{i-1} \right] > \eta \right) =0, \end{aligned}$$

  then for any $\eta >0$

  $$\begin{aligned} \lim _{n,\varepsilon } P \left( \left\| \sum _{i=1}^n \chi ^{n,\varepsilon }_i \right\| > \eta \right) = 0. \end{aligned}$$

• (ii) If

  $$\begin{aligned} \lim _{M\rightarrow \infty } \sup _{n,\varepsilon } P \left( \sum _{i=1}^n E \left[ \Vert \chi ^{n,\varepsilon }_i \Vert \, \Big | \, \mathcal {G}^{n,\varepsilon }_{i-1} \right] > M \right) = 0, \end{aligned}$$

  then

  $$\begin{aligned} \lim _{M\rightarrow \infty } \sup _{n,\varepsilon } P \left( \left\| \sum _{i=1}^n \chi ^{n,\varepsilon }_i \right\| > M \right) = 0. \end{aligned}$$

Proof

Since for any $\eta ,\eta '>0$

$$\begin{aligned}{} & {} P \left( \left\| \sum _{i=1}^n \chi ^{n,\varepsilon }_i \right\| > \eta , ~ \sum _{i=1}^n E \left[ \Vert \chi ^{n,\varepsilon }_i \Vert \, \Big | \, \mathcal {G}^{n,\varepsilon }_{i-1} \right] \le \eta ' \right) \\{} & {} \quad \le \frac{1}{\eta } E \left[ \sum _{i=1}^n \Vert \chi ^{n,\varepsilon }_i\Vert ~ 1_{\left\{ \sum _{i=1}^n E \left[ \Vert \chi ^{n,\varepsilon }_i \Vert \, \Big | \, \mathcal {G}^{n,\varepsilon }_{i-1} \right] \le \eta ' \right\} } \right] \nonumber \\{} & {} \quad \le \frac{1}{\eta } E \left[ \left( E \left[ \Vert \chi ^{n,\varepsilon }_n\Vert \, \Big | \, \mathcal {G}^{n,\varepsilon }_{n-1} \right] + \sum _{i=1}^{n-1} \Vert \chi ^{n,\varepsilon }_i \Vert \right) ~ 1_{\left\{ \sum _{i=1}^n E \left[ \Vert \chi ^{n,\varepsilon }_i \Vert \, \Big | \, \mathcal {G}^{n,\varepsilon }_{i-1} \right] \le \eta ' \right\} } \right] \\{} & {} \quad \le \frac{1}{\eta } E \left[ \eta ' + \sum _{i=1}^{n-1} \left( \Vert \chi ^{n,\varepsilon }_i \Vert - E \left[ \Vert \chi ^{n,\varepsilon }_i \Vert \, \Big | \, \mathcal {G}^{n,\varepsilon }_{i-1} \right] \right) 1_{\left\{ E \left[ \Vert \chi ^{n,\varepsilon }_i \Vert \, \Big | \, \mathcal {G}^{n,\varepsilon }_{i-1} \right] \le \eta ' \right\} } \right] < \frac{\eta '}{\eta }, \end{aligned}$$

we obtain

$$\begin{aligned} P \left( \left\| \sum _{i=1}^n \chi ^{n,\varepsilon }_i \right\|> \eta \right) \le \frac{\eta '}{\eta } + P \left( \sum _{i=1}^n E \left[ \Vert \chi ^{n,\varepsilon }_i \Vert \, \Big | \, \mathcal {G}^{n,\varepsilon }_{i-1} \right] > \eta ' \right) . \end{aligned}$$

Thus, the assertions (i) and (ii) follows. $\square$

Remark A.2

When $\mathcal {X}=\mathbb {R}$, this lemma can be shown by the same argument in the proof of Lemma 9 in Genon-Catalot and Jacod (1993). However, the argument does not work in general, since we may not have Lenglart’s inequality (e.g., Lemma 3.30 in Jacod and Shiryaev Jacod and Shiryaev (2003)) when $\mathcal {X}$ is a Banach space.

Remark A.3

We have an immediate consequence from this lemma that

$$\begin{aligned} \sum _{i=1}^n E \left[ \Vert \chi ^{n,\varepsilon }_i \Vert \, \Big | \, \mathcal {G}^{n,\varepsilon }_{i-1} \right] =o_p(r_{n,\varepsilon }) \quad&\Longrightarrow \quad \sum _{i=1}^n \chi ^{n,\varepsilon }_i = o_p(r_{n,\varepsilon }), \\ \sum _{i=1}^n E \left[ \Vert \chi ^{n,\varepsilon }_i \Vert \, \Big | \, \mathcal {G}^{n,\varepsilon }_{i-1} \right] =O_p(r_{n,\varepsilon }) \quad&\Longrightarrow \quad \sum _{i=1}^n \chi ^{n,\varepsilon }_i = O_p(r_{n,\varepsilon }), \end{aligned}$$

where $r_{n,\varepsilon }\in \mathbb {R}$.