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.
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1 Introduction
This paper is concerned with the following stochastic differential equation (SDE):
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
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
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
Notation 2.2
A function \(\psi \) on \(\mathbb {R}\times \mathbb {R}\times \Theta _3\) is of the form
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
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
Assumption 2.6
The familiy \(\left\{ {c(\cdot ,\alpha )}\right\} _{\sigma \in \bar{\Theta }_3}\) satisfies
with some positve constants \(c_1\) and \(c_2\). In this paper, without loss of generality, we may assume
Assumption 2.7
If \(\mu \ne \mu _0\), \(\sigma \ne \sigma _0\) or \(\alpha \ne \alpha _0\), then
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
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 \(\alpha \in \Theta _3\)
are continuous at every points in \(I_{x_0}\), and there exist \(\delta >0\) and \(C>0\) such that
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 \(I_{x_0}^\delta \).
-
(ii.a)
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
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 \(\alpha \in \Theta \), \(i=1,\dots ,d_3\)
are continuous at every points \(x\in I_{x_0}\), and there exist \(\delta >0\) such that
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 \(I_{x_0}^\delta \).
-
(ii.a)
Remark 2.1
Instead of Assumptions 2.5 and 2.6, the following stronger assumptions are often used:
(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,
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
where \(\varepsilon >0\).
Notation 2.4
Denote
where \(n\in \mathbb {N}\), \(\varepsilon >0\).
Notation 2.5
Define random times
Notation 2.6
Define events \(J^{n,\varepsilon }_{k,i}\) \((k=1,\dots ,n,~i=0,1,2)\) by
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
where \(n\in \mathbb {N}\), \(\varepsilon >0\), \(\rho \in (0,1/2)\). Then, put
where \(n\in \mathbb {N}\), \(\varepsilon >0\), \(\rho \in (0,1/2)\). Furthermore, for sufficiently small \(\delta >0\), we may put
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
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
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):
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:
with
Then, the quasi-maximum likelihood estimator is given by
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:
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,
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
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
and
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 \(p\ge 2\),
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
where C depends only on p, a, b, c and \(f_{\alpha _0}\).
Proof
For any \(p\ge 2\), we have
where C depends only on p. Then, it follows from the Lipschitz continuity of \(a(\cdot ,\mu _0)\) that
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
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
where C depends only on p and c. Here, we have
where C depends only on p and \(f_{\alpha _0}\), and
where C depends only on p and \(f_{\alpha _0}\). Thus,
where C depends only on p, c and \(f_{\alpha _0}\). By using the Burkholder-Davis-Gundy inequality,
where C depends only on p and \(f_{\alpha _0}\). From (4.1), (4.2), (4.3), (4.4) and (4.5),
where C depends only on p, a, b, c and \(f_{\alpha _0}\). By Gronwall’s inequality,
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 \(p\ge 2\)
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
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 \(p\ge 1\)
as \(\varepsilon \rightarrow 0\), \(\lambda _\varepsilon \rightarrow \infty \) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\), and
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,
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 \(p\in [1,\infty )\),
where C depends only on p, a and b, and
where C depends only on p, a, b, c and \(f_{\alpha _0}\).
Proof
For \(t\in [t_{k-1},\tau _{k})\) and \(p\ge 2\),
where C depnds only on p, a and b. By using Gronwall’s inequality, we obtain
where C depnds only on p, a and b. Similarly,
where C depnds only on p, a and b. From Lemma 4.1, we have
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
Then, for any \(p\in (2,\infty )\),
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
as \(n\rightarrow \infty \), \(\varepsilon \rightarrow 0\), \(\lambda _\varepsilon \rightarrow \infty \) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\). It follows from Lemma A.3 that
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 \(\rho \in (0,1/2)\)
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
Also, it follows from
and Lemmas 4.2, 4.5 and A.2 that
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 \(\rho \in (0,1/2)\)
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 \(p\in [2/(1-2\rho ),\infty )\)
as \(n\rightarrow \infty \), \(\varepsilon \rightarrow 0\), \(\lambda _\varepsilon \rightarrow \infty \) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\).
Proof
Since
it follows from Lemmas 4.4 and 4.7 that for \(p\ge 2\) and \(\rho \in (0,1/2)\)
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
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 \(\rho \in (0,1/2)\)
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 \(p\in [2,\infty )\)
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
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 \(p\in [0,\infty )\)
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
as \(\varepsilon \rightarrow 0\),
as \(n\rightarrow \infty \) and \(\varepsilon \rightarrow 0\), uniformly in \(\theta \in \Theta \), since for any \(\eta >0\)
Similarly, from Lemma 4.8, when
as \(\varepsilon \rightarrow 0\),
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
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
as \(\varepsilon \rightarrow 0\), then
as \(n\rightarrow \infty \) and \(\varepsilon \rightarrow 0\), uniformly in \(\theta \in \Theta \).
Proof
Since from Lemma 4.3
as \(n\rightarrow \infty \), \(\varepsilon \rightarrow 0\) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\), for any \(\eta >0\)
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
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,
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 \(p\in [2,\infty )\)
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
For any \(\eta >0\)
It follows from Lemmas 4.3, 4.4 and 4.9 that
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
$$\begin{aligned} r_{n,\varepsilon } := \frac{1}{\varepsilon n^{1-1/p}} + \frac{1}{n^{1/2-1/p}}. \end{aligned}$$
-
(i.a)
-
(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
$$\begin{aligned} r_{n,\varepsilon } := \frac{1}{\varepsilon n^{1-1/p-q\rho }} + \frac{1}{n^{1/2-1/p-q\rho }}. \end{aligned}$$
-
(ii.a)
Then,
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
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
where \(c_1\) and \(c_2\) are the constants from Assumption 2.6. In this proof, we may simply write the maps
and we denote the following event by \(\tilde{D}^{n,\varepsilon ,\rho }_{k,1}\)
Since
under either of the assumptions (i.c) or (ii.e), we obtain from Lemma 4.10 that for any non-random \(r_{n,\varepsilon }'>0\) (\(n\in \mathbb {N}, \varepsilon >0\)),
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
as \(n\rightarrow \infty \), \(\varepsilon \rightarrow 0\), \(\lambda _\varepsilon \rightarrow \infty \) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\), uniformly in \(\theta \in \Theta \).
Put
By using Taylor’s theorem under either of the assumptions (i.a) or (ii.a), we have
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
as \(n\rightarrow \infty \), \(\varepsilon \rightarrow 0\), \(\lambda _\varepsilon \rightarrow \infty \) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\). Indeed, for any \(M>0\)
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
we obtain from the assumption (ii.b) that
Since
it follows from Lemma A.3 (ii), Lemmas 4.4 and 4.6 and the assumption (ii.d) that
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
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
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
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 \(\zeta \in [0,1]\)
and that on \(\tilde{D}^{n,\varepsilon ,\rho }_{k,1}\)
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 \(\theta \in \Theta \)
are continuous at every points in \(I_{x_0}\), and that there exist \(\delta >0\), \(C>0\) and \(q\ge 0\) such that
Then,
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 \(\theta \in \Theta \)
as \(n\rightarrow \infty \), \(\varepsilon \rightarrow 0\), \(\lambda _\varepsilon \rightarrow \infty \) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\), and that
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 \(\theta \in \Theta \)
as \(n\rightarrow \infty \), \(\varepsilon \rightarrow 0\), \(\lambda _\varepsilon \rightarrow \infty \) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\). Put
Then, by the same argument in the proof of Lemma 4.10, it follows from Lemma 4.3 that
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 \(\theta \in \Theta \)
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
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)Footnote 1. From the assumption (114.11), we obtain
and
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
where C is defined as (114.11) and for \(q\ge 1\)
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,
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
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:
as \(n\rightarrow \infty \), \(\varepsilon \rightarrow 0\), \(\lambda _\varepsilon \rightarrow \infty \) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\).
At second, we show that
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
it holds from Morrey’s inequality (see, e.g., Theorem 5 in Evans (2010), Section 5.6) that for \(q\in (d,\infty )\)
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
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
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
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 \(q'\in (1,d/(d-1))\)
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 \(q\in (d,\infty )\)
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
where \(C_2\) depends only on q. By using Lemmas 4.4 and 4.7, for any \(p>2\) we obtain
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 \(j=1,\dots ,d\)
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
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,
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
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
Similarly to the proof of (4.12), it follows that
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
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
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
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
as \(n\rightarrow \infty \), \(\varepsilon \rightarrow 0\), \(\lambda _\varepsilon \rightarrow \infty \) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\), and
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
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
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
is bounded in probability, it follows from Lemmas 4.1, 4.8 and 4.9 and the linearity of b that
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
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
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
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:
and
Lemma 4.16
Under Assumptions 2.1 to 2.6, 2.8 and 2.10, the following convergences are holds.
For \(\ell =1,2\)
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,
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 \(\rho \in (0,1/2)\)
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,
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
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
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
and by the same argument as in the proof of Lemma 4.15, we obtain
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
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,
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
it holds from Lemmas 4.4 and 4.7 that for any \(p\ge 1\)
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
it follows from Lemmas 4.4 and 4.7 that for any \(p\ge 1\)
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
The last equality holds from the fact that
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,
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
as \(n\rightarrow \infty \), \(\varepsilon \rightarrow 0\), \(\lambda _\varepsilon \rightarrow \infty \) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\), we obtain
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 \(p>1\)
as \(n\rightarrow \infty \), \(\varepsilon \rightarrow 0\), \(\lambda _\varepsilon \rightarrow \infty \) and \(\varepsilon \lambda _\varepsilon \rightarrow 0\), we obtain from Lemma 4.4 that
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
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
it follows from Lemmas 4.4 and 4.7 that for any \(p\ge 1\)
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,
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,
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
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
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
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,
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
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,
where \(\varphi (x,y,\alpha ) := \exp \psi (x,y,\alpha )\). Since
where
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.,
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 \((x,y,\alpha )\in I_{x_0}^\delta \times D_+\times \Theta _3\),
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
Then,
Since
we have
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
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
for \(\alpha \in \Theta _3\). Then,
Since
for \(z>0\) and \(\alpha \in \Theta _3\), we have
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
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
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
for \(\alpha \in \Theta _3\). Then,
Since
for \(z>0\) and \(\alpha \in \Theta _3\), we have
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
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
for \(\alpha \in \Theta _3\). Then,
Since
for \(z>0\) and \(\alpha \in \Theta _3\), we have
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
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
and that there exists a constant \(C>0\) such that
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
for \(\alpha \in \Theta _3\). Then,
Since
for \(z>0\) and \(\alpha \in \Theta _3\), we have
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
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
and that there exists a constant \(C>0\) such that
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
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
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
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.
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.
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Acknowledgements
This work is partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) #21K03358 and JST CREST #PMJCR14D7, Japan. Also, the authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.
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A Appendix
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.,
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
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
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
that is,
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.,
then
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
and therefore,
This implies the conclusion. \(\square \)
Remark A.1
By the proof of Lemma A.2, it also follows that for any \(C_1>0\),
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 \(\eta ,\eta '>0\)
we obtain
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
where \(r_{n,\varepsilon }\in \mathbb {R}\).
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Kobayashi, M., Shimizu, Y. Threshold estimation for jump-diffusions under small noise asymptotics. Stat Inference Stoch Process 26, 361–411 (2023). https://doi.org/10.1007/s11203-023-09286-y
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DOI: https://doi.org/10.1007/s11203-023-09286-y