Central Limit Theorems of Local Polynomial Threshold Estimators for Diffusion Processes with Jumps

TL;DR

This paper establishes central limit theorems for local polynomial threshold estimators of diffusion process volatility with jumps, addressing challenges in statistical inference due to the estimator's structure.

Contribution

It provides a novel proof for the CLTs of local polynomial threshold estimators, especially in the local linear case, for diffusion processes with jumps.

Findings

Proved CLTs for local polynomial threshold estimators with jumps.

Addressed the breakdown of classical CLTs due to estimator structure.

Introduced a new proof method for these estimators.

Abstract

Central limit theorems play an important role in the study of statistical inference for stochastic processes. However, when the nonparametric local polynomial threshold estimator, especially local linear case, is employed to estimate the diffusion coefficients of diffusion processes, the adaptive and predictable structure of the estimator conditionally on the $\sigma-$field generated by diffusion processes is destroyed, the classical central limit theorem for martingale difference sequences can not work. In this paper, we proved the central limit theorems of local polynomial threshold estimators for the volatility function in diffusion processes with jumps. We believe that our proof for local polynomial threshold estimators provides a new method in this fields, especially local linear case.

Equations

$$d{X}_t=\mu (X_t)dt+\sigma (X_t)d{W}_t$$

$$\underset{\delta \to 0}{\lim }E[\frac{(X_{t+\delta }-X_t{)}^2}{\delta }|X_t=x]={\sigma }^2(x),$$

$${\breve{\sigma }}_n^2(x)=\frac{{\sum }_{i=1}^{n}K(\frac{X_{t_{i-1}}-x}{h_n})\frac{(X_{t_i}-X_{t_{i-1}}{)}^2}{\delta }}{{\sum }_{i=1}^{n}K(\frac{X_{t_{i-1}}-x}{h_n})},$$

$$d{X}_t=\mu (X_{t-})dt+\sigma (X_{t-})d{W}_t+d{J}_t,$$

$${\bar{\sigma }}_n^2(x)=\frac{{\sum }_{i=1}^{n}K(\frac{X_{t_{i-1}}-x}{h_n})\frac{(X_{t_i}-X_{t_{i-1}}{)}^2}{\delta }{I}_{\{({\Delta }_{i}X{)}^2\le \vartheta (\delta )\}}}{{\sum }_{i=1}^{n}K(\frac{X_{t_{i-1}}-x}{h_n})}.$$

$$\mathbb{E}(Yf(Z_n))⟶\int Y(\omega )f(x)\eta (d\omega ,dx)$$

$$\mathbb{E}(Yf(Z_n))⟶\tilde{\mathbb{E}}(Yf(Z)),$$

$$Y_n\stackrel{P}{⟶}Y\Rightarrow (Y_n,Z_n)\stackrel{\mathcal{S}-\mathcal{L}}{⟶}(Y,Z),$$

$$\sup_{0\leq s\leq t}\big{|}\sum_{i=1}^{[s/\Delta_{n}]}\mathbb{E}\big{[}\zeta_{n,i}|\mathcal{F}_{(i-1)\Delta_{n}}\big{]}-B_{s}\big{|}\stackrel{{\scriptstyle p}}{{\longrightarrow}}0,$$

$$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sum_{i=1}^{[t/\Delta_{n}]}\big{(}\mathbb{E}\big{[}\zeta_{n,i}^{2}|\mathcal{F}_{(i-1)\Delta_{n}}\big{]}-\mathbb{E}^{2}\big{[}\zeta_{n,i}|\mathcal{F}_{(i-1)\Delta_{n}}\big{]}\big{)}-C_{t}\stackrel{{\scriptstyle p}}{{\longrightarrow}}0,$$

$$\sum_{i=1}^{[t/\Delta_{n}]}\mathbb{E}\big{[}\zeta_{n,i}^{4}|\mathcal{F}_{(i-1)\Delta_{n}}\big{]}\stackrel{{\scriptstyle p}}{{\longrightarrow}}0.$$

$$\sum_{i=1}^{[t/\Delta_{n}]}\mathbb{E}\big{[}\zeta_{n,i}\Delta_{n}^{i}H|\mathcal{F}_{(i-1)\Delta_{n}}\big{]}\stackrel{{\scriptstyle p}}{{\longrightarrow}}0,$$

$$\sum _{i=1}^{[t/{\Delta }_n]}{\zeta }_{n,i}\stackrel{\mathcal{S}-\mathcal{L}}{⟶}{B}_t+M_t,$$

$$\sum _{i=1}^{n}K\left(\frac{X_{t_{i-1}}-x}{h_n}\right)\frac{(X_{t_i}-X_{t_{i-1}}{)}^2}{\delta }{I}_{\{({\Delta }_{i}X{)}^2\le \vartheta (\delta )\}}$$

$$\sum _{i=1}^{n}K\left(\frac{X_{t_{i-1}}-x}{h}\right)\left\{\frac{\delta S_{n,2}}{h^2}-(\frac{X_{t_{i-1}}-x}{h})\frac{\delta S_{n,1}}{h}\right\}\frac{({\Delta }_{i}X{)}^2{I}_{\{({\Delta }_{i}X{)}^2\le \vartheta (\delta )\}}}{\delta }$$

$$d{X}_t=\mu (X_{t-})dt+\sigma (X_{t-})d{W}_t+d{J}_t,t\in [0,T]$$

$$J_{1t}={\int }_0^t{\int }_{\mathcal{R}}x\cdot m(dx,du)=\sum _{l=1}^{N_t}{\gamma }_l,$$

$${\tilde{J}}_{2s}:={\int }_0^s{\int }_{|x|\le 1}x[m(dt,dx)-\nu (dx)dt]$$

$$\alpha :=\inf \left\{\delta \ge 0:{\int }_{|x|\le 1}|x{|}^{\delta }\nu (dx)<+\infty \right\},$$

$$m(x)$$

$$\arg\min_{\beta_{0},\beta_{1},\cdots,\beta_{p}}\sum_{i=0}^{n-1}\Big{\{}Y_{t_{i}}-\sum_{j=0}^{p}\beta_{j}(X_{t_{i}}-x)^{j}\Big{\}}^{2}K_{h_{n}}(X_{t_{i}}-x),$$

$$\hat{\beta }:=({\hat{\beta }}_0,{\hat{\beta }}_1,\cdot \cdot \cdot ,{\hat{\beta }}_p{)}^T=S_n^{-1}{Q}_n,$$

$$S_n=\left(\begin{array}{ccc}{S}_{n,0}& \cdot \cdot \cdot & S_{n,p}\\ \cdot \cdot \cdot & & \cdot \cdot \cdot \\ S_{n,p}& \cdot \cdot \cdot & S_{n,2p}\end{array}\right),Q_n=\left(\begin{array}{c}{Q}_{n,0}\\ \cdot \cdot \cdot \\ Q_{n,p}\end{array}\right),$$

$$S_{n,k}=\frac{1}{h}\sum _{i=1}^{n}K\left(\frac{X_{i\delta }-x}{h}\right){\left(X_{i\delta }-x\right)}^k$$

$$Q_{n,k}=\frac{1}{h}\sum _{i=1}^{n}K\left(\frac{X_{i\delta }-x}{h}\right){\left(X_{i\delta }-x\right)}^k\frac{(X_{t_{i+1}}-X_{t_i}{)}^2}{\delta }{I}_{\{({\Delta }_{i}X{)}^2\le \vartheta (\delta )\}}.$$

$${\hat{\sigma }}_n^2(x)=\frac{{\sum }_{i=1}^{n}K(\frac{X_{t_{i-1}}-x}{h})\{\frac{\delta S_{n,2}}{h^2}-(\frac{X_{t_{i-1}}-x}{h})\frac{\delta S_{n,1}}{h}\}\frac{({\Delta }_{i}X{)}^2{I}_{\{({\Delta }_{i}X{)}^2\le \vartheta (\delta )\}}}{\delta }}{{\sum }_{i=1}^{n}K(\frac{X_{t_{i-1}}-x}{h})\{\frac{\delta S_{n,2}}{h^2}-(\frac{X_{t_{i-1}}-x}{h})\frac{\delta S_{n,1}}{h}\}}$$

$${\hat{\sigma }}_n^2(x{)}^{Num}$$

$$\frac{{\hat{\sigma }}_n^2(x{)}^{Num}}{L_X(T,x)}$$

$$|\mu (x)-\mu (y)|+|\sigma (x)-\sigma (y)|\le L_n|x-y|,$$

$$|\mu (x)|+|\sigma (x)|\le C(1+|x|),$$

Full text

62M10,

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\startlocaldefs\endlocaldefs

t1This research work is support by the National Natural Science Foundation of China (No. 11371317, 11171303) and the Fundamental Research Fund of Shandong University (No. 2016GN019). t2Corresponding author

1 Introduction

Volatility is an important feature of financial markets, which is directly related to market uncertainty and risk. It is not only an effective indicator of quality and efficiency for financial market, but also a core variable for portfolio theory, asset price modeling, arbitrage price modeling and option price formula. Hence, how to effectively describe the dynamic behavior of volatility for financial market has always been the core problem.

Estimating the price volatility of financial assets exactly is fundamental for the financial risk management, which has long been the focus of the theoretical study and empirical application such as risk management, asset pricing, proprietary trading and portfolio managements. In finance, “tick data” are recorded at every transaction time (sampled at very high frequency), so we do get huge amounts of data on the prices or return rates of various assets and so on. In this context, it gives a new challenge to study the estimators for the process, which characterizes the prices or returns of various assets and so on. With the development of financial statistical methods, using real-time transaction data to estimate asset return volatility has become a hot topic. In high frequency context, more and more statisticians and economists are interested in the nonparametric inference for diffusion coefficients of stochastic processes which characterize the dynamics of option prices, interest rates, exchange rates and inter alia.

In this paper, we assume that all processes are defined on a filtered probability space $(\Omega,(\mathscr{F}_{t})_{t\in[0,T]},\mathscr{F},P)$, satisfying the usual conditions (Jacod and Shiryaev [18]). A diffusion processes can be represented by the solution of following stochastic differential equation:

[TABLE]

where $W_{t}$ is a standard Brownian motion. Diffusion processes play an important role in the study of mathematical financial. Especially, many models in economics and finance, like those for an interest rate or an asset price, involve diffusion processes. Assuming that the process (1) is observed at $n+1$ discrete time observations $\{X_{0},X_{t_{1}},...,X_{t_{n-1}},X_{t_{n}}\}$ with $t_{i}-t_{i-1}=\delta$,  for $i=1,...,n$ and $t_{n}=T$, based on the infinitesimal conditional moment restriction

[TABLE]

the nonparametric estimators for volatility function $\sigma^{2}(x)$ can be constructed by the nonparametric regression method. The Nadaraya-Watson estimators is a natural choice, we can estimate $\sigma^{2}(x)$ through

[TABLE]

where $K(\cdot)$ is a kernel function, and $h_{n}$ is bandwidth. Bandi and Phillips [7] obtained the central limit theorems for $\breve{\sigma}^{2}_{n}(x)-\sigma^{2}(x)$ by Knight’s embedding theorem ( Revuz and Yor [27]). There are many methods to improve the statistical behaviors of $\breve{\sigma}^{2}_{n}(x)$. Local polynomial smoothing method is a popular method for improving Nadaraya-Watson estimator. Fan and Zhang ([14]) first estimated $\sigma^{2}(x)$ through local polynomial method, however, the asymptotic normality was not obtained by them (they only computed the bias and variance for the estimator of the diffusion coefficient).

Recently, diffusion processes with jumps as an intension of continuous-time ones have been studied by more and more statisticians and economists since the financial phenomena can be better characterized (see Johannes [19], Aït-Sahalia and Jacod [1], Bandi and Nguyen [6]). It is natural to consider the following model:

[TABLE]

where $J_{t}$ is a pure jump semimartingale. The jumps $J_{t}=J_{1t}+\tilde{J}_{2t}$ consist of large and infrequent jump component $J_{1t}$ (finite activity) as well as small and frequent jump component with finite variation $\tilde{J}_{2t}$ (infinite activity). Ordinarily, $J_{1t}$ is assumed compound Poisson processes and $\tilde{J}_{2t}$ is assumed to be Lévy. One can refer to Bandi and Nguyen [6] for doubly stochastic compound Poisson process, Madan [20] for Variance Gamma process, Carr et al. [9] for the CGMY model with $Y<1$, Cont and Tandov [11] for $\alpha-$stable or tempered stable process with $\alpha<1$. Disentangling the jump component from observations is essential for risk management, one can refer to Andersen et al. [4] and Corsi et al. [10]. In presence of jump component $J_{1t},$ Bandi and Nguyen [6], Johannes [19] constructed nonparametric estimation for $\sigma^{2}(x)$ based on estimation of infinitesimal moments and provided central limiting theory.

Under the influence of jumps, especially $\tilde{J}_{2t}$, how to estimate $\sigma^{2}(x)$ is an interesting problem. For finite activity case, Mancini [22] showed that due to the continuity modulus of the Brownian motion paths, it is possible to disentangle in which intervals jumps occur when the interval between two observations shrinks for $T<\infty$. This property allows one to identify the jump component asymptotically and remove it from X. Mancini and Renò [23] showed that this methodology is robust to enlarging time span ($T\rightarrow\infty$) and to the presence of infinite activity jumps $\tilde{J}_{2t}$. For more knowledge of this aspect, one also can refer to Barndorff-Nielsen and Shephard [8], Mancini [21], Aït-Sahalia and Jacod [1] for alternative approaches to disentangle jumps from diffusion based on power and multipower variation.

Mancini and Renò [23] combined the Nadaraya-Watson estimator and threshold method to eliminate the impact of jumps. They estimated $\sigma^{2}(x)$ through

[TABLE]

In the context of nonparametric estimator with finite-dimensional auxiliary variables, local polynomial smoothing become the “golden standard”, see Fan [12], Wand and Jones [28]. The local polynomial estimator is known to share the simplicity and consistency of the kernel estimators such as Nadaraya-Watson or Gasser-Müller estimators. Moreover, when the convergence rates are concerned, local polynomial estimator possesses simple bias representation and corrects the boundary bias automatically. However, when the nonparametric local polynomial threshold estimator is employed to estimate volatility function $\sigma^{2}(x)$ for better bias properties instead of Nadaraya Watson estimator, the adaptive and predictable structure of estimator is destroyed, so the classical central limit theorem for martingale difference sequences can not work. In this paper, we will discuss this problem and prove the central limit theorem for local linear threshold estimator for the diffusion coefficient $\sigma^{2}(x).$

The remainder of this paper is organized as follows. Stable convergence and its property is shown in section 2. Section 3 introduces our model, local polynomial threshold estimator and main results. The proofs of the results will be collected in section 4.

2 Stable convergence and its property

In this section, firstly, we will define the stable convergence in law and mention its property, secondly, we will show limit theorem for partial sums of triangular arrays of random variables, one can refer to Jacod and Shiryaev [18] or Jacod [17] for more details.

1) Stable convergence in law.

This notation was firstly introduced by Rényi [26], which in the same reason we need here for the proof, and exposited by Aldous and Eagleson [3].

A sequence of random variables $Z_{n}$ defined on the probability space $(\Omega,\mathcal{F},\mathbb{P}),$ taking their values in the state space $(E,\mathcal{E}),$ assumed to be Polish. We say that $Z_{n}$ stably converges in law if there is a probability measure $\eta$ on the product $(\Omega\times E,\mathcal{F}\times\mathcal{E}),$ such that $\eta(A\times E)=\mathbb{P}(A)$ for all $A\in\mathcal{F}$ and

[TABLE]

for all bounded continuous functions $f$ on $E$ and bounded random variables $Y$ on $(\Omega,\mathcal{F}).$

Take $\tilde{\Omega}=\Omega\times E$, $\tilde{\mathcal{F}}=\mathcal{F}\times\mathcal{E}$ and endow $(\tilde{\Omega},\tilde{\mathcal{F}})$ with the probability $\eta,$ and put $Z(\omega,x)=x,$ on the extension $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$ of $({\Omega},{\mathcal{F}},{\mathbb{P}})$ with the expectation $\tilde{\mathbb{E}}$ we have

[TABLE]

then we say that $Z_{n}$ converges stably to $Z,$ denoted by $\stackrel{{\scriptstyle\mathcal{S}-\mathcal{L}}}{{\longrightarrow}}.$

The stable convergence implies the following crucial property, which is fundamental for the mixed normal distribution with random variance of the local polynomial estimator, detailed in the proof of Theorem 1 and 2.

$\bf{Lemma~{}2.1.}$  if $Z_{n}\stackrel{{\scriptstyle\mathcal{S}-\mathcal{L}}}{{\longrightarrow}}Z$ and if $Y_{n}$ and $Y$ are variables defined on $(\Omega,\mathcal{F},\mathbb{P})$ and with values in the same Polish space F, then

[TABLE]

which implies that $Y_{n}\times Z_{n}\stackrel{{\scriptstyle\mathcal{S}-\mathcal{L}}}{{\longrightarrow}}Y\times Z$ through the continuous function $g(x,y)=x*y.$

2) Convergence of triangular arrays.

In this part, we give the available convergence criteria for stable convergence of partial sums of triangular arrays, one can refer to Jacod [17] (P17-Lemma 4.4).

$\bf{Lemma~{}2.2.}$  [Jacod’s stable convergence theorem] A sequence of $\mathbb{R}-$valued variables $(\zeta_{n,i}:i\geq 1)$ defined on the filtered probability space $(\Omega,\mathcal{F},(\mathcal{F})_{t\geq 0},\mathbb{P})$ is $\mathcal{F}_{i\Delta_{n}}-$measurable for all $n,i.$ Assume there exists a continuous adapted $\mathbb{R}-$valued process of finite variation $B_{t}$ and a continuous adapted and increasing process $C_{t}$, for any $t>0,$ we have

[TABLE]

[TABLE]

[TABLE]

Assume also

[TABLE]

where either H is one of the components of Wiener process $W$ or is any bounded martingale orthogonal (in the martingale sense) to $W$ and $\Delta_{n}^{i}H=H_{i\Delta_{n}}-H_{(i-1)\Delta_{n}}.$

Then the processes

[TABLE]

where $M_{t}$ is a continuous process defined on an extension \big{(}\widetilde{\Omega},\widetilde{P},\widetilde{\mathcal{F}}\big{)} of the filtered probability space \big{(}{\Omega},{P},{\mathcal{F}}\big{)} and which, conditionally on the the $\sigma-$filter $\mathcal{F}$, is a centered Gaussian $\mathbb{R}-$valued process with \widetilde{E}\big{[}M_{t}^{2}|\mathcal{F}\big{]}=C_{t}.

Remark 2.1. As Jacod [17] mentioned that the key assumption of Lemma 2.2 is that for all $n,~{}i$ the variable $\zeta_{n,i}$ is $\mathcal{F}_{i\Delta_{n}}-$measurable. For Nadaraya-Watson estimator, the triangular arrays of numerator in (5):

[TABLE]

is $\mathcal{F}_{i\Delta_{n}}-$measurable, so Mancini and Renò [23] can employ Lemma 2.2 to prove the stable convergence for numerator of $\bar{\sigma}^{2}_{n}(x).$ However, for local linear estimator, the triangular arrays of numerator in (15):

[TABLE]

with $S_{n,k}=\frac{1}{h}\sum_{i=1}^{n}K\left(\frac{X_{i\delta}-x}{h}\right)\left(X_{i\delta}-x\right)^{k}$, for $k=1,2$ is not $\mathcal{F}_{i\Delta_{n}}-$measurable due to $S_{n,k}$, so we can not directly employ Lemma 2.2 to show the stable convergence for it. Fortunately, we could deal with the problem under some techniques with the help of Lemma 2.1, one can refer to the third part or the detailed proof for some understanding the methodology.

3 Setting and Main results

Recall that a diffusion process with jumps can be defined by the following stochastic differential equation (4):

[TABLE]

where $\mu(x)$ and $\sigma(x)$ are smooth functions, $W=\{W_{t},t\geq 0\}$ is a standard Brownian motion, where $J_{t}$ is a pure jump semimartingale. The jumps $J_{t}=J_{1t}+\tilde{J}_{2t}$ consist of large and infrequent jump component $J_{1t}$ (finite activity) as well as small and frequent jump component with finite variation $\tilde{J}_{2t}$ (infinite activity). $J_{1t}$ is a finite activity (FA) pure jump semimartingale (e.g. driven by a doubly stochastic compound Poisson process with jump intensity $\lambda(\cdot)$ in $L^{1}(\Omega\times[0,+\infty))$), independent of $\{W_{t},t\geq 0\}$.

Generally, since $J_{1t}$ is any FA pure jump semimartingale ,which we can write as

[TABLE]

where $m$ is the jump random measure of $J_{1t}$, the jump intensity $\lambda(\cdot)$ is a stochastic process, and $N_{t}:=\int_{0}^{t}\int_{\mathscr{R}}{1\cdot m(dx,du)}$ is a.s. finite.

$\tilde{J}_{2t}$ is assumed to be a pure jump Lévy process of type

[TABLE]

with $\nu\{|x|\leq 1\}=+\infty,$ where $\nu$ is the Lévy measure of $\tilde{J}_{2}.$ Cont and Tandov [11] discussed the $Blumenthal-Getoor~{}index$ for any Lévy process:

[TABLE]

which measure how frenetic the jump activity. Here we only consider the case $\alpha<1$, which implies $\tilde{J}_{2}$ has finite variation, that is, $\sum_{s\leq T}\Delta\tilde{J}_{2s}<\infty.$ In this case, Protter [25] showed that there exists the local time $L_{t}(x)$, which is continuous in $t$ and càdlàg in $x$, and the occupation time formula keeps true.

As a nonparametric methodology, the local polynomial estimator has received increasing attention and become a powerful and useful diagnostic tool for data analysis making use of the observation information to estimate corresponding functions and its derivatives without assuming the function form. The estimator is obtained by locally fitting $p$-th polynomial to the data via weighted least squares. The procedure of weighted local polynomial regression is conducted as follows: under some smoothness conditions of the curve $m(x)$, we can expand $m(x)$ in a neighborhood of the point $x_{0}$ as follows:

[TABLE]

where $\beta_{j}=\frac{m^{(j)}(x_{0})}{j~{}!}.$

Thus, the problem of estimating infinite dimensional $m(x)$ is equivalent to estimating the $p$-dimensional parameter $\beta_{0},\beta_{0},\cdots,\beta_{p}.$

When we want to estimate $\sigma^{2}(x)$ in model (4) from the discrete time observations $\{X_{0},X_{t_{1}},...,X_{t_{n-1}},X_{t_{n}}\}$, with $t_{i}-t_{i-1}=\delta$, we can consider a weighted local polynomial regression through the threshold method to eliminate the impact of jumps:

[TABLE]

where $Y_{t_{i}}=\frac{(X_{t_{i+1}}-X_{t_{i}})^{2}}{\delta}I_{\{(\Delta_{i}X)^{2}\leq\vartheta(\delta)\}}$ and $K_{h_{n}}(\cdot)=\frac{1}{h_{n}}K(\frac{\cdot}{h_{n}})$ is kernel function with $h_{n}$ the bandwidth, $\vartheta(\delta)$ is a threshold function.

Under the algebra calculus (one can refer to Fan and Gijbels [13]), we obtain the solution to this minimization problem (13) is

[TABLE]

with

[TABLE]

where

[TABLE]

and

[TABLE]

As Fan [12] showed, since this methodology is mainly conducted by means of locally fitting $p$-th polynomial, the degree is not allowed higher, usually $\tau+1$ and rarely $\tau+3,$ where $\tau$ is the degree of unknown function we need to estimate in $\hat{\beta}.$ What we are interested in estimating $\sigma^{2}(x)$ is $\hat{\beta}_{0},$ that is $\tau=0,$ it is reasonable for us to discuss $p=1$ in this paper, which is the local linear estimator.

In fact, we can write the solutions $\hat{\beta}_{0}$ of (13) with $p=1$ for (14), that is,

[TABLE]

where $\Delta_{i}X=X_{t_{i}}-X_{t_{i-1}}$, $S_{n,k}=\frac{1}{h}\sum_{i=1}^{n}K(\frac{X_{i\delta}-x}{h})(X_{i\delta}-x)^{k}$,  for $k=1,2$.

In fact, there are many papers on local linear estimator in regression analysis and time series analysis, more details can be found in Fan and Gijbels [13]. The primary purpose of the present paper is to establish central limit theorems for $\hat{\sigma}^{2}_{n}(x)-\sigma^{2}(x).$

The triangular arrays of numerator of local linear estimator in (15) is

[TABLE]

We have shown in remark 2.1 that we can not directly employ Lemma 2.2 to show the stable convergence for the numerator. With the help of lemma 4.3, we obtain $\frac{\delta S_{n,2}}{h^{2}}\stackrel{{\scriptstyle\mathrm{a.s.}}}{{\longrightarrow}}~{}\frac{K_{1}^{2}L_{X}(T,x)}{\sigma^{2}(x)}$ and $\frac{\delta S_{n,1}}{h}\stackrel{{\scriptstyle\mathrm{a.s.}}}{{\longrightarrow}}~{}\frac{K_{1}^{1}L_{X}(T,x)}{\sigma^{2}(x)}.$ Hence,

[TABLE]

Obviously, the triangular arrays $\zeta_{n,i}$ is $\mathcal{F}_{i\Delta_{n}}-$measurable, so we can utilize lemma 2.2 to prove the stable convergence in law for $\sum_{i=1}^{n}\zeta_{n,i}.$ From lemma 4.3, we know that \frac{1}{h}\sum_{i=1}^{n}K\big{(}\frac{X_{t_{i-1}}-x}{h}\big{)}\delta~{}\stackrel{{\scriptstyle\mathrm{a.s.}}}{{\longrightarrow}}~{}\frac{L_{X}(T,x)}{\sigma^{2}(x)}, which implies that we can prove the stable convergence in law for $\hat{\sigma}^{2}_{n}(x)^{Num}$ by means of the property as lemma 2.1, more details can be sketched in the proof of Theorem 1.

Assume that $\mathfrak{D}=(l,u)$ with $-\infty\leq l<u\leq\infty$ is the range of the process $X$. We will use notation “$\stackrel{{\scriptstyle p}}{{\rightarrow}}$” to denote “convergence in probability”, “$\stackrel{{\scriptstyle a.s.}}{{\rightarrow}}$” to denote “convergence almost surely”, “$\Rightarrow$” to denote “convergence in distribution” and “$\stackrel{{\scriptstyle\mathcal{S}-\mathcal{L}}}{{\longrightarrow}}$” to denote “stable convergence in law”. We impose the following assumptions throughout the paper.

Assumption 1**.**

For model (4), the coefficients $\mu_{t}$ and $\sigma_{t}$ are progressively measurable process with càdlàg paths and the following polynomial growth:

(i) For each $n\in\mathbb{N},$ there exists a positive constant $L_{n}$ such that for any $|x|\leq n,|y|\leq n$,

[TABLE]

(ii)There exists a positive constant C, such that for all $x\in\mathbb{R}$,

[TABLE]

(iii) $\sigma^{2}$ is strictly positive and $\sigma^{{}^{\prime}}$ is bounded.

Remark 3.1**.**

This assumption (i) and (ii) guarantees the existence and uniqueness of a strong solution to $X$ in Eq.(4) on our filtered probability space

$(\Omega,(\mathscr{F}_{t})_{t\in[0,T]},\mathscr{F},P)$, which is adapted with càdlàg paths on $[0,T]$, see Ikeda and Watanabe **[16]** for more details.

Assumption 2**.**

The kernel function K $(\cdot):\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is a continuous differentiable and bounded density function with bounded compact support, such that $\int_{0}^{\infty}K(u)du=1$ and $\int_{0}^{\infty}{|K^{{}^{\prime}}(u)|^{2}}du<\infty$. Denote $K_{i}^{j}=\int_{0}^{\infty}{u^{j}K^{i}(u)}du<\infty.$

Remark 3.2**.**

The one-sided and asymmetric kernel function is mentioned in assumption 4 (ii) of Bandi and Nguyen [6]. Fan and Zhang [14] proposed that the one-sided kernel function will make prediction easier, such as the Epanechnikov kernel $K(u)=\frac{3}{4}(1-u^{2})I_{(u<0)}.$

Assumption 3**.**

A bandwidth parameter is a sequence of real number $h_{n}$ such that as $n\longrightarrow\infty,$ we have $h_{n}\longrightarrow 0,$ $\frac{\delta{ln(\frac{1}{\delta})}}{h^{2}}\rightarrow 0.$

For model (4), under the assumptions, we build the corresponding theorems of local linear threshold estimators (15) for different jump cases.

$\bf{Finite~{}Activity~{}Jumps~{}(FA~{}case)}:$

In (4), if we assume that $J_{t}=J_{1,t}=\sum_{l=1}^{N_{t}}\gamma_{l}$, where $N_{t}$ is a doubly stochastic Poisson process with an intensity process $\lambda(X_{t-})$, we have the following result.

Theorem 1**.**

*Under Assumptions 1, 2, 3 and we also assume that

(1) as $\delta\rightarrow 0$ both the threshold function $\vartheta(\delta)$ and $\frac{\delta ln(\frac{1}{\delta})}{\vartheta(\delta)}$ tend to 0;

(2) $\frac{h_{n}^{5}}{\delta_{n,T}}=O_{p}(1),$ then we can obtain*

[TABLE]

where $\delta_{n,T}=\frac{T}{n}$, V_{x}=\frac{K_{2}^{0}\cdot\big{(}K_{1}^{2}\big{)}^{2}+K_{2}^{2}\cdot\big{(}K_{1}^{1}\big{)}^{2}-2K_{2}^{1}\cdot K_{1}^{2}\cdot K_{1}^{1}}{(K_{1}^{2}-(K_{1}^{1})^{2})^{2}} and $M\mathscr{N}(0,U^{2})$ is a random variable having a mixed normal law with the characteristic function $\phi(u)=E[e^{-\frac{U^{2}u^{2}}{2}}].$

Corollary 1. Under Assumptions 1, 2, 3 and we also assume that

(1) as $\delta\rightarrow 0$ both the threshold function $\vartheta(\delta)$ and $\frac{\delta ln(\frac{1}{\delta})}{\vartheta(\delta)}$ tend to 0;

(2) $\frac{h_{n}^{5}\hat{L}_{X}(T,x)}{\delta_{n,T}}=O_{p}(1),$ then we can obtain

[TABLE]

where \hat{L}_{X}(T,x)=\frac{1}{h}\sum_{i=1}^{n}K\big{(}\frac{X_{t_{i-1}}-x}{h}\big{)}\delta.

Remark 3.3**.**

According to Lemma 4.3, we know that $\hat{L}_{X}(T,x)\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}\frac{L_{X}(T,x)}{\sigma^{2}(x)},$ so we can deduce corollary 1 by means of lemma 2.1 easily with the property that the stable convergence implies convergence in distribution.

$\bf{Infinite~{}Activity~{}Jumps~{}(IA~{}case)}:$

Furthermore, if we assume $J_{t}=J_{1t}+\tilde{J}_{2t}$, we have the following result

Theorem 2**.**

*Under Assumptions 1, 2, 3 and we also assume that:

(1) $\alpha<1$ and $\int_{|x|\leq\varepsilon}x^{2}\nu(dx)=O(\varepsilon^{2-\alpha}),$ as $\varepsilon\rightarrow 0$;

(2) $h=\delta^{\phi}$ with $\frac{h_{n}^{5}}{\delta_{n,T}}=O_{p}(1),$ $\vartheta(\delta)=\delta^{\eta},\eta\in(0,1)$, with $\eta/2>\phi,~{}(1-\alpha\eta)-1/2+\phi/2>0$ and $\eta(1-\alpha/2)-1/2+\phi/2>0,$ then we can obtain*

[TABLE]

Corollary 2. Under Assumptions 1, 2, 3 and we also assume that:

(1) $\alpha<1$ and $\int_{|x|\leq\varepsilon}x^{2}\nu(dx)=O(\varepsilon^{2-\alpha}),$ as $\varepsilon\rightarrow 0$;

(2) $h=\delta^{\phi}$ with $\frac{h_{n}^{5}\hat{L}_{X}(T,x)}{\delta_{n,T}}=O_{p}(1),$ $\vartheta(\delta)=\delta^{\eta},\eta\in(0,1)$, with $\eta/2>\phi,~{}(1-\alpha\eta)-1/2+\phi/2>0$ and $\eta(1-\alpha/2)-1/2+\phi/2>0,$ then we can obtain

[TABLE]

Remark 3.4**.**

For the local polynomial estimator (13) of order $p$ with $\hat{\beta}_{0}=\hat{\sigma}^{2}_{n}(x)$, under Assumptions 1, 2, 3 and some mild conditions for the bandwidth $h_{n}$ and the threshold function $\vartheta(\delta)$, we can obtain

[TABLE]

where $(\sigma^{2})^{(p+1)}$ denotes the $(p+1)$th derivative of $\sigma^{2}$, $\beta=(\sigma^{2},(\sigma^{2})^{(1)},\cdot\cdot\cdot,(\sigma^{2})^{(p)}/p!)^{T},$ $H=diag(1,h_{n},\cdot\cdot\cdot,h_{n}^{p}),$ $S=(K^{i+j-2}_{1})$ and $S^{*}=(K^{i+j-2}_{2})$ for $(1\leq i\leq p+1;1\leq j\leq p+1),$ $c_{p}=(K^{p+1}_{1},\cdot\cdot\cdot,K^{2p+1}_{1})^{T}.$

Remark 3.5**.**

In Mancini and Renò [23], they only considered the case of fixed time span $~{}T=1$ with $\delta_{n,T}=\frac{1}{n}$ in Theorem 3.2 and 4.1, and the convergence rate was $\sqrt{nh_{n}}$ for the stable convergence in law and $\sqrt{nh_{n}\hat{L}_{X}(T,x)}$ for convergence in distribution. Bandi and Phillips [7] studied the limiting distribution of the diffusion estimator in model (1) for the case of time span $T\longrightarrow\infty$ with $\delta_{n,T}=\frac{T}{n}$ in Theorem 5, and the convergence rate was $\sqrt{\frac{h_{n}\hat{L}_{X}(T,x)}{\delta_{n,T}}}$ for convergence in distribution. In this paper, under two-dimensional asymptotics in both the time span $T\longrightarrow\infty$ and the sampling interval $\delta_{n,T}=\frac{T}{n}\longrightarrow 0,$ we derive the local nonparametric estimator of the diffusion functions for nonstationary model (1) with convergence rate of $\sqrt{\frac{h_{n}\hat{L}_{X}(T,x)}{\delta_{n,T}}}$ for convergence in distribution. We extend the result of Bandi and Phillips [7] to the diffusion with jumps model (4), especially, the infinite activity jumps. Meanwhile, we extend the result of Mancini and Renò [23] in third directions: first, showing the local polynomial approach to reduce the finite sample bias, which also extends the result in Moloche [24] to the diffusion with jumps, second, considering two-dimensional asymptotics in both the time span $T\longrightarrow\infty$ and the sampling interval $\delta_{n,T}=\frac{T}{n}\longrightarrow 0,$ third, posing weak conditions to the bandwidth parameter $h_{n}$ not allowing for $nh_{n}^{3}\longrightarrow 0$, which results in the precise bias representation for the estimator of diffusion function.

Remark 3.6**.**

If posing weak conditions to the bandwidth parameter $h_{n}$ not allowing for $nh_{n}^{3}\longrightarrow 0,$ the bias is $h_{n}\cdot(\sigma^{2})^{{}^{\prime}}K_{1}^{1}$ for asymmetric kernels, or $h_{n}^{2}\cdot((\sigma^{2})^{{}^{\prime}}\frac{s^{{}^{\prime}}(x)}{s(x)}+\frac{1}{2}(\sigma^{2})^{{}^{\prime\prime}})K_{1}^{2}$ for symmetric kernels in Mancini and Renò [23], where $s(x)$ is the natural scale funcion, while the bias is $\frac{1}{2}\frac{(\sigma^{2})^{{}^{\prime\prime}}(x)[(K_{1}^{2})^{2}-K_{1}^{1}K_{1}^{3}]}{K_{1}^{2}-(K_{1}^{1})^{2}}\cdot h^{2}$ in this paper with the asymmetric kernel. Hence, the bias in the local linear case is smaller than the one in the Nadaraya-Watson case in comparison to the results between this paper and Mancini and Renò [23] whether the kernel function $K(\cdot)$ is symmetric or not.

Remark 3.7**.**

It is very important to consider the choice of the bandwidth $h_{n}$ for the nonparametric estimation. There are many rules of thumb on selecting the bandwidth, one can refer to Bandi, Corradi and Moloche [5], Fan and Gijbels [13], Aït-Sahalia and Park [2]. Here it would be nice to calculate the optimal bandwidth based on the mean square error (MSE). The optimal bandwidth of local threshold nonparametric estimator for model ([om]) based corollary 1 or 2 is given

[TABLE]

In contrary to Bandi and Nguyen [6], they pointed out if $h_{n}^{5}\hat{L}_{X}(T,x)=O_{a.s.}(1)$, then the features of the nonrandom bias term imply an asymptotic mean-squared error of order $h_{n}^{4}+\frac{1}{h_{n}\hat{L}_{X}(T,x)}$ and, in consequence, optimal bandwidth sequences of order

[TABLE]

for $\sigma^{2}(x)+\lambda(x)\mathbb{E}_{Y}[c^{2}(x,y)]$ (P297, equation (13)) in diffusion model with compound Poisson finite activity jumps. Hence, the optimal bandwidth in our paper converges to zero faster than that in Bandi and Nguyen [6] for diffusion function. To the best of our knowledge, the optimal bandwidth are not yet derived in the context of local threshold nonparametric inference for diffusion with jumps, especially infinite jumps.

Remark 3.8**.**

Compared with Hanif [15], this paper considers the local threshold nonparametric estimation for the diffusion function $\sigma^{2}(x)$ by disentangling jumps from the observations. It provides a new method to estimate the components of quadratic variation separately, especially the volatility contributed by the Brownian part. With the techniques of lemma 2.1 and 4.3, we deal with the adaptive and predictable structure of the local nonparametric threshold estimator conditionally on the $\sigma-$field generated by diffusion processes, so the lemma 2.2 of stable convergence in law can be utilized for the estimators. To some extend, the results for the diffusion with finite and infinite activity jumps in Theorem 1 and 2 effectively solve the conjecture for discontinuous variations proposed in the conclusion part of Ye et al. [29], the two-step estimation procedure of the volatility function in which is a part of (21) in this paper.

4 The proof of main results

We recall that $\delta=\frac{T}{n},t_{i}=i\delta,X=Y+J,~{}dY_{t}=\mu(X_{t})dt+\sigma(X_{t})dW_{t}.$  Denote for an integer $l$,

$\Delta_{i}\hat{Y}=\Delta_{i}XI_{\{(\Delta_{i}X)^{2}\leq\vartheta(\delta)\}},~{}\Delta_{i}\hat{J}=\Delta_{i}XI_{\{(\Delta_{i}X)^{2}>\vartheta(\delta)\}},$

$\Delta_{i}\hat{N}=I_{\{(\Delta_{i}X)^{2}>\vartheta(\delta)\}},\qquad K_{j}^{k}(u)=K^{j}(u)*u^{k}.$

For any bounded process Z we denote by $\bar{Z}=\sup_{u\in[0,T]}|Z_{u}|.$ Throughout this article, we use $C$ to denote a generic constant, which may vary from line to line. By $\sigma\cdot W$ we denote the stochastic integral of $\sigma$ with respect to $W$. We denote by \big{(}\tau_{j}\big{)}_{j\in\mathbb{N}} the jump instants of $J_{t}$ and by $\tau^{(i)}$ the instant of the first jump in $(t_{i-1},t_{i}]$, if $\Delta_{i}N\geq 1$.

Before proving our results, we first present some lemmas.

Lemma 4.1**.**

(Mancini and Renò [23]) Assume that $J_{t}=\sum_{l=1}^{N_{t}}\gamma_{l}$, where $N_{t}$ is a doubly stochastic Poisson process with an intensity process $\lambda(X_{t-})$. If $\lambda(\cdot)$ is bounded, then uniformly for all $i=1,\cdots,n$,

[TABLE]

[TABLE]

Lemma 4.2**.**

(Mancini and Renò [23]) Define $\prod^{(n)}=\{\frac{i}{n},i=1,\cdot\cdot\cdot,n\}$ the partitions of [0, 1] on which the sums are constructed. There exists a subsequence $\prod^{(n_{k})}=\{\frac{i}{n_{k}},i=1,\cdot\cdot\cdot,n_{k}\},$ with $\delta_{k}=\frac{1}{n_{k}},$ such that a.s. for sufficiently small $\delta_{k},$ for all $i=1,\cdot\cdot\cdot,n_{k},$ on the set $\{(\Delta_{i}\tilde{J_{2}})^{2}\leq\vartheta(\delta_{k})\},$ we have

[TABLE]

Lemma 4.3**.**

Under Assumptions 1, 2, 3, we have

[TABLE]

where $K_{1}^{k}L_{X}(T,x)~{}=~{}L_{X}(T,x)\int_{R_{+}}K(u)u^{k}du$ for all $x$, as $\delta~{}and~{}h\to 0.$

Proof.

For simplicity, we set $T=1$. Write

[TABLE]

By the occupation time formula,

[TABLE]

which converges to $\frac{K_{1}^{k}L_{X}(T,x)}{\sigma^{2}(x)}$ almost surly.

For each $n$, we define the random sets

[TABLE]

and

[TABLE]

Then

[TABLE]

Noticing that $K(\cdot)$ is bounded supported, \frac{1}{h}\sum_{i\in I_{1,n}}\int_{t_{i-1}}^{t_{i}}K\big{(}\frac{X_{s-}-x}{h}\big{)}\big{(}\frac{X_{s-}-x}{h}\big{)}^{k}ds is dominated by $N_{1}\frac{2C\delta}{h}\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}0$.

\frac{1}{h}\sum_{i\in I_{1,n}}\int_{t_{i-1}}^{t_{i}}{\Big{(}K\big{(}\frac{X_{t_{i-1}}-x}{h}\big{)}\cdot\big{(}\frac{X_{t_{i-1}}-x}{h}\big{)}^{k}}ds can be written using the mean-value theorem, and it is a.s. dominated by

[TABLE]

where $\tilde{X}_{is}$ is some point between $X_{i\delta}$ and $X_{s}$ for $i\in I_{0,n}$. Using the property of uniform boundedness of the increments of X paths when J $\equiv$ 0 (indicated as the UBI property), (17) can be a.s. dominated by

[TABLE]

Since \frac{1}{h}\sum_{i\in I_{1,n}}\int_{t_{i-1}}^{t_{i}}{\Big{|}{K_{1}^{k}}^{{}^{\prime}}\big{(}\frac{\tilde{X}_{is}-x}{h}\big{)}\Big{|}\frac{(\delta ln{\frac{1}{\delta}})^{\frac{1}{2}}}{h}}ds\leq CN_{1}\frac{(\delta ln{\frac{1}{\delta}})^{\frac{1}{2}}}{h}\frac{\delta}{h}\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}0, so (18) has the same limit as

[TABLE]

The inequality follows from (using the Cauchy-Schwarz inequality and $K(\cdot)$ bounded support denoted as $[0,C]$)

[TABLE]

Thus

[TABLE]

We obtain this lemma. ∎

Lemma 4.4**.**

Under Assumptions 1, 2, 3 we have

[TABLE]

as $\delta\rightarrow 0$.

Proof.

Here, we consider the case of $J_{t}=J_{1t}+\tilde{J}_{2t}.$

On $\{(\Delta_{i}X)^{2}\leq\vartheta(\delta_{k})\},$ we have $|\Delta_{i}J|-|\Delta_{i}Y|\leq|\Delta_{i}X|\leq\sqrt{\vartheta(\delta_{k})},$ and by UBI property of $Y_{t}$, for small $\delta_{k},~{}|\Delta_{i}J|\leq 2\sqrt{\vartheta(\delta_{k})},$

[TABLE]

However, by the bounded support of $K(\cdot)$ and the UBI property,

[TABLE]

where $N_{1}$ denotes the number of jumps in [0, 1], thus,

[TABLE]

It is sufficient to prove

[TABLE]

which can be similarly proved using lemma 4.2 as the technical details for Lemma 3 in Mancini and Renò ([23]) with $K(\cdot)(\cdot)^{k}$ instead of $K(\cdot)$. ∎

4.1 The proof of Theorem 1

Set $T=1$ and $\sigma(X_{s})=:\sigma_{s},\sigma(X_{(i-1)\delta})=:\sigma_{i-1}$. Theorem 1 in Mancini ([22]) means that $I_{\{(\Delta_{i}X)^{2}\leq\vartheta(\delta)\}}(\omega)=I_{\{\Delta_{i}N=0\}}(\omega)$), then

[TABLE]

Similar to the proof of Lemma 4.3, last term is

[TABLE]

By Jacod’s stable convergence theorem with the help of lemmas 2.1 and 4.3, we first show that the numerator of

[TABLE]

converges stably in law to $M_{1}$ with the asymptotic bias \frac{1}{2}(\sigma^{2})^{{}^{\prime\prime}}(x)[(K_{1}^{2})^{2}-K_{1}^{1}K_{1}^{3}]\cdot\Big{(}\frac{L_{X}(T,x)}{\sigma^{2}(x)}\Big{)}^{2}\cdot h^{2}, where $M_{1}$ is a Gaussian martingale defined on an extension \big{(}\tilde{\Omega},\tilde{P},\tilde{\mathscr{F}}\big{)} of our filtered probability space and having $\tilde{E}[M_{1}^{2}|\mathscr{F}]=\frac{2}{\sigma^{2}(x)}\cdot L_{X}^{3}(T,x)\cdot V_{x}^{{}^{\prime}}$ with V_{x}^{{}^{\prime}}=K_{2}^{0}\cdot\big{(}K_{1}^{2}\big{)}^{2}+K_{2}^{2}\cdot\big{(}K_{1}^{1}\big{)}^{2}-2K_{2}^{1}\cdot K_{1}^{2}\cdot K_{1}^{1}.

Using the Itô formula on $(\Delta_{i}Y)^{2}$, we have

[TABLE]

First Step: the stable convergence in law for the numerator of the estimator.

For the term $\sum_{i=1}^{n}q_{i}.$ Divide $q_{i}$ by $L_{X}(T,x)$ as $q_{i}^{{}^{\prime}}.$ For simplicity in the detailed proof, denote

[TABLE]

So there exists an integer $m$ such that $K^{\divideontimes}_{i-1}=K^{\star}_{i-1}+o_{\mathrm{a.s.}}(1)$ when $n>m$. In the following proof, we will substitute $K(\frac{X_{t_{i-1}}-x}{h})\cdot\frac{K_{1}^{2}}{\sigma^{2}(x)}-K(\frac{X_{t_{i-1}}-x}{h})(\frac{X_{t_{i-1}}-x}{h})\cdot\frac{K_{1}^{1}}{\sigma^{2}(x)}$ for $K^{\star}_{i-1}$ for the sample sizes $n>m$ and assume the sample sizes $n>m$.

In fact,

[TABLE]

where $K^{\star}_{i-1}$ is measurable with respect to the $\sigma$-algebra generated by $\{X_{u},0\leq u\leq t_{i-1}\}.$

Jacod’s stable convergence theorem tell us that the following arguments,

[TABLE]

implies $\sum_{i=1}^{n}q_{i}^{{}^{\prime}}\stackrel{{\scriptstyle st}}{{\rightarrow}}M_{1}$, where either $H=W$ or $H$ is any bounded martingale orthogonal (in the martingale sense) to $W$ and $E_{i-1}[~{}\cdot~{}]=E[~{}\cdot~{}|X_{t_{i-1}}].$ Remark that $\mu$ is assumed to be càdlàg, therefore we know that it is locally bounded on $[0,T]$. By localizing, we can assume that $\mu$ is a.s. bounded on $[0,T].$

For $S_{1}$,

[TABLE]

by the measurability $K^{\star}_{i-1}$ with respect to $\sigma\{X_{u},0\leq u\leq t_{i-1}\}$ in the second equation, the martingale property of stochastic integral in the third equation, the UBI property in the second inequation, the expression of $K^{\star}_{i-1}$, the assumption 3.3 and the boundness of $\mu$ in the third inequation.

For $S_{2}$,

[TABLE]

By Hölder and Burkholder-Davis-Gundy inequality, $S_{2,3}$ is larger than the others (which has the lowest infinitesimal order). Here we only deal with the dominant one, others are neglected. For $S_{2,3}$, it consists of three terms by an expansion of ${(Y_{s}-Y_{(i-1)\delta})^{2}}$, of which we only need to consider the lowest infinitesimal order one. Due to Hölder and Burkholder-Davis-Gundy inequality again, it is sufficient to prove the convergence in probability of

[TABLE]

To show it, we can prove the following five arguments:

(D1) \frac{n}{h}\sum_{i=1}^{n}{K_{i-1}^{\star}}^{2}\int_{t_{i-1}}^{t_{i}}{E_{i-1}\big{[}\big{(}\int_{(i-1)\delta}^{s}\sigma_{u}dW_{u}\big{)}^{2}(\sigma_{s}^{2}-\sigma_{i-1}^{2})}\big{]}ds\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}0;

(D2) \frac{1}{h}\sum_{i=1}^{n}{K_{i-1}^{\star}}^{2}\int_{t_{i-1}}^{t_{i}}{\big{(}nE_{i-1}\big{[}\int_{(i-1)\delta}^{s}\sigma_{u}^{2}du\big{]}\sigma_{i-1}^{2}-\frac{\sigma_{i-1}^{4}}{2}}\big{)}ds\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}0;

(D3) $\frac{1}{h}\sum_{i=1}^{n}{K_{i-1}^{\star}}^{2}\int_{t_{i-1}}^{t_{i}}(\sigma_{i-1}^{4}-\sigma_{s}^{4})\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}0;$

(D4) \frac{1}{h}\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\big{(}{K_{i-1}^{\star}}^{2}\frac{\sigma_{s}^{4}}{2}-{K_{s}^{\star}}^{2}\frac{\sigma_{s}^{4}}{2}\big{)}ds\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}0;

(D5) \frac{4}{h}\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}\big{(}{K_{s}^{\star}}^{2}\frac{\sigma_{s}^{4}}{2}\big{)}ds\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}\frac{2}{\sigma^{2}(x)}\cdot L_{X}(T,x)\big{[}K_{2}^{0}\cdot\big{(}K_{1}^{2}\big{)}^{2}+K_{2}^{2}\cdot\big{(}K_{1}^{1}\big{)}^{2}-2K_{2}^{1}\cdot K_{1}^{2}\cdot K_{1}^{1}\big{]}.

$\bf{For~{}~{}D1:}$

Applying the mean-value theorem for $\sigma^{2}$, neglecting the terms with $i\in I_{1,n}$ similarly as that in Proposition 3.1 and bounding $|X_{s}-X_{(i-1)\delta}|$ by the UBI property when $i\in I_{0,n}$, for the sum in (D1) we can reach

[TABLE]

$\bf{For~{}~{}D2:}$

By the Taylor expansion,

[TABLE]

For the sum in (D2) we can obtain

[TABLE]

$\bf{For~{}~{}D3:}$

Neglecting the terms with $i\in I_{1,n}$ proceeding as Lemma 4.3, (D3) is a.s. dominated by

[TABLE]

$\bf{For~{}~{}D4:}$

Since $\sigma_{s}^{4}$ is bounded almost surely, similar to the proof of Lemma 4.3 and $\bf{D1}$, $\bf{D4}$ is obtained.

$\bf{For~{}~{}D5:}$

Using the occupation time formula, we obtain

[TABLE]

For $S_{3}$, let us come back to the proof of $S_{3}$. Using BDG and Hölder inequalities, we have

[TABLE]

For $S_{4}$,

Set $\Delta_{i}Z:=\int_{t_{i-1}}^{t_{i}}(Y_{s}-Y_{i-1})dY_{s}.$ If $H=W,$ then

[TABLE]

by using the Hölder inequality.

If H is orthogonal to W, then

[TABLE]

provided the boundness of H such that $\Delta_{i}H\leq C.$

Second Step: the asymptotic bias for the numerator of the estimator.

We now prove the following three results for

[TABLE]

that is,

[TABLE]

where

[TABLE]

Firstly,

[TABLE]

By the Taylor expansion for $\sigma^{2}_{i-1}-\sigma^{2}(x)$ in $A_{2_{n,T}}$ up to order 2,

[TABLE]

where $\theta$ is a random variable satisfying $\theta\in[0,1].$

For $A_{2_{n,T}}$ , by Lemma 4.3,

[TABLE]

Furthermore, we use the mean-value theorem to $\sigma^{2}_{s}-\sigma^{2}_{i-1}$ for $A_{1_{n,T}},$ then

[TABLE]

by the UBI property of $i\in I_{0,n}$.

Result about $B_{1_{n,T}}$ can be obtained using $K(u)\cdot u$ instead of $K(u)$ similarly as $A_{1_{n,T}}.$

Under a simple calculus,

[TABLE]

For $C_{n,T}$ using the Taylor expansion for $\sigma^{2}_{i-1}-\sigma^{2}(x)$ up to order 2, we have

[TABLE]

Similarly as the proof of Lemma 4.3, we obtain

[TABLE]

[TABLE]

so we have

[TABLE]

We complete the proof for Theorem 1.

4.2 The proof of Theorem 2.

It proceeds basically along the same idea as the detailed procedure of Lemma 4.3, which gives the result for $X_{t}$ with finite activity jumps (FA case). As is shown in the proof of Lemma 4.3, it is sufficient to prove

[TABLE]

for $X_{t}$ with finite and infinite activity jumps (IA case). Hence, we only need to check that the contribution for

[TABLE]

given by the IA jumps is negligible in the following part based on the result of Lemma 4.3 for FA case.

According to the assumption of $\alpha<1,$ which means $J$ has finite variation, we can obtain

[TABLE]

Denote $X_{0,t}=\int_{0}^{t}(\mu_{s}+C)ds+\int_{0}^{t}\sigma_{s}dW_{s},$ we can split $X_{t}=X_{0,t}+J_{1,t}+J_{2,t}.$ For $i\in I_{1,n}=\{i\in\{1,2,\cdot\cdot\cdot n\}:\Delta_{i}N\neq 0\},$ where $N$ is the counting process with respect to $J_{1,t},$ we have

[TABLE]

Therefore, we fix $J_{1}$ and regard K\big{(}\frac{X_{s-}-x}{h}\big{)}\big{(}\frac{X_{s-}-x}{h}\big{)}^{k} as a two variable function F(a,b):=K\Big{(}\frac{a+J_{1s}+b-x}{h}\Big{)}\Big{(}\frac{a+J_{1s}+b-x}{h}\Big{)}^{k} evaluated at $a=X_{0,s}$ and $b=J_{2,s}.$

For a function $F(a,b)$ with two variables, by the Taylor expansion, we have

[TABLE]

where $F_{a}$ denotes the first partial derivative of the function $F.$

Using the expansion equation for K\big{(}\frac{X_{t_{i-1}}-x}{h}\big{)}\cdot\big{(}\frac{X_{t_{i-1}}-x}{h}\big{)}^{k}-{{K\big{(}\frac{X_{s-}-x}{h}\big{)}}\big{(}\frac{X_{s-}-x}{h}\big{)}^{k}} around $(a_{0},b_{0})=(X_{0s},J_{2s})$ with $(a,b)=(X_{0,t_{i-1}}J_{2,t_{i-1}})$ and $F(\cdot)=K(\cdot)(\cdot)^{k}$, we reach

[TABLE]

where $\tilde{X}_{0s},\tilde{J}_{2s}$ are the suitable points to give the Lagrange remainder for the Taylor expansion and $F_{a}$ denotes the derivative for $K(\cdot)(\cdot)^{k}$ with respect to the first variable for simplicity.

According to the Taylor expansion, we have

[TABLE]

We now show that the five terms give a negligible contribution.

For $\Pi_{1_{n,T}}$.

Using the UBI property of $X_{0}$ and the occupation time formula, we get

[TABLE]

For $\Pi_{2_{n,T}}$.

Using the UBI property of $X_{0}$, the boundedness of $F_{ab}(\cdot,\cdot)$ and $E[|\tilde{J}_{2,s}-\tilde{J}_{2,t_{i-1}}|^{2}]=O(\delta)$ with Hölder inequality for $s\in[t_{i-1},t_{i}]$ (one can refer to this equation in the Proof of Theorem 4 for Mancini and Renò [23]), we have

[TABLE]

hence, it is shown that $\Pi_{2_{n,T}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}0.$

For $\Pi_{3_{n,T}}$.

Using the UBI property of $X_{0}$ and the occupation time formula, we have

[TABLE]

For $\Pi_{4_{n,T}}$.

Using the UBI property of $X_{0}$, the boundedness of $F_{aab}(\cdot,\cdot)$ and $E[|\tilde{J}_{2,s}-\tilde{J}_{2,t_{i-1}}|^{2}]=O(\delta)$ with Hölder inequality for $s\in[t_{i-1},t_{i}]$, it can be shown

[TABLE]

so we have prove $\Pi_{4_{n,T}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}0.$

For $\Pi_{5_{n,T}}$.

Using the boundedness of $F_{b}(\cdot,\cdot)$ and $E[|\tilde{J}_{2,s}-\tilde{J}_{2,t_{i-1}}|^{2}]=O(\delta)$ with Hölder inequality for $s\in[t_{i-1},t_{i}]$, we can prove

[TABLE]

so we have $\Pi_{5_{n,T}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}0.$

From the above five parts, we get

[TABLE]

so we have

[TABLE]

with the result \frac{1}{h}\int_{0}^{1}{{K\big{(}\frac{X_{s-}-x}{h}\big{)}}\big{(}\frac{X_{s-}-x}{h}\big{)}^{k}}ds\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}\frac{K_{1}^{k}L_{X}(T,x)}{\sigma^{2}(x)} in the detailed proof of Lemma 4.3

For the consistency and asymptotic normality for $\hat{\sigma}^{2}_{n}(x),$ we follow the same procedures as that in the detailed proof of Theorem 1, it is sufficient to check that the contribution given by the IA jumps $\tilde{J}_{2}$ is negligible at each step based on the result of Theorem 1 for FA case. Write

[TABLE]

To prove the result it is sufficient to show that the numerator tends stably in law to random variable $M_{1}.$

Recall the following fact, if $Z_{n}\stackrel{{\scriptstyle\mathcal{S}-\mathcal{L}}}{{\longrightarrow}}Z$ and if $Y_{n}$ and $Y$ are variables defined on $(\Omega,\mathcal{F},\mathbb{P})$ and with values in the same Polish space F, then

[TABLE]

which implies that $Y_{n}+Z_{n}\stackrel{{\scriptstyle\mathcal{S}-\mathcal{L}}}{{\longrightarrow}}Y+Z.$ Hence, we need to prove that

[TABLE]

For (4.2), we have

[TABLE]

From the detailed proof for $X_{t}$ with finite activity jump (FA case) in Theorem 1, we have shown that

[TABLE]

where $M_{1}$ is a Gaussian martingale defined on an extension \big{(}\tilde{\Omega},\tilde{P},\tilde{\mathscr{F}}\big{)} of our filtered probability space and having $\tilde{E}[M_{1}^{2}|\mathscr{F}]=\frac{2}{\sigma^{2}(x)}\cdot L_{X}^{3}(T,x)\cdot V_{x}^{{}^{\prime}}$ with V_{x}=K_{2}^{0}\cdot\big{(}K_{1}^{2}\big{)}^{2}+K_{2}^{2}\cdot\big{(}K_{1}^{1}\big{)}^{2}-2K_{2}^{1}\cdot K_{1}^{2}\cdot K_{1}^{1}.

In the following part, we will verify the fact that $\mathcal{S}_{1}$ converges stably to $M_{1}$, the other terms tend to zero in probability for $X_{t}$ with infinite activity jump (IA case) similarly as the result in Theorem 1, that is, the contribution given by infinite activity jumps can be negligible.

For $\mathcal{S}_{1}$.

It consists of two terms

[TABLE]

For the first term $\sum_{i=1}^{n}q_{i}=L_{T}(T,x)\sum_{i=1}^{n}q_{i}^{{}^{\prime}}$, $\sum_{i=1}^{n}q_{i}^{{}^{\prime}}$ is composed of four parts such as $S_{1},~{}S_{2},~{}S_{3},~{}S_{4}$.

$S_{1},~{}S_{3},~{}S_{4}$ these three parts can be dealt as the FA jumps case in Theorem 1. For $S_{2},$ in $D1,~{}D3,$ we similarly expand $\sigma^{2}_{s},~{}\sigma^{4}_{s}$ up to the first order respectively. Using the Hölder, BDG inequality and the IA jump component contribution with $E[|\tilde{J}_{2,s}-\tilde{J}_{2,t_{i-1}}|^{2}]=O(\delta)$, we can obtain the convergence of $D1,~{}D3$ to 0 in probability. $D2,~{}D5$ can be proved with the similar procedure as that in Theorem 1 using the occupation time formula. $D3$ can be dealt by the similar steps.

For $\mathcal{S}_{2}$.

The sum of the terms with $\Delta_{i}N\neq 0$ is $O_{p}(\sqrt{\frac{\delta}{h_{n}}})\rightarrow 0$ by Lemma 4.2. For the sum of the terms with $(\Delta_{i}\tilde{J_{2}})^{2}>4\vartheta(\delta_{k}),$ we consider

[TABLE]

For $\mathcal{S}_{2}^{{}^{\prime}}$, we have

[TABLE]

using the Hölder inequality with $q$ close to 1, the BDG inequality. Hence we prove the convergence of $\mathcal{S}_{2}$ to 0 in probability.

For $\mathcal{S}_{3}$,

We consider

[TABLE]

For $\mathcal{S}_{3}^{{}^{\prime}}$, we have

[TABLE]

using Hölder inequality with $q$ close to 1 andthe BDG inequality. Hence, we prove the convergence of $\mathcal{S}_{3}$ to 0 in probability.

For $\mathcal{S}_{4}$, We consider

[TABLE]

For $\mathcal{S}_{4}^{{}^{\prime}}$, we have

[TABLE]

Hence we prove the convergence of $\mathcal{S}_{4}$ to 0 in probability.

We complete the proof for Theorem 2.