Realized Laplace transforms for pure jump semimartingales with presence of microstructure noise

Abstract

This paper considers the estimation of integrated Laplace transform of local ‘volatility’ by using noisy high-frequency data. We allow for the presence of microstructure noise under a pure jump semimartingale over a fixed time interval [0, t]. We propose an efficient estimator for the integrated Laplace transform of volatility via applying the pre-averaging method. Under some mild conditions on the Lévy density, the asymptotic properties of the estimator including consistency and asymptotic normality are established. Simulation studies further confirm our theoretical results.

Appendix: Proofs

In the following, C denotes a generic constant, we let $t_i^n:=i\Delta _n$, $g_j^n:=g\left( \frac{j}{k_n}\right)$, $E_i^n[\cdot ]:=E[\cdot |\mathcal {F}_{i\Delta _n}]$.

Proof of Theorem 1

First step, we use Taylor expansion to separate latent process $X_t$ and noise $\varepsilon _t$, then select a suitable $k_n$ to weaken the noise effect.

$$\begin{aligned}&\cos \left( (2u)^{\frac{1}{\beta }}\left( \phi _\beta ^n k_n\Delta _n\right) ^{-\frac{1}{\beta }}\Delta _i^n\overline{U}\right) \\&\quad = \cos \left( (2u)^{\frac{1}{\beta }}\left( \phi _\beta ^n k_n\Delta _n\right) ^{-\frac{1}{\beta }}\Delta _i^n\overline{X}\right) \\&\qquad -\,\sin \left( (2u)^{\frac{1}{\beta }}\left( \phi _\beta ^n k_n\Delta _n\right) ^{-\frac{1}{\beta }}\Delta _i^n\overline{X}\right) \frac{(2u)^{\frac{1}{\beta }}}{\left( \phi _\beta ^n k_n\Delta _n\right) ^{\frac{1}{\beta }}}\Delta _i^n\overline{\varepsilon }\\&\qquad -\,\frac{1}{2}\cos (x^*)\left( (2u)^{\frac{1}{\beta }}\left( \phi _\beta ^n k_n\Delta _n\right) ^{-\frac{1}{\beta }}\Delta _i^n\overline{\varepsilon }\right) ^2\\&\quad :=\cos \left( (2u)^{\frac{1}{\beta }}\left( \phi _\beta ^n k_n\Delta _n\right) ^{-\frac{1}{\beta }}\Delta _i^n\overline{X}\right) +R_i^{(1)}+R_i^{(2)}. \end{aligned}$$

For noise $\varepsilon$, from Assumption 4, we have $E(\varepsilon _t)=0$, $\mathrm {Var}(\varepsilon _t)=\omega _t^2\le C$ and $\varepsilon _t$ are independent,

$$\begin{aligned} \Delta _i^n\overline{\varepsilon }= & {} \sum _{j=1}^{k_n-1}g_j^n(\varepsilon _{i+j}-\varepsilon _{i+j-1})\\= & {} \sum _{j=0}^{k_n-1}\left( g_j^n-g_{j+1}^n\right) \varepsilon _{i+j}. \end{aligned}$$

When $0<r\le 2$, applying Hölder inequality, we have

$$\begin{aligned} E\left( \left| \Delta _i^n \overline{\varepsilon }\right| ^r\right) \le Ck_n^{-\frac{r}{2}}. \end{aligned}$$

Then it can be easily shown that

$$\begin{aligned}&E\left[ \left( \sum _{i=0}^{n-k_n+1}\Delta _nR_i^{(1)}\right) ^2\right] \\&\quad \le \Delta _n^2k_n\sum _{i=0}^{n-k_n+1}E\left[ \sin ^2\left( (2u)^{\frac{1}{\beta }}\left( \phi _{\beta }^nk_n\Delta _n\right) ^{-\frac{1}{\beta }}\Delta _i^n\overline{X}\right) \right. \\&\qquad \left. \cdot (2u)^{\frac{2}{\beta }}\left( \phi _{\beta }^nk_n\Delta _n\right) ^{-\frac{2}{\beta }}\left( \Delta _i^n\overline{\varepsilon }\right) ^2\right] \\&\quad \le Ck_n^{-\frac{2}{\beta }}\Delta _n^{1-\frac{2}{\beta }},\\&E\left[ \left| \sum _{i=0}^{n-k_n+1}\Delta _nR_i^{(2)}\right| \right] \le Ck_n^{-\frac{2}{\beta }-1}\Delta _n^{-\frac{2}{\beta }}. \end{aligned}$$

To ensure the effect of noise ignorable, we need $k_n=O(n^{\frac{2}{\beta +2}+\tau }),0<\tau <\frac{\beta }{\beta +2}$.

Note that $X_t$ has an integral form representation:

$$\begin{aligned} X_t = X_0+\int _0^t {\alpha }_s\mathrm {d}s+\int _0^t\int _\mathbb {R}\sigma _{s}x\tilde{\mu }(\mathrm {d}s,\mathrm {d}x) +Y_t \end{aligned}$$

where $\mu$ is homogenous Poisson measure with compensator $\nu (\mathrm {d}x) = \frac{A}{|x|^{\beta +1}}\mathrm {d}x$ and $\tilde{\mu } = \mu -\nu$, according to the Lévy density $\nu (x)=\frac{A}{|x|^{\beta +1}}$. We define $L_t = \int _0^t\int _{\mathbb {R}}x \tilde{\mu }(\mathrm {d}s, \mathrm {d}x)$. Next we make a decomposition as follows:

$$\begin{aligned}&V_T(U,\Delta _n, \beta , u, g) - \int _{0}^{T}e^{-u|\sigma _s|^\beta } ds \nonumber \\&\quad = \sum _{i=0}^{n-k_n+1} \Delta _n \cos \left( (2u)^{\frac{1}{\beta }}\left( \phi _\beta ^n\Delta _n k_n\right) ^{-\frac{1}{\beta }}\Delta _i^n\overline{X}\right) \nonumber \\&\qquad - \int _{0}^{T}e^{-u|\sigma _t|^\beta } dt + \sum _{i=0}^{n-k_n+1}\Delta _n\left( R_i^{(1)}+R_i^{(2)}\right) \nonumber \\&\quad = \sum _{i=0}^{n-k_n+1} \Delta _n\left[ \cos \left( \frac{(2u)^{\frac{1}{\beta }}}{\left( \phi _\beta ^n\Delta _n k_n\right) ^{\frac{1}{\beta }}}\sigma _{t_i^n}\Delta _i^n\overline{L}\right) - e^{-u|\sigma _{t_i^n}|^{\beta }}\right] \nonumber \\&\qquad + \sum _{i=0}^{n-k_n+1} \int _{t_i^n}^{t_{i+1}^n} \left( e^{-u|\sigma _{t_i^n}|^\beta }-e^{-u|\sigma _{s}|^\beta }\right) ds \nonumber \\&\qquad + \sum _{i=0}^{n-k_n+1} \Delta _n \left[ \cos \left( (2u)^{\frac{1}{\beta }}\left( \phi _\beta ^n\Delta _n k_n\right) ^{-\frac{1}{\beta }}\Delta _i^n\overline{X}\right) \right. \nonumber \\&\left. \qquad - \cos \left( (2u)^{\frac{1}{\beta }}\left( \phi _\beta ^n\Delta _n k_n\right) ^{-\frac{1}{\beta }}\sigma _{t_i^n}\Delta _i^n\overline{L}\right) \right] \nonumber \\&\qquad - \int _{(n-k_n)\Delta _n}^{T} e^{-u|\sigma _s|^\beta } ds + \sum _{i=0}^{n-k_n+1}\Delta _n \left( R_i^{(1)}+R_i^{(2)}\right) \nonumber \\&\quad {:=} \sum _{i=0}^{n-k_n+1} \xi _{i,u}^{(1)} + \sum _{i=0}^{n-k_n+1} \xi _{i,u}^{(2)} + \sum _{i=0}^{n-k_n+1} \xi _{i,u}^{(3)} + Re_u \nonumber \\&\qquad + \sum _{i=0}^{n-k_n+1} \Delta _n \left( R_i^{(1)}+R_i^{(2)}\right) , \end{aligned}$$

(7)

where the definitions of $\xi _{i,u}^{(1)}$, $\xi _{i,u}^{(2)}$, $\xi _{i,u}^{(3)}$ and $Re_u$ are clear.

For the term $\xi _{i,u}^{(1)}$, by the self-similarity of the stable process $L_t$, we have $\Delta _i^nL_t=\Delta _i^nZ_t=\Delta _n^{\frac{1}{\beta }}S$, where S has characteristic function $E(e^{iuS})=e^{-\frac{|u|^{\beta }}{2}}$, then we can conclude that

$$\begin{aligned} \left\{ \begin{array}{ll} &{} E_i^n\left( \cos \left( \frac{(2u)^{\frac{1}{\beta }}}{(\phi _\beta ^n\Delta _n k_n)^{\frac{1}{\beta }}}\sigma _{t_i^n} \Delta _i^n\overline{L}\right) -e^{-u|\sigma _{t_i^n}|^\beta }\right) = 0,\\ &{} E_i^n\left( \cos \left( \frac{(2u)^{\frac{1}{\beta }}}{(\phi _\beta ^n\Delta _n k_n)^{\frac{1}{\beta }}}\sigma _{t_i^n} \Delta _i^n\overline{L}\right) -e^{-u|\sigma _{t_i^n}|^\beta }\right) ^2\\ &{}~~= G_\beta (u^{\frac{1}{\beta }}|\sigma _{t_i^n}|)_0^n, \\ &{} E_i^n\left( \cos \left( \frac{(2u)^{\frac{1}{\beta }}}{(\phi _\beta ^n\Delta _n k_n)^{\frac{1}{\beta }}}\sigma _{t_i^n} \Delta _i^n\overline{L}\right) -e^{-u|\sigma _{t_i^n}|^\beta }\right) ^4\le C, \end{array} \right. \end{aligned}$$

(8)

where $G_\beta (x)_0^n=\frac{e^{-2^\beta x^\beta }-2e^{-2x^\beta }+1}{2}$.

For the term $\xi _{i,u}^{(2)}$, we first subtract and add a midterm $e^{-u|\sigma _{t_i^n}+\int _{t_i^n}^s\tilde{\sigma }_r\mathrm {d}W_r|^{\beta }}$ and then take Taylor expansion of $e^{-u|\sigma _{t_i^n}+\int _{t_i^n}^s\tilde{\sigma }_r\mathrm {d}W_r|^{\beta }}$ at point $\sigma _{t_i^n}$. It follows that

$$\begin{aligned} \xi _{i,u}^{(2)}= & {} \int _{t_i^n}^{t_{i+1}^n}e^{-u|\sigma _{t_i^n}|^{\beta }}u\cdot \text {sign}(\sigma _{t_i^n}) \beta |\sigma _{t_i^n}|^{\beta -1}\int _{t_i^n}^s\tilde{\sigma }_r\mathrm {d}W_r\mathrm {d}s\\&+\,\int _{t_i^n}^{t_{i+1}^n}\left[ \left( e^{-u|\sigma _s^*|^{\beta }}u\cdot \text {sign}(\sigma ^*) \beta |\sigma _s^*|^{\beta -1}\right. \right. \\&\left. \left. -\,e^{-u|\sigma _{t_i^n}|^{\beta }}u\cdot \text {sign}(\sigma _{t_i^n}) \beta |\sigma _{t_i^n}|^{\beta -1}\right) \int _{t_i^n}^s\tilde{\sigma }_r\mathrm {d}W_r\right] \mathrm {d}s\\&+\,\int _{t_i^n}^{t_{i+1}^n}(e^{-u|\sigma _{t_i^n}+\int _{t_i^n}^s\tilde{\sigma }_r\mathrm {d}W_r|^{\beta }}-e^{-u|\sigma _s|^{\beta }})\mathrm {d}s\\:= & {} \xi _{i,u}^{(2)}(1)+\xi _{i,u}^{(2)}(2)+\xi _{i,u}^{(2)}(3), \end{aligned}$$

where $\sigma _s^*$ is between $\sigma _{t_i^n}$ and $\sigma _{t_i^n}+\int _{t_i^n}^s\tilde{\sigma }_r\mathrm {d}W_r$. Since $E_{i}^n\left( \xi _{i,u}^{(2)}(1)\right) = 0$, using Cauchy Schwarz inequality, Itô isometry, and Assumption 3, we can get

$$\begin{aligned} E_i^n\left( \xi _{i,u}^{(2)}(1)^2\right)\le & {} CE_i^n\left( \int _{t_i^n}^{t_{i+1}^n}\int _{t_i^n}^s\tilde{\sigma }_r\mathrm {d}W_r\mathrm {d}s\right) ^2\le C\Delta _n^3. \end{aligned}$$

(9)

To consider $\xi _{i,u}^{(2)}(2)$, we use the similar arguments in Todorov and Tauchen (2012c). By Cauchy–Schwarz inequality, Itô isometry, Assumption 3, Burkholder–Davis–Gundy inequality, then we have

$$\begin{aligned}&E|\xi _{i,u}^{(2)}(2)|\nonumber \\&\quad \le C\Delta _n^{\frac{3}{2}}\left[ E\left( \sup _{s\in [t_i^n,t_{i+1}^n]}\left( e^{-u|\sigma _s^*|^{\beta }}u\cdot \text {sign}(\sigma ^*) \beta |\sigma _s^*|^{\beta -1}\right. \right. \right. \nonumber \\&\qquad \left. \left. \left. -e^{-u|\sigma _{t_i^n}|^{\beta }}u\cdot \text {sign}(\sigma _{t_i^n}) \beta |\sigma _{t_i^n}|^{\beta -1}\right) ^2\right) \right] ^{\frac{1}{2}}\nonumber \\&\quad \le C\Delta _n^{1+\frac{\beta }{2}}. \end{aligned}$$

(10)

By Taylor expansion at $|\sigma _{t_i^n}+\int _{t_i^n}^s\tilde{\sigma }_r\mathrm {d}W_r|^{\beta }$, and using inequality $||x-y|^p-|x|^p|\le Cp(|y|^p+|x|^{p-1}|y|), p>1$, we obtain,

$$\begin{aligned}&E_i^n\left( \left| \xi _{i,u}^{(2)}(3)\right| \right) \nonumber \\&\quad =E_i^n\left| \int _{t_i^n}^{t_{i+1}^n}ue^{-u|\sigma _s^{**}|^{\beta }}\left( |\sigma _s|^{\beta }-|\sigma _{t_i^n}+\int _{t_i^n}^s\tilde{\sigma _r}\mathrm {d}W_r|^{\beta }\right) \mathrm {d}s\right| \nonumber \\&\quad \le C\Delta _n^2, \end{aligned}$$

(11)

where $\sigma _s^{**}$ is between $\sigma _{t_i^n}+\int _{t_i^n}^s\tilde{\sigma }_r\mathrm {d}W_r$ and $\sigma _s=\sigma _{t_i^n}+\int _{t_i^n}^s\tilde{\alpha }_r\mathrm {d}r+\int _{t_i^n}^s\tilde{\sigma }_r\mathrm {d}W_r$.

For the term $\xi _{i,u}^{(3)}$, we first subtract and add a same term, $\cos \left( (2u)^{\frac{1}{\beta }}(\phi _{\beta }^n\Delta _nk_n)^{-\frac{1}{\beta }}\left( \sum _{j=1}^{k_n-1}g_j^n(\int _{(i+j-1)\Delta _n}^{(i+j)\Delta _n}\alpha _s ds +\int _{(i+j-1)\Delta _n}^{(i+j)\Delta _n}\sigma _{s} d{L_s})\right) \right)$. Then, by trigonometric function $\cos (x)-\cos (y)=-2\sin (\frac{1}{2}(x+y))\sin (\frac{1}{2}(x-y))$ and second-order Taylor expansion separately, we have

$$\begin{aligned}&\sum _{i=0}^{n-k_n+1}\xi _{i,u}^{(3)}\\= & {} \sum _{i=0}^{n-k_n+1}\left[ -2\Delta _n\sin \left( \frac{1}{2}\frac{(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n\right. \right. \\&~~\left. \cdot \left( 2\int _{t_{i+j-1}^n}^{t_{i+j}^n}\alpha _sds+2\int _{t_{i+j-1}^n}^{t_{i+j}^n}\sigma _{s}dL_s+\Delta _{i+j}^nY\right) \right) \\&~~\cdot \sin \left( \frac{1}{2}\frac{(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n\Delta _{i+j}^nY\right) \\&-\Delta _n\sin \left( \frac{(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n \sigma _{t_i^n} \Delta _{i+j}^nL \right) \\&~~\cdot \frac{(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}\alpha _sds\\&-\Delta _n\sin \left( \frac{(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n \sigma _{t_i^n} \Delta _{i+j}^nL \right) \\&~~\cdot \frac{(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}(\sigma _s-\sigma _{t_i^n})dL_s\\&-\Delta _n\cos (\tilde{\chi })\left( \frac{(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\left( \sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}\alpha _sds\right. \right. \\&\left. \left. \left. +\sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}(\sigma _s-\sigma _{t_i^n})dL_s\right) \right) ^2\right] \\:= & {} \sum _{i=0}^{n-k_n+1}\left[ \xi _{i,u}^{(3)}(1)+\xi _{i,u}^{(3)}(2)+\xi _{i,u}^{(3)}(3)+\xi _{i,u}^{(3)}(4)\right] , \end{aligned}$$

where $\tilde{\chi }$ denoting some value between $(2u)^{\frac{1}{\beta }}(\phi _{\beta }^n\Delta _nk_n)^{-\frac{1}{\beta }} \sum _{j=1}^{k_n-1}g_j^n\sigma _{t_i^n}\Delta _{i+j}^nL$ and $(2u)^{\frac{1}{\beta }}(\phi _{\beta }^n\Delta _nk_n)^{-\frac{1}{\beta }} \sum _{j=1}^{k_n-1}g_j^n\cdot (\int _{t_{i+j-1}^n}^{t_{i+j}^n}\alpha _s\mathrm {d}s+\int _{t_{i+j-1}^n}^{t_{i+j}^n}\sigma _{s-} d{L_s})$. Using the basic inequality $|\sin (x)|\le |x|$ and Assumption 2, we have

$$\begin{aligned}&E\left| \xi _{i,u}^{(3)}(1)\right| \le C\Delta _nE\left| (2u)^{\frac{1}{\beta }}\left( \phi _\beta ^n\Delta _nk_n\right) ^{-\frac{1}{\beta }}\sum _{j=1}^{k_n-1}g_j^n\Delta _{i+j}^nY\right| \nonumber \\&\quad \le C\Delta _n^{1+\frac{1}{2\beta '}} k_n^{1-\frac{1}{\beta }}. \end{aligned}$$

(12)

We next divide $\xi _{i,u}^{(3)}(2)$ into two parts,

$$\begin{aligned}&\xi _{i,u}^{(3)}(2)\\= & {} -\frac{\Delta _n(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\sin \left( \frac{(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n \sigma _{t_i^n} \Delta _{i+j}^nL \right) \\&~~\cdot \left( \sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}\alpha _{i\Delta _n}ds\right) \\&-\frac{\Delta _n(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\sin \left( \frac{(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n \sigma _{t_i^n} \Delta _{i+j}^nL \right) \\&~~\cdot \left( \sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}(\alpha _s-\alpha _{i\Delta _n})ds\right) \\:= & {} \xi _{i,u}^{(3)}(2,a)+\xi _{i,u}^{(3)}(2,b). \end{aligned}$$

Noting that $E_i^n(\Delta _i^nL)=0$ and by Assumption 3, Cauchy Schwarz inequality, we have

$$\begin{aligned}&E_i^n\left[ \big |\xi _{i,u}^{(3)}(2,a)\big |^2\right] \le C(\Delta _nk_n)^{3-\frac{2}{\beta }},\\&E_i^n\left[ \big |\xi _{i,u}^{(3)}(2,b)\big |\right] \le C(\Delta _nk_n)^{\frac{3}{2}-\frac{1}{\beta }}. \end{aligned}$$

Then we can obtain

$$\begin{aligned} E_i^n\left[ \left| \sum _{i=0}^{n-k_n+1}\xi _{i,u}^{(3)}(2)\right| \right] \le C(\Delta _nk_n)^{\frac{3}{2}-\frac{1}{\beta }}. \end{aligned}$$

(13)

For $\xi _{i,u}^{(3)}(3)$, we first split

$$\begin{aligned} \sigma _s-\sigma _{i\Delta _n}=\sigma _{1s}+\sigma _{2s}, \end{aligned}$$

where

$$\begin{aligned}&\sigma _{1s}=\int _{t_i^n}^s\tilde{\alpha }_udu+\int _{t_i^n}^s(\tilde{\sigma }_u-\tilde{\sigma }_{i\Delta _n})dW_u,\\&\sigma _{2s}=\int _{t_i^n}^s\tilde{\sigma }_{i\Delta _n}dW_u. \end{aligned}$$

Then we have the inequality

$$\begin{aligned}&E_i^n\left| \xi _{i,u}^{(3)}(3)\right| \\&\quad \le CE_i^n\left| \frac{\Delta _n}{(\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}\int _{|x|\ge \Delta _n^{\frac{1}{\beta }}}\sigma _{1s}x\tilde{\mu }(\mathrm {d}s,\mathrm {d}x)\right| \\&\qquad +\,CE_i^n\left| \frac{\Delta _n}{(\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}\int _{|x|<\Delta _n^{\frac{1}{\beta }}}\sigma _{1s}x\tilde{\mu }(\mathrm {d}s,\mathrm {d}x)\right| \\&\qquad +\,CE_i^n\left| \frac{\Delta _n}{(\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}\int _{|x|\ge \Delta _n^{\frac{1}{\beta }}}\sigma _{2s}x\tilde{\mu }(\mathrm {d}s,\mathrm {d}x)\right| \\&\qquad +\,CE_i^n\left| \frac{\Delta _n}{(\Delta _nk_n)^{\frac{1}{\beta }}}\sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}\int _{|x|<\Delta _n^{\frac{1}{\beta }}}\sigma _{2s}x\tilde{\mu }(\mathrm {d}s,\mathrm {d}x)\right| . \end{aligned}$$

Because $L_s$ is a martingale, using Cauchy–Schwarz inequality and Assumption 3, we get

$$\begin{aligned}&E_i^n\left| \Delta _n^{1-\frac{1}{\beta }}k_n^{-\frac{1}{\beta }}\sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}\int _{|x|\ge \Delta _n^{\frac{1}{\beta }}}\sigma _{1s}x\tilde{\mu }(\mathrm {d}s,\mathrm {d}x)\right| \\&\quad \le C\Delta _n^2k_n^{\frac{3}{2}-\frac{1}{\beta }}. \end{aligned}$$

For the second term, Cauchy–Schwarz inequality, Burkholder–Davis–Gundy inequality and Assumption 3 imply

$$\begin{aligned}&E_i^n\left| \Delta _n^{1-\frac{1}{\beta }}k_n^{-\frac{1}{\beta }}\sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}\int _{|x|<\Delta _n^{\frac{1}{\beta }}}\sigma _{1s}x\tilde{\mu }(\mathrm {d}s,\mathrm {d}x)\right| \\&\quad \le \left[ C\sum _{j=1}^{k_n-1}(g_j^n)^2E_i^n\left\langle \int _{t_{i+j-1}^n}^{t_{i+j}^n}\int _{|x|<\Delta _n^{\frac{1}{\beta }}}\sigma _{1s}x\tilde{\mu }(\mathrm {d}s,\mathrm {d}x)\right\rangle \right. \\&\qquad \left. \cdot \Delta _n^{2-\frac{2}{\beta }}k_n^{-\frac{2}{\beta }}\right] ^\frac{1}{2} \le C\Delta _n^2k_n^{\frac{3}{2}-\frac{1}{\beta }}. \end{aligned}$$

Similarly, we have

$$\begin{aligned}&E_i^n\left| \Delta _n^{1-\frac{1}{\beta }}k_n^{-\frac{1}{\beta }}\sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}\int _{|x|\ge \Delta _n^{\frac{1}{\beta }}}\sigma _{2s}x\tilde{\mu }(\mathrm {d}s,\mathrm {d}x)\right| \\&\quad \le C\Delta _n^{\frac{3}{2}}k_n^{1-\frac{1}{\beta }},\\&E_i^n\left| \Delta _n^{1-\frac{1}{\beta }}k_n^{-\frac{1}{\beta }}\sum _{j=1}^{k_n-1}g_j^n\int _{t_{i+j-1}^n}^{t_{i+j}^n}\int _{|x|<\Delta _n^{\frac{1}{\beta }}}\sigma _{2s}x\tilde{\mu }(\mathrm {d}s,\mathrm {d}x)\right| \\&\quad \le C\Delta _n^{\frac{3}{2}}k_n^{1-\frac{1}{\beta }}. \end{aligned}$$

Martingale property of $L_s$ yields

$$\begin{aligned} E_i^n\left| \sum _{i=0}^{n-k_n+1}\xi _{i,u}^{(3)}(3)\right| \le C\Delta _nk_n^{\frac{3}{2}-\frac{1}{\beta }}. \end{aligned}$$

(14)

By Assumption 3, we obtain

$$\begin{aligned} E_i^n\left| \xi _{i,u}^{(3)}(4)\right| \le C\Delta _n^{3-\frac{2}{\beta }}k_n^{2-\frac{2}{\beta }}. \end{aligned}$$

(15)

For the fourth term $Re_u$, it can be easily shown that

$$\begin{aligned} E|\hbox {Re}_u|\le Ck_n\Delta _n \end{aligned}$$

(16)

uniformly in u. Combining (7), (8)–(16) and the bound of $R_i$, we have

$$\begin{aligned}&E|V_T(U,\Delta _n, \beta , u, g) - \int _{0}^{T}e^{-u|\sigma _s|^\beta } ds|\nonumber \\\le & {} C\left( (\Delta _nk_n)^{\frac{1}{2}}+(\Delta _nk_n)^{2-\frac{2}{\beta }}+\frac{1}{(\Delta _nk_n)^{\frac{2}{\beta }}{k_n}}\right) , \end{aligned}$$

(17)

which gives the desired result. $\square$

Proof of Theorem 2

First, we separately deal with the main term and the noise term of

$$\begin{aligned}&\frac{1}{\sqrt{\Delta _nk_n}}\left( V(U,\Delta _n,u,g)-\int _0^Te^{-u|\sigma _s|^{\beta }}\mathrm {d}s\right) \\&\quad =\frac{1}{\sqrt{\Delta _nk_n}}\left( \sum _{i=0}^{n-k_n+1}(\xi _{i,u}^{(1)}+\xi _{i,u}^{(2)}+\xi _{i,u}^{(3)})+Re(u)\right. \\&\qquad \left. +\sum _{i=0}^{n-k_n+1}\Delta _n(R_i^{(1)}+R_i^{(2)})\right) . \end{aligned}$$

To find an upper bound of noise term, we have

$$\begin{aligned}&E\left[ \frac{1}{\sqrt{\Delta _nk_n}}\left| \sum _{i=0}^{n-k_n+1}\Delta _n\left( R_i^{(1)}+R_i^{(2)}\right) \right| \right] \\&\quad \le Ck_n^{-\frac{1}{\beta }-\frac{1}{2}}\Delta _n^{-\frac{1}{\beta }}+Ck_n^{-\frac{2}{\beta }-\frac{3}{2}}\Delta _n^{-\frac{2}{\beta }-\frac{1}{2}}. \end{aligned}$$

By selecting $k_n=O\left( n^{\frac{2}{2+\beta }+\tau }\right) , ~ \tau \in \left( \frac{\beta ^2}{(\beta +2)(3\beta +4)},\frac{\beta }{\beta +2}\right)$, we can show that the noise term converges in probability to zero.

For the second term, invoking Theorem 1 and (9)–(11), we get

$$\begin{aligned} E\left( \frac{1}{\sqrt{\Delta _nk_n}}\sum _{i=0}^{n-k_n+1}\xi _{i,u}^{(2)}\right) ^2\le C\Delta _n^{-1+\beta }k_n^{-1}\rightarrow 0. \end{aligned}$$

For the third term, from (12)–(14), we have

$$\begin{aligned}&E\left| \frac{1}{\sqrt{\Delta _nk_n}}\sum _{i=0}^{n-k_n+1}\left( \xi _{i,u}^{(3)}(1)+\xi _{i,u}^{(3)}(2)+\xi _{i,u}^{(3)}(3)\right) \right| \\&\quad \le C\Delta _n^{-\frac{1}{2}+\frac{1}{2\beta '}}k_n^{\frac{1}{2}-\frac{1}{\beta }}+C(\Delta _nk_n)^{1-\frac{1}{\beta }}+C(\Delta _nk_n)^{\frac{1}{2}}k_n^{\frac{1}{2}-\frac{1}{\beta }} \end{aligned}$$

which converges to 0, when $n\rightarrow \infty$. It follows from (15) that

$$\begin{aligned} E\left| \frac{1}{\sqrt{\Delta _nk_n}}\sum _{i=0}^{n-k_n+1}\xi _{i,u}^{(3)}(4)\right| \le C(\Delta _nk_n)^{\frac{3}{2}-\frac{2}{\beta }}. \end{aligned}$$

To let this term converge to 0 as $n\rightarrow \infty$, we need $\beta >\frac{4}{3}$. When $n\rightarrow \infty$,

$$\begin{aligned} E\left| \frac{1}{\sqrt{\Delta _nk_n}}\hbox {Re}(u)\right| \le C\sqrt{\Delta _nk_n} \rightarrow 0 . \end{aligned}$$

So under the conditions of Theorem 2,

$$\begin{aligned}&E\Bigg |\frac{1}{\sqrt{\Delta _nk_n}}\Bigg [\sum _{i=0}^{n-k_n+1}\left( \xi _{i,u}^{(2)}+\xi _{i,u}^{(3)}+\Delta _nR_i^{(1)}+\Delta _nR_i^{(2)}\right) \nonumber \\&\quad +\hbox {Re}(u)\Bigg ]\Bigg |\rightarrow 0,\quad n\rightarrow \infty . \end{aligned}$$

To prove the central limit theorem, we only need to prove that as $\Delta _n \rightarrow 0$,

$$\begin{aligned} \frac{1}{\sqrt{\Delta _nk_n}}\sum _{i=0}^{n-k_n+1}\xi _{i,u}^{(1)}{\mathop {\longrightarrow }\limits ^{\mathcal {L}_s}} \sqrt{\int _0^T \int _0^1G_\beta \left( u^{\frac{1}{\beta }}|\sigma _s|\right) _r \mathrm{d}r \mathrm{d}s}\times \mathcal{N}. \end{aligned}$$

To this end, we apply the technique of “big blocks, small blocks” as introduced in Jacod et al. (2009). The limiting distribution of the proposed estimator stems from the “big blocks”. Whereas the “small blocks” does not affect the final asymptotical behavior, which is eventually proven to be negligible but ensure the conditional independence between the “big” blocks. For a given integer p, we let $i_n(p)=\left[ \frac{[T/\Delta _n]-k_n+1}{(p+1)k_n} \right] - 1$ and for $i=0,\ldots , i_n(p)$, write $a_i(p)=i(p+1)k_n+1$, $b_i(p)=i(p+1)k_n +pk_n$ and denote the i-th “big” block by $\mathrm {A}_i=\{k: a_i(p)\le k \le b_i(p), k\in N^{+}\}$ and the i-th “small block” by $\mathrm {B}_i =\{k: b_i(p)< k < a_{i+1}(p), k\in N^{+}\}$. We define

$$\begin{aligned}&\zeta _i(p,1) = \sum _{j\in \mathrm {A}_i} \xi _{j,u}^{(1)},\quad \zeta '_i(p,1) = E_{a_i(p)-1}^n\left( \zeta _i(p,1) \right) , \\&\zeta _i(p,2)=\sum _{j\in \mathrm {B}_i} \xi _{j,u}^{(1)},\quad \zeta '_i(p,2) = E_{a_i(p)-1}^n\left( \zeta _i(p,2) \right) , \end{aligned}$$

where $\xi _{j,u}^{(1)}$ is defined in (7). Then, we denote

$$\begin{aligned}&M(p) = \sum _{i=0}^{i_n(p)} \left( \zeta _i(p,1)-\zeta '_i(p,1)\right) ,\quad M'(p) =\sum _{i=0}^{i_n(p)} \zeta '_i(p,1), \\&N(p) = \sum _{i=0}^{i_n(p)} \left( \zeta _i(p,2) - \zeta '_i(p,2) \right) ,\quad N'(p) = \sum _{i=0}^{i_n(p)}\zeta '_i(p,2), \\&C(p) = \sum _{i=i_n(p)+1}^{[T/\Delta _n]-k_n+1} \xi _{i,u}^{(1)} . \end{aligned}$$

Since the convergence rate of V is $(\Delta _nk_n)^{\frac{1}{2}}$. Therefore, we have the following decomposition,

$$\begin{aligned}&\frac{1}{\sqrt{\Delta _n k_n}}\left( \sum _{i=0}^{n-k_n+1}\xi _{i,u}^{(1)} - \int _{0}^{T}e^{-u|\sigma _s|^{\beta }} ds \right) \\&\quad = \frac{1}{\sqrt{\Delta _n k_n}}\left[ M(p) + M'(p) + N(p) + N'(p) + C(p)\right] .\nonumber \end{aligned}$$

(18)

Following similar steps in Jacod et al. (2009), we obtain that for any $\epsilon >0$,

$$\begin{aligned}&P\left( |M'(p)| + |N(p)| + | N'(p)| + |C(p)| >(\Delta _n k_n)^{\frac{1}{2}} \epsilon \right) \\&\quad \rightarrow 0. \end{aligned}$$

Hence, it suffices to prove

$$\begin{aligned} (\Delta _n k_n)^{-\frac{1}{2}} M(p) {\mathop {\longrightarrow }\limits ^{\mathcal {L}_s}} \sqrt{\int _0^T\int _0^1G_{\beta }\left( u^{\frac{1}{\beta }}|\sigma _s|\right) \mathrm{d}r\mathrm{d}s}\times \mathcal {N}. \nonumber \\ \end{aligned}$$

(19)

To this end, we apply the martingale central limit theorem which is presented in Theorem IX.7.28 in Jacod and Shiryayev (2003), with which we need to verify the following properties:

$$\begin{aligned}&\frac{1}{\Delta _nk_n}\sum _{i=0}^{i_n(p)} E_{a_i(p)-1}^n\left( \left( \zeta _i(p,1) - \zeta '_{i}(p,1)\right) ^2 \right) \nonumber \\&\quad {\mathop {\longrightarrow }\limits ^{P}} \int _{0}^{T}\int _{0}^{1}G_{\beta }\left( {u}^{\frac{1}{\beta }}|\sigma _{s}|\right) _r \mathrm{d}r \mathrm{d}s, \end{aligned}$$

(20)

$$\begin{aligned}&(\Delta _nk_n)^{-2}\sum _{i=0}^{i_n(p)} E_{a_i(p)-1}^n\left( \zeta _i^4(p,1) \right) {\mathop {\longrightarrow }\limits ^{P}} 0, \end{aligned}$$

(21)

$$\begin{aligned}&(\Delta _nk_n)^{-\frac{1}{2}}\sum _{i=0}^{i_n(p)}E_{a_i(p)-1}^n\left( \zeta _i(p,1)\Delta H(p)_i\right) {\mathop {\longrightarrow }\limits ^{P}} 0, \end{aligned}$$

(22)

where, $\Delta H(p)_i=H_{b_i(p)\Delta _n} - H_{a_i(p)\Delta _n}$, H is a bounded martingale defined on the original probability space. We remark that in deriving formula (20), we let p be fixed first and then tend to infinity.

From (8), we get ${\zeta '_i}(p,1) = 0$. From self-similarity property of stable process, we know $\Delta _i^nL=\Delta _n^{\frac{1}{\beta }}S_1$, where $S_1$ is a stable process with characteristic function $E(e^{iuS_1})=e^{-|u|^\beta /2}$. Applying these arguments and using the basic equalities that $\cos (a+b)=\cos (a)\cos (b)-\sin (a)\sin (b)$, $\sin (a)\cos (b)=\left( \sin (a+b)+\sin (a-b)\right) /2$ and $\cos (a)=(e^{ia}+e^{-ia})/2$, we can obtain

$$\begin{aligned}&\frac{1}{\Delta _nk_n}\sum _{i=0}^{i_n(p)}E_{a_i(p)-1}^n\left( \left( \zeta _i(p,1) - \zeta '_{i}(p,1)\right) ^2 \right) \nonumber \\&\quad =\frac{1}{\Delta _nk_n}\sum _{i=0}^{i_n(p)}E_{a_i(p)-1}^n\left[ \left( \sum _{j\in A_i}\Delta _n\left( \cos (\frac{(2u)^{\frac{1}{\beta }}}{(\phi _{\beta }^n\Delta _nk_n)^{\frac{1}{\beta }}}\cdot \sigma _{t_i^n}\Delta _i^n\overline{L})\right. \right. \right. \nonumber \\&\qquad \left. \left. \left. -e^{-u|\sigma _{t_i^n}|^{\beta }}\right) \right) ^2\right] \nonumber \\&\quad = \frac{\Delta _n}{k_n}\sum _{i=0}^{i_n(p)}\bigg ( \sum _{j=a_i(p)}^{b_i(p)} G_{\beta }({u}^{\frac{1}{\beta }}|\sigma _{j\Delta _n}|)_{0}^n\nonumber \\&\qquad + 2\sum _{j_1=a_i(p)}^{b_i(p)}\sum _{j_2>j_1} G_{\beta }\left( {u}^{\frac{1}{\beta }}|\sigma _{j_1\Delta _n}|\right) _{j_2-j_1}^n + O_p(\Delta _n^{\frac{1}{2}}k_n^{\frac{5}{2}}p) \bigg ) \nonumber \\&\quad =\frac{\Delta _n}{k_n}\sum _{i=0}^{i_n(p)} \sum _{j=a_i(p)}^{b_i(p)}\bigg ( 2\sum _{s=0}^{k_n-1} G_{\beta }({u}^{\frac{1}{\beta }}|\sigma _{j\Delta _n}|)_{s}^n- G_{\beta }({u}^{\frac{1}{\beta }}|\sigma _{j\Delta _n}|)_{0}^n\bigg ) \nonumber \\&\qquad + O_p\left( \frac{p}{p+1}(\Delta _n k_n)^{\frac{1}{2}}\right) \nonumber \\&\quad = 2 \frac{\Delta _n}{k_n} \sum _{j=0}^{[ T/\Delta _n ]-1} \sum _{s=0}^{k_n-1} G_{\beta }({u}^{\frac{1}{\beta }}|\sigma _{j\Delta _n}|)_{s}^n+O_p((p+1)\Delta _n) \nonumber \\&\qquad +O_p(\frac{p}{(p+1)k_n})+ O_p\left( \frac{p}{p+1}(\Delta _nk_n)^{\frac{1}{2}}\right) + O_p\left( \frac{1}{p+1}\right) ,\nonumber \\ \end{aligned}$$

(23)

where $\phi _\beta ^n=k_n^{-1}\sum _{j=0}^{k_n-1}(g(\frac{j}{k_n}))^\beta$ and

$$\begin{aligned}&G_{\beta }(x)_{s}^n \\&\quad = \frac{1}{2}\left[ e^{-\frac{x^\beta }{\phi _\beta ^nk_n}\left( \sum _{l=1}^{s}(g_l^n)^\beta +\sum _{l=k_n-s}^{k_n-1}(g_l^n)^\beta \right) }\right. \nonumber \\&\qquad \cdot \left( e^{-\frac{x^\beta }{\phi _\beta ^nk_n}\sum _{l=s+1}^{k_n-1}(g_l^n+g_{l-s}^n)^\beta }+e^{-\frac{x^\beta }{\phi _\beta ^nk_n}\sum _{l=s+1}^{k_n-1}(g_l^n-g_{l-s}^n)^\beta }\right) \nonumber \\&\qquad \left. -2e^{-2x^\beta }\right] . \end{aligned}$$

On the other hand, because for any $r\in (0,1)$, $|G'_{\beta }(x)_r| \le C$, applying Taylor expansion,

$$\begin{aligned}&E\left( \sum _{i=0}^{[T/\Delta _n] -1} \sum _{s=1}^{k_n} \int _{t_i^n}^{t_{i+1}^n} \int _{(s-1)/k_n}^{s/k_n} \Big | G_{\beta }\left( {u}^{\frac{1}{\beta }}|\sigma _{t}|\right) _r \right. \nonumber \\&\left. \quad - G_{\beta }\left( {u}^{\frac{1}{\beta }}|\sigma _{t_i^n}|\right) _s^n\Big |\mathrm{d}r\mathrm{d}t \right) \le C\left( \sqrt{\Delta _n} + k_n^{-1}\right) \rightarrow 0. \end{aligned}$$

(24)

Collecting (23) and (24) yields the property (20). Following the arguments in calculating (23), (24) and applying the steps in the proof of Theorem 1, we obtain (21). Then we need to show (22). When H is discontinuous martingale. Using the property of predictable quadratic variation, quadratic variation and Itô isometry, we can prove that $\sum _{i=1}^{i_n(p)}E_{a_i(p)-1}^n\left( (\Delta _nk_n)^{-\frac{1}{2}}\zeta _i(p,1) \Delta H(p)_i \right)$ converges in probability to 0. The detail is similar to the proof in Todorov and Tauchen (2012c). As pure jump and continuous martingales are orthogonal, we have $E_{a_i(p)-1}^n(\zeta _i(p,1)\Delta H(p)_i)=0$ when H is a continuous martingale. Then proof of Theorem 2 is completed. $\square$