Estimating the quadratic variation (QV) using high-frequency financial data is studied in this article, and this work makes two major contributions: first, the fundamental Itô isometry $E\left[{\left({\int }_b^{a}f(t,\omega )d{B}_t\right)}^2\right]=E\left[{\int }_b^a{f}^2(t,\omega )dt\right]$ is generalized to some extent, and as an application example, the widely known convergence property of realized volatility (RV) estimator of QV is analyzed by alternatively utilizing this generalized Itô isometry; second, we intuitively establish two types of new estimators of QV which permit volatility varying with time. To construct such estimators, the RV combined with realized bipower variation and RV with realized quarticity are employed, respectively. By applying the generalized Itô isometry, we further show that each of the proposed estimators can converge to QV, at a rate of O(n^−1/2), almost surely and in mean square sense, both of which are stronger than the existing convergence in probability. Moreover, the error of approximation is also provided for each estimation. In addition, the obtained convergence property for both types of estimators is demonstrated by empirical investigations based on high-frequency data of IBM stock.

中文翻译：

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使用高频数据的二次变化的非参数估计

本文研究了使用高频金融数据估计二次方差（QV），这项工作做出了两个主要贡献：第一，基本的 Itô 等距$乙\left[{\left({\int }_{乙}^{A}F（t,\omega ）d{乙}_t\right)}^2\right]=乙\left[{\int }_{乙}^A{F}^2（t,\omega ）dt\right]$在一定程度上得到了推广，作为一个应用实例，交替利用这种广义的 Itô 等距分析了广为人知的 QV 实现波动率 (RV) 估计量的收敛性；其次，我们直观地建立了两种新的 QV 估计量，允许波动性随时间变化。为了构造这样的估计器，分别采用与实现的双功率变化相结合的 RV 和具有实现的四次性的 RV。通过应用广义 Itô 等距，我们进一步表明，每个提出的估计量都可以以O ( n ^−1/2 )的速率收敛到 QV，几乎可以肯定并且在均方意义上，这两者都比现有的更强概率收敛。此外，还为每个估计提供了近似误差。此外，基于 IBM 股票高频数据的实证研究证明了两种类型估计器所获得的收敛特性。