We consider the problem of estimating volatility for high-frequency data when the observed process is the sum of a continuous It\^o semimartingale and a noise process that locally behaves like fractional Brownian motion with Hurst parameter H. The resulting class of processes, which we call mixed semimartingales, generalizes the mixed fractional Brownian motion introduced by Cheridito [Bernoulli 7 (2001) 913-934] to time-dependent and stochastic volatility. Based on central limit theorems for variation functionals, we derive consistent estimators and asymptotic confidence intervals for H and the integrated volatilities of both the semimartingale and the noise part, in all cases where these quantities are identifiable. When applied to recent stock price data, we find strong empirical evidence for the presence of fractional noise, with Hurst parameters H that vary considerably over time and between assets.
翻译:我们考虑高频数据波动的估计问题,当观察过程是一个连续的It ⁇ o半成像和噪音过程的总和时,即当地行为类似于分形布朗运动和Hurst参数H。 由此产生的过程类别,我们称之为混合半成形,将Cheridito[Bernoulli 7(2001) 913-934] 提出的混合成形布朗运动概括为基于时间和随机变化的波动。根据变化功能的中央限值,我们为H和半成形和噪音部分的综合挥发性得出一致的估测和零成信心间隔,在所有这些情况下,这些数量都是可以识别的。当应用到最近的股票价格数据时,我们发现有有力的实证证据表明存在分数噪音,而赫斯特参数H在时间上和资产之间有很大差异。