Mixed semimartingales: Volatility estimation in the presence of fractional noise

Abstract

We consider the problem of estimating volatility for high-frequency data when the observed process is the sum of a continuous Ito 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.