Consider the sum $Y=B+B(H)$ of a Brownian motion $B$ and an independent fractional Brownian motion $B(H)$ with Hurst parameter $H\in(0,1)$. Surprisingly, even though $B(H)$ is not a semimartingale, Cheridito proved in [Bernoulli 7 (2001) 913--934] that $Y$ is a semimartingale if $H>3/4$. Moreover, $Y$ is locally equivalent to $B$ in this case, so $H$ cannot be consistently estimated from local observations of $Y$. This paper pivots on a second surprise in this model: if $B$ and $B(H)$ become correlated, then $Y$ will never be a semimartingale, and $H$ can be identified, regardless of its value. This and other results will follow from a detailed statistical analysis of a more general class of processes called mixed semimartingales, which are semiparametric extensions of $Y$ with stochastic volatility in both the martingale and the fractional component. In particular, we derive consistent estimators and feasible central limit theorems for all parameters and processes that can be identified from high-frequency observations. We further show that our estimators achieve optimal rates in a minimax sense. The estimation of mixed semimartingales with correlation is motivated by applications to high-frequency financial data contaminated by rough noise.

翻译：令人惊讶的是,即使B(H)美元不是半配方,Cheridito 美元在[Bernoulli 7(2001) 913-934] 中证明,如果美元是半配方,美元是半配方,如果H3/4美元。此外,美元在当地相当于美元,因此无法从当地对Y美元的观察中连续估算美元。本模型中第二个惊喜的纸块对冲值是:如果B(H)美元和美元(H)不是半配方,那么,Cheridito在[Bernoulli 7(2001) 913-934]中证明,如果美元是半配方美元,那么美元是一种半配方美元半配方美元。此外,美元在当地相当于美元,因此,从当地对Y的观察中无法持续估算。如果B(H)美元和B(H)美元不是半调方美元,那么,那么,美元就是比方对等值高的货币对等值,我们就能通过高比值的货币和平比值参数来进一步显示。