Consider the sum $Y=B+B(H)$ of a Brownian motion $B$ and an independent fractional Brownian motion $B(H)$ with Hurst parameter $H\in(0,1)$. Even though $B(H)$ is not a semimartingale, it was shown in [\textit{Bernoulli} \textbf{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 another unexpected feature 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 \emph{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.

翻译：考虑布朗运动 $B$ 与独立的 Hurst 参数为 $H\in(0,1)$ 的分数布朗运动 $B(H)$ 之和 $Y=B+B(H)$。尽管 $B(H)$ 不是半鞅，但 [\textit{Bernoulli} \textbf{7} (2001) 913--934] 中证明当 $H>3/4$ 时 $Y$ 是半鞅。此外，在此情况下 $Y$ 局部等价于 $B$，因此无法从 $Y$ 的局部观测中一致地估计 $H$。本文揭示了该模型的另一个意外特征：若 $B$ 与 $B(H)$ 具有相关性，则 $Y$ 将不再是半鞅，且无论 $H$ 取何值均可被识别。此结论及其他结果源于对一类更广义过程（称为\emph{混合半鞅}）的详细统计分析，该类过程是 $Y$ 的半参数扩展，在鞅分量与分数分量中均包含随机波动性。特别地，我们针对所有可通过高频观测识别的参数与过程，推导出一致估计量及可行的中心极限定理。我们进一步证明，所提估计量在极小极大意义下达到了最优收敛速率。