论文标题
混合半明星的速率最佳估计
Rate-optimal estimation of mixed semimartingales
论文作者
论文摘要
考虑一下Brownian Motion $ b $的总和$ y = b+b（h）$，以及带有Hurst参数$ h \ in（0,1）$的独立分数Brownian Motion $ b（h）$。即使$ b（h）$不是半明星，但它在[\ textit {bernoulli} \ textbf {7}（2001）（2001）913--934]中显示，如果$ y $是semimartingale，如果$ h> 3/4 $。此外，在这种情况下，$ y $在本地等同于$ b $，因此$ h $不能从$ y $的本地观察值中始终如一地估算。本文在该模型中的另一个意外功能上旋转：如果$ b $和$ b（h）$变得相关，那么$ y $永远不会是半木马，并且无论其价值如何，都可以识别出$ h $。该结果和其他结果将取决于对\ emph {混合半段}的更通用过程的详细统计分析，该过程是$ y $的半摩米术扩展，在martingale和分数组件中均具有随机波动。特别是，我们为所有参数和过程提供了一致的估计器和可行的中心限制定理，这些参数和过程可以从高频观察中识别。我们进一步表明，我们的估计器从最小程度上实现了最佳速率。
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.
