For a semi-martingale $X_t$, which forms a stochastic boundary, arate-optimal estimator for its quadratic variation $\langle X, X \rangle_t$ isconstructed based on observations in the vicinity of $X_t$. The problem isembedded in a Poisson point process framework, which reveals an interestingconnection to the theory of Brownian excursion areas. We derive $n^{-1/3}$ asoptimal convergence rate in a high-frequency framework with $n$ observations(in mean). We discuss a potential application for the estimation of theintegrated squared volatility of an efficient price process $X_t$ fromintra-day order book quotes.
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机译：对于形成随机边界的半-$ X_t $，根据其在$ X_t $附近的观测值，构造其二次变化$ \ langle X的比率最优估计量，X \ rangle_t $。该问题被嵌入到Poisson点过程框架中，该框架揭示了与布朗漂移区理论的有趣联系。我们在高频框架中以$ n $个观测值（均值）得出$ n ^ {-1/3} $最优收敛速度。我们讨论了从日内订单簿报价中估算有效价格过程$ X_t $的综合平方波动率的潜在应用。
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