论文标题
强大的Faber-基于抗动力学的离散观察结果
Robust Faber--Schauder approximation based on discrete observations of an antiderivative
论文作者
论文摘要
我们研究了从离散观察到其抗但是$ f $的连续功能$ f $的faber-schauder系数的问题。例如，当资产价格轨迹的综合波动率估算波动性的粗糙度时，这一问题是在金融数学中引起的。我们的方法始于数学通过分段二次样条插值来制定重建问题。然后，我们提供封闭形式的解决方案和深入的误差分析。这些结果导致了一些令人惊讶的观察结果，这也对二次样条插值本身的古典主题进行了新的启示：它们表明，该方法的众所周知的不稳定性可以仅位于估计的faber- schauder系数的最后一代中，这些系数遭受了非网站的损失和对初始值的强烈依赖。相比之下，所有其他faber-示波器系数仅在局部取决于数据，独立于初始值，并且允许统一误差界限。因此，我们得出的结论是，可以通过简单地从估计的faber-schauder系数中删除最终代系数来获得我们问题的强大且良好的估计器。
We study the problem of reconstructing the Faber--Schauder coefficients of a continuous function $f$ from discrete observations of its antiderivative $F$. For instance, this question arises in financial mathematics when estimating the roughness of volatility from the integrated volatility of an asset price trajectory. Our approach starts with mathematically formulating the reconstruction problem through piecewise quadratic spline interpolation. We then provide a closed-form solution and an in-depth error analysis. These results lead to some surprising observations, which also throw new light on the classical topic of quadratic spline interpolation itself: They show that the well-known instabilities of this method can be located exclusively within the final generation of estimated Faber--Schauder coefficients, which suffer from non-locality and strong dependence on the initial value. By contrast, all other Faber--Schauder coefficients depend only locally on the data, are independent of the initial value, and admit uniform error bounds. We thus conclude that a robust and well-behaved estimator for our problem can be obtained by simply dropping the final-generation coefficients from the estimated Faber--Schauder coefficients.
