Robust Faber–Schauder Approximation Based on Discrete Observations of an Antiderivative
研究从反导数的离散观测中重建连续函数Faber-Schauder系数的问题，发现二次样条插值的不稳定性仅存在于最后一代系数，剔除后可得稳健估计，对金融波动率粗糙度估计有直接应用。
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 nonlocality 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. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2017-04054 and RGPIN-2024-03761].