Robust Faber--Schauder approximation based on discrete observations of an antiderivative

TL;DR

This paper develops a robust method for reconstructing Faber--Schauder coefficients of a function from discrete observations of its antiderivative, revealing that excluding the final coefficients improves stability and accuracy.

Contribution

It introduces a closed-form solution and error analysis for reconstructing Faber--Schauder coefficients, highlighting the local dependence and instability of the final coefficients.

Findings

Final-generation coefficients cause instability and depend on initial value.

Excluding final coefficients yields a more stable and accurate estimator.

The approach applies to financial mathematics, such as estimating volatility from integrated data.

Abstract

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.