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

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.

Figures (3)

• Figure 1: Graphical illustration of the instability of the solution to problem \ref{['eq inverse Faber']} with $F(t)=1-\cos\pi t$ and $n=3$. While $F$ and $\hat{F}_n$ (left, dotted and dashed respectively) and $f$ and $\hat{f}_n$ (right, dotted and dashed respectively) are basically indistinguishable when taking $\hat{f}_0=0$, the value $\hat{f}_0=4$ yields wildly oscillating functions $\hat{F}_n$ and $\hat{f}_n$ (solid lines).
• Figure 2: Graphical illustrations of the data dependence of the coefficients $\vartheta^{(n)}_{m,k}$. In the bottom-left graph, we plot the approximating functions $\hat{f}_{4,1}$ and $\hat{f}_{4,2}$ in dotted and solid lines respectively, and the contribution $\vartheta^{(4)}_{2,3}e_{2,3}$ is highlighted in red. In the bottom-right graph, we plot the approximating functions $\hat{f}_{4,3}$ and $\hat{f}_{4,4}$ in dotted and solid lines respectively, and the contribution $\vartheta^{(4)}_{4,9}e_{4,9}$ is highlighted in blue. Respective data points of the function $F$ that contribute to the estimation of $\vartheta^{(4)}_{2,3}$ and $\vartheta^{(4)}_{4,9}$ are highlighted by circles in the corresponding upper graphs. All data points that do not contribute to the computation of the coefficients in question are represented by cross marks.
• Figure 3: Graphical illustration of the persistence of integrated Faber--Schauder basis and instability of the Fourier basis; The left graph plots $F(t) = t$ (solid), $F_1(t) = 2^{n/2+2}\psi_{n,k}(t)$ (dotted) and $F_2(t) = 2^{n/2+2}\psi_{n,k}(t) + 4\sum_{m = n+2}^{n_0}\sum_{k = 0}^{2^{m-1}-1}(-1)^k\psi_{m,k}(t) + 2\sum_{m = n+2}^{n_0}\sum_{k = 2^{m-1}}^{2^{m}-1}(-1)^k\psi_{m,k}(t)$ (dashed) with $n = 1$ and $n_0 = 4$. Clearly, values of $F_1$ and $F_2$ coincide with those of $F$ over dyadic partition $\mathbb{T}_{n+1}$, and both $F_1$ and $F_2$ yield well-behaved approximations; the right graph plots $F(t) = t$ (solid), $\widetilde{F}_{n,1}(t)$ (dashed), $\widetilde{F}_{n,3}(t)$ (dotted) and $\widetilde{F}_{n,7}(t)$ (dotdashed) with $n = 2$. Here, $\widetilde{F}_{n,7}$ yields a wildly oscillating approximation of $F$.

Theorems & Definitions (39)

• Theorem 2.1
• Remark 2.2
• Example 2.3
• Proposition 2.4
• Theorem 2.5
• Theorem 2.6
• Corollary 2.7
• Theorem 2.8
• Lemma 2.9
• Theorem 2.10