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Hölder regularity and roughness: construction and examples

Erhan Bayraktar, Purba Das, Donghan Kim

TL;DR

The paper develops a Schauder-based framework to construct processes on a finite interval with prescribed roughness and specified marginal moments. It extends Ciesielski's isomorphism to balanced, complete refining partition sequences, linking Hölder regularity to Schauder coefficients and enabling a pathwise estimator. It then uses this framework to build fake (fractional) Brownian motions with the same finite joint moments as real fBM, showing that non-Gaussian processes can mimic Gaussian marginals and be statistically hard to distinguish. The results also highlight that the critical Hölder exponent and the variation index can differ, informing rough-volatility modeling and offering a flexible method to generate diverse, model-free training data for finance applications.

Abstract

We study how to construct a stochastic process on a finite interval with given `roughness' and finite joint moments of marginal distributions. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of Hölder regularity of a function in terms of its Schauder coefficients. Using this characterization we provide a better (pathwise) estimator of Hölder exponent. As an additional application, we construct fake (fractional) Brownian motions with some path properties and finite moments of marginal distributions same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.

Hölder regularity and roughness: construction and examples

TL;DR

The paper develops a Schauder-based framework to construct processes on a finite interval with prescribed roughness and specified marginal moments. It extends Ciesielski's isomorphism to balanced, complete refining partition sequences, linking Hölder regularity to Schauder coefficients and enabling a pathwise estimator. It then uses this framework to build fake (fractional) Brownian motions with the same finite joint moments as real fBM, showing that non-Gaussian processes can mimic Gaussian marginals and be statistically hard to distinguish. The results also highlight that the critical Hölder exponent and the variation index can differ, informing rough-volatility modeling and offering a flexible method to generate diverse, model-free training data for finance applications.

Abstract

We study how to construct a stochastic process on a finite interval with given `roughness' and finite joint moments of marginal distributions. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of Hölder regularity of a function in terms of its Schauder coefficients. Using this characterization we provide a better (pathwise) estimator of Hölder exponent. As an additional application, we construct fake (fractional) Brownian motions with some path properties and finite moments of marginal distributions same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.
Paper Structure (11 sections, 11 theorems, 45 equations, 2 figures)

This paper contains 11 sections, 11 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Schauder representation along general partition sequence
  3. Sequence of partitions
  4. Generalized Haar basis associated with a finitely refining partition sequence
  5. Schauder representation of a continuous function
  6. Hölder exponent and Schauder coefficients
  7. Characterization of Hölder exponent
  8. Pathwise Hölder regularity estimator
  9. Critical Hölder exponent and variation index
  10. Fake Brownian motion
  11. Construction of fake Brownian processes

Key Result

Lemma 2.6

Let $\pi = (\pi^n)_{n \in \mathbb{N}}$ be a refining sequence of partitions.

Figures (2)

  • Figure 1: Estimating Hölder regularity
  • Figure 2: Function $x$, a sample path of $Y$, and their quadratic variations in Example \ref{['ex.sqrt(m)']} and Remark \ref{['rem.make random']}.

Theorems & Definitions (33)

  • Definition 2.1: Refining sequence of partitions
  • Definition 2.2: Finitely refining sequence of partitions
  • Definition 2.3: Balanced sequence of partitions
  • Definition 2.4: Complete refining sequence of partitions
  • Remark 2.5: Notation
  • Lemma 2.6
  • Definition 2.7: Generalized Haar basis
  • Definition 2.8: Schauder function
  • Proposition 2.9: Theorem 3.8 of das2021
  • Lemma 2.10: Continuous limit
  • ...and 23 more