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On Construction, Properties and Simulation of Haar-Based Multifractional Processes

Antoine Ayache, Andriy Olenko, Nemini Samarakoon

TL;DR

This work introduces Gaussian Haar-based multifractional processes (GHBMP) by representing a time-varying roughness through Haar wavelet expansions, yielding a centered Gaussian process with non-stationary increments and a broad class of Hurst functions $H(t)$. The authors establish $L^{2}$-convergence, finite second moments, and Hölder continuity up to $\underline{H}$, and they prove a precise almost-sure characterization: $\alpha_{X}(t) = H(t)$ for all $t$ on a probability-one set, via complementary lower- and upper-bound analyses using Haar-based increments and Gaussian-tail techniques. Simulation studies validate the theory across constant, linear, oscillatory, and discontinuous Hurst functions, and an $H$-estimation procedure based on generalized quadratic variations demonstrates accurate recovery of $H(t)$; results suggest practical guidance on truncation level $J$ for accurate modelling. The approach combines computational efficiency with theoretical rigor, enabling robust replication and inference in settings with non-stationary roughness and abrupt changes in regularity.

Abstract

Multifractional processes extend the concept of fractional Brownian motion by replacing the constant Hurst parameter with a time-varying Hurst function. This extension allows for modulation of the roughness of sample paths over time. The paper introduces a new class of multifractional processes, the Gaussian Haar-based multifractional processes (GHBMP), which is based on the Haar wavelet series representations. The resulting processes cover a significantly broader set of Hurst functions compared to the existing literature, enhancing their suitability for both practical applications and theoretical studies. The theoretical properties of these processes are investigated. Simulation studies conducted for various Hurst functions validate the proposed model and demonstrate its applicability, even for Hurst functions exhibiting discontinuous behaviour.

On Construction, Properties and Simulation of Haar-Based Multifractional Processes

TL;DR

This work introduces Gaussian Haar-based multifractional processes (GHBMP) by representing a time-varying roughness through Haar wavelet expansions, yielding a centered Gaussian process with non-stationary increments and a broad class of Hurst functions . The authors establish -convergence, finite second moments, and Hölder continuity up to , and they prove a precise almost-sure characterization: for all on a probability-one set, via complementary lower- and upper-bound analyses using Haar-based increments and Gaussian-tail techniques. Simulation studies validate the theory across constant, linear, oscillatory, and discontinuous Hurst functions, and an -estimation procedure based on generalized quadratic variations demonstrates accurate recovery of ; results suggest practical guidance on truncation level for accurate modelling. The approach combines computational efficiency with theoretical rigor, enabling robust replication and inference in settings with non-stationary roughness and abrupt changes in regularity.

Abstract

Multifractional processes extend the concept of fractional Brownian motion by replacing the constant Hurst parameter with a time-varying Hurst function. This extension allows for modulation of the roughness of sample paths over time. The paper introduces a new class of multifractional processes, the Gaussian Haar-based multifractional processes (GHBMP), which is based on the Haar wavelet series representations. The resulting processes cover a significantly broader set of Hurst functions compared to the existing literature, enhancing their suitability for both practical applications and theoretical studies. The theoretical properties of these processes are investigated. Simulation studies conducted for various Hurst functions validate the proposed model and demonstrate its applicability, even for Hurst functions exhibiting discontinuous behaviour.

Paper Structure

This paper contains 7 sections, 17 theorems, 164 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On GHBMP and its Hölder continuity
  4. Lower bound for Hölder exponent
  5. Upper bound for Hölder exponent
  6. Simulation studies
  7. Conclusion and future studies

Key Result

Theorem 3.1

For all fixed $t \in [0,1]$ it holds Therefore, the GHBMP processes are properly defined in $L_2$-sense.

Figures (2)

  • Figure 1: Realisations of GHBMP and corresponding Hurst functions
  • Figure 2: Boxplots of differences of estimates and $H(t)=0.5-0.4\sin(6\pi t)$

Theorems & Definitions (40)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 3.1
  • Lemma 1
  • Remark 1
  • proof : Proof of Lemma \ref{['lemma1']}
  • Remark 2
  • Lemma 2
  • ...and 30 more