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Rough Hurst function estimation

Fabian Mies, Benedikt Wilkens

TL;DR

This work advocates Itô-mBm as a tractable nonstationary extension of fractional Brownian motion by showing that the local Hurst function $H_t$ can be estimated with standard nonparametric rates for any smoothness level $\eta>0$, in contrast to classical mBm which requires $\eta>1$. It introduces two local estimators based on kernel-weighted local polynomial regression of second-order increments, achieving convergence rates that adapt to the local smoothness, and derives an integrated-Hurst estimator $\widehat{\mathcal{H}}(u)$ with a functional central limit theorem, enabling Changepoint and goodness-of-fit testing that are robust to volatility nuisance. The results are grounded in a functional CLT for locally stationary time series and a multiplier bootstrap framework, providing feasible inference procedures. Collectively, the paper enhances inference for nonstationary fractional models and positions Itô-mBm as a preferred baseline model for statistical analysis of rough, time-varying Hurst behavior, with practical tests for constancy and shape of $H_t$.

Abstract

The fractional Brownian motion (fBm) is parameterized by the Hurst exponent $H\in(0,1)$, which determines the dependence structure and regularity of sample paths. Empirical findings suggest that the Hurst exponent may be non-constant in time, giving rise to the so-called multifractional Brownian motion (mBm). The Itô-mBm is an alternative to the classical mBm, and has been shown to admit more intuitive sample path properties in case the Hurst function is rough. In this paper, we show that the Itô-mBm also allows for a simplified statistical treatment compared to the classical mBm. In particular, estimation of the local Hurst parameter $H(t)$ with Hölder exponent $η>0$ achieves rates of convergence which are standard in nonparametric regression, whereas similar results for the classical mBm only hold for the smoother regime $η>1$. Furthermore, we derive an estimator of the integrated Hurst exponent $\int_0^t H(s)\, ds$ which achieves a parametric rate of convergence, and use it to construct goodness-of-fit tests.

Rough Hurst function estimation

TL;DR

This work advocates Itô-mBm as a tractable nonstationary extension of fractional Brownian motion by showing that the local Hurst function can be estimated with standard nonparametric rates for any smoothness level , in contrast to classical mBm which requires . It introduces two local estimators based on kernel-weighted local polynomial regression of second-order increments, achieving convergence rates that adapt to the local smoothness, and derives an integrated-Hurst estimator with a functional central limit theorem, enabling Changepoint and goodness-of-fit testing that are robust to volatility nuisance. The results are grounded in a functional CLT for locally stationary time series and a multiplier bootstrap framework, providing feasible inference procedures. Collectively, the paper enhances inference for nonstationary fractional models and positions Itô-mBm as a preferred baseline model for statistical analysis of rough, time-varying Hurst behavior, with practical tests for constancy and shape of .

Abstract

The fractional Brownian motion (fBm) is parameterized by the Hurst exponent , which determines the dependence structure and regularity of sample paths. Empirical findings suggest that the Hurst exponent may be non-constant in time, giving rise to the so-called multifractional Brownian motion (mBm). The Itô-mBm is an alternative to the classical mBm, and has been shown to admit more intuitive sample path properties in case the Hurst function is rough. In this paper, we show that the Itô-mBm also allows for a simplified statistical treatment compared to the classical mBm. In particular, estimation of the local Hurst parameter with Hölder exponent achieves rates of convergence which are standard in nonparametric regression, whereas similar results for the classical mBm only hold for the smoother regime . Furthermore, we derive an estimator of the integrated Hurst exponent which achieves a parametric rate of convergence, and use it to construct goodness-of-fit tests.

Paper Structure

This paper contains 11 sections, 13 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Local nonparametric estimation
  3. Integrated parameter estimation
  4. Testing for constant Hurst exponent
  5. Application to goodness-of-fit testing
  6. Discussion
  7. Multiplier invariance principle
  8. Proofs
  9. Preliminaries
  10. Local nonparametric estimation
  11. Integrated parameter estimation

Key Result

theorem 1

Suppose that $v\mapsto \theta_v = (\sigma_v, H_v)$ is Hölder continuous with exponent $\eta\in(0,l+1]$, and that $0<\underline{H} \leq H_v \leq \overline{H} < 1$ for all $v$, and $0<\underline{\sigma}^2\leq \sigma_v^2 \leq \overline{\sigma}^2<\infty$. If $b_n \ll n^{q-1}$ for some $q\in(0,1)$, then The bound holds uniformly for $\theta \in [\underline{\sigma}^2,\overline{\sigma}^2] \times [\under

Theorems & Definitions (31)

  • Remark 1
  • theorem 1
  • Remark 2
  • theorem 2
  • Remark 3
  • Remark 4
  • Remark 5
  • theorem 3
  • Remark 6
  • theorem 4
  • ...and 21 more