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Rough Hurst function estimation


Preprint


Fabian Mies, Benedikt Wilkens
Stochastic Processes and their Applications, accepted, 2026


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APA   Click to copy
Mies, F., & Wilkens, B. (2026). Rough Hurst function estimation. Stochastic Processes and Their Applications, accepted. https://doi.org/10.1016/j.spa.2026.105143


Chicago/Turabian   Click to copy
Mies, Fabian, and Benedikt Wilkens. “Rough Hurst Function Estimation.” Stochastic Processes and their Applications accepted (2026).


MLA   Click to copy
Mies, Fabian, and Benedikt Wilkens. “Rough Hurst Function Estimation.” Stochastic Processes and Their Applications, vol. accepted, 2026, doi:10.1016/j.spa.2026.105143.


BibTeX   Click to copy

@article{fabian2026a,
  title = {Rough Hurst function estimation},
  year = {2026},
  journal = {Stochastic Processes and their Applications},
  volume = {accepted},
  doi = {10.1016/j.spa.2026.105143},
  author = {Mies, Fabian and Wilkens, Benedikt}
}

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 for various phenomena, 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 this paper, we show that 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 $\eta>0$ achieves rates of convergence which are standard in nonparametric regression, whereas similar results for the classical mBm only hold for $\eta>1$. 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.

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