On local path behavior of Surgailis multifractional processes

Authors:
A. Ayache and F. Bouly

Journal:
Theor. Probability and Math. Statist. **106** (2022), 3-26

MSC (2020):
Primary 60G17, 60G22; Secondary 60H05

DOI:
https://doi.org/10.1090/tpms/1162

Published electronically:
May 16, 2022

MathSciNet review:
4438441

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Abstract:

Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the *classical* Multifractional Brownian Motion (MBM) $\{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}$ of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter ${\mathcal {H}}$ of the well-known Fractional Brownian Motion by a deterministic function ${\mathcal {H}}(t)$ having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two *non-classical* Gaussian multifactional processes denoted by $\{X(t)\}_{t\in \mathbb {R}}$ and $\{Y(t)\}_{t\in \mathbb {R}}$.

In our article, under a rather weak condition on the functional parameter ${\mathcal {H}}(\cdot )$, we show that $\{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}$ and $\{X(t)\}_{t\in \mathbb {R}}$ as well as $\{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}$ and $\{Y(t)\}_{t\in \mathbb {R}}$ only differ by a part which is locally more regular than $\{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}$ itself. On one hand this result implies that the two *non-classical* multifractional processes $\{X(t)\}_{t\in \mathbb {R}}$ and $\{Y(t)\}_{t\in \mathbb {R}}$ have exactly the same local path behavior as that of the *classical* MBM $\{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}$. On the other hand it allows to obtain from discrete realizations of $\{X(t)\}_{t\in \mathbb {R}}$ and $\{Y(t)\}_{t\in \mathbb {R}}$ strongly consistent statistical estimators for values of their functional parameter.

References
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References
- A. Ayache,
*Continuous Gaussian multifractional processes with random pointwise Hölder regularity*, J. Theoret. Probab. **26** (2013), no. 1, 72–93. MR **3023836**
- A. Ayache,
*Multifractional stochastic fields*, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. MR **3839281**
- A. Ayache, S. Jaffard, and M. S. Taqqu,
*Wavelet construction of generalized multifractional processes*, Rev. Mat. Iberoam. **23** (2007), no. 1, 327–370. MR **2351137**
- A. Ayache and J. Lévy Véhel,
*On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion*, Stochastic Process. Appl. **111** (2004), no. 1, 119–156. MR **2049572**
- P. Balança,
*Some sample path properties of multifractional Brownian motion*, Stochastic Process. Appl. **125** (2015), no. 10, 3823–3850. MR **3373305**
- J.-M. Bardet and D. Surgailis,
*Nonparametric estimation of the local Hurst function of multifractional Gaussian processes*, Stochastic Process. Appl. **123** (2013), no. 3, 1004–1045. MR **3005013**
- A. Benassi, S. Jaffard, and D. Roux,
*Elliptic Gaussian random processes*, Rev. Mat. Iberoamericana **13** (1997), no. 1, 19–90. MR **1462329**
- S. M. Berman,
*Gaussian sample functions: Uniform dimension and Hölder conditions nowhere*, Nagoya Math. J. **46** (1972), 63–86. MR **307320**
- S. M. Berman,
*Local nondeterminism and local times of Gaussian processes*, Indiana Univ. Math. J. **23** (1973/74), 69–94. MR **317397**
- E. Herbin,
*From $N$ parameter fractional Brownian motions to $N$ parameter multifractional Brownian motions*, Rocky Mountain J. Math. **36** (2006), no. 4, 1249–1284. MR **2274895**
- I. Karatzas and S. E. Shreve,
*Brownian motion and stochastic calculus*, Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1987. MR **917065**
- D. Khoshnevisan,
*Multiparameter processes*, Springer Monographs in Mathematics, Springer-Verlag, New York, 2002. MR **1914748**
- R. Peltier and J. Lévy Véhel,
*Multifractional Brownian motion: definition and preliminary results*, Rapport de recherche INRIA (1995), no. 2645.
- L. D. Pitt,
*Local times for Gaussian vector fields*, Indiana Univ. Math. J. **27** (1978), no. 2, 309–330. MR **471055**
- D. Surgailis,
*Nonhomogeneous fractional integration and multifractional processes*, Stochastic Process. Appl. **118** (2008), no. 2, 171–198. MR **2376898**
- Y. Xiao,
*Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields*, Probab. Theory Related Fields **109** (1997), no. 1, 129–157. MR **1469923**

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Additional Information

**A. Ayache**

Affiliation:
Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France

Email:
antoine.ayache@univ-lille.fr

**F. Bouly**

Affiliation:
Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France

Email:
florent.bouly@univ-lille.fr

Keywords:
Gaussian processes,
variable Hurst parameter,
local and pointwise Hölder regularity,
local self-similarity

Received by editor(s):
June 25, 2021

Accepted for publication:
October 18, 2021

Published electronically:
May 16, 2022

Additional Notes:
This work has been partially supported by the Labex CEMPI (ANR-11-LABX-0007-01) and the GDR 3475 (Analyse Multifractale et Autosimilarité).

Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv