Statistical inference for models driven by $n$-th order fractional Brownian motion

Authors:
Hicham Chaouch, Hamid El Maroufy and Mohamed El Omari

Journal:
Theor. Probability and Math. Statist. **108** (2023), 29-43

MSC (2020):
Primary 62F12; Secondary 60G15, 60G18

DOI:
https://doi.org/10.1090/tpms/1185

Published electronically:
May 2, 2023

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Abstract: We consider the following stochastic integral equation $X(t)=\mu t + \sigma \int _0^t \varphi (s) dB_H^n(s)$, $t\geq 0$, where $\varphi$ is a known function and $B^n_H$ is the $n$-th order fractional Brownian motion. We provide explicit maximum likelihood estimators for both $\mu$ and $\sigma ^2$, then we formulate explicitly a least squares estimator for $\mu$ and an estimator for $\sigma ^2$ by using power variations method. The consistency and asymptotic normality are established for those estimators when the number of observations or the time horizon is sufficiently large.

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References
- R. Belfadli, K. Es-Sebaiy, and Y. Ouknine,
*Parameter estimation for fractional Ornstein–Uhlenbeck processes: Non-ergodic case,* Frontiers in Science and Engineering, **1** (2011), no. 1, 1–16.
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*Statistics for long-memory processes*, Monographs on Statistics and Applied Probability, 61, Chapman and Hall, New York, 1994. MR **1304490**
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*Algorithm 1008: Multicomplex number class for Matlab, with a focus on the accurate calculation of small imaginary terms for multicomplex step sensitivity calculations*, ACM Trans. Math. Software **46** (2020), no. 2, Art. 18, 26 pp. MR **4103644**
- J.M. Corcuera, D. Nualart, and J.H.C. Woerner,
*Power variation of some integral fractional processes*, Bernoulli **12** (2006), no. 4, 713–735. MR **2248234**
- M. El Omari,
*Mixtures of higher-order fractional Brownian motions*, Comm. Statist. Theory Methods, (2021), 1–16.
- M. El Omari,
*An $\alpha$-order fractional Brownian motion with Hurst index $H\in (0,1)$ and $\alpha \in \mathbb {R}_+$*, Sankhya A **85** (2023), no. 1, 572–599. MR **4540809**
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*Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation*, Acta Math. Sci. Ser. B (Engl. Ed.) **31** (2011), no. 5, 1851–1859. MR **2884954**
- Y. Hu, and D. Nualart,
*Parameter estimation for fractional Ornstein–Uhlenbeck processes*, Statist. Probab. Lett. **80** (2010), no. 11-12, 1030–1038. MR **2638974**
- Y. Hu, D. Nualart, and H. Zhou,
*Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter*, Stat. Inference Stoch. Process. **22** (2019), no. 1, 111–142. MR **3918739**
- R. Jennane and R. Harba,
*Fractional Brownian motion: A model for image texture*, EUSIPCO, Signal Processing, **3** (1994), 1389–1392.
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*Statistical analysis of the fractional Ornstein–Uhlenbeck type process*, Stat. Inference Stoch. Process. **5** (2002), no. 3, 229–248. MR **1943832**
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*Using multicomplex variables for automatic computation of high-order derivatives*, ACM Trans. Math. Software **38** (2012), no. 3, Art. 16, 21 pp. MR **2923553**
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* Fractional Brownian motion: A maximum likelihood estimator and its application to image texture*, IEEE Transactions on medical imaging, **5**(1986), no. 3, 152–161.
- B. Mandelbrot and J. W. Van Ness,
*Fractional Brownian motions, fractional noises and applications*, SIAM Rev. **10** (1968), 422–437. MR **242239**
- D. Nualart,
*The Malliavin calculus and related topics*, Probability and its Applications, Springer, New York, 1995. MR **1344217**
- D. Nualart and S. Ortiz-Latorre.
*Central limit theorems for multiple stochastic integrals and Malliavin calculus*, Stochastic Process. Appl. **118** (2008), no. 4, 614–628. MR **2394845**
- A. P. Pentland,
*Fractal-based description of natural scenes*, IEEE transactions on pattern analysis and machine intelligence, **6** (1984), 661–674.
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*$n$th-order fractional Brownian motion and fractional Gaussian noises*, IEEE Transactions on Signal Processing, **49** (2001), no. 5, 1049–1059.
- V. Pipiras and M.S. Taqqu,
*Integration questions related to fractional Brownian motion*, Probab. Theory Related Fields **118** (2000), no. 2, 251–291. MR **1790083**
- J.R. Schott,
*Matrix analysis for statistics*, third edition, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ, 2017. MR **3497549**
- T. Sottinen, and L. Viitasaari,
*Transfer principle for $n$th order fractional Brownian motion with applications to prediction and equivalence in law*, Theory Probab. Math. Statist. (2019), No. 98, 199–216. MR **3824687**
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* Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, NJ, 1970. MR **0290095**
- K. Tanaka,
*Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process*, Stat. Inference Stoch. Process. **18** (2015), no. 3, 315–332. MR **3395610**
- K. Tanaka,
*Comparison of the LS-based estimators and the MLE for the fractional Ornstein–Uhlenbeck process*, Stat. Inference Stoch. Process. **23** (2020), no. 2, 415–434. MR **4123930**
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*Matrix computations*, third edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins Univ. Press, Baltimore, MD, 1996. MR **1417720**
- W. Xiao and J. Yu,
* Asymptotic theory for estimating drift parameters in the fractional Vasicek model*, Econometric Theory **35** (2019), no. 1, 198–231. MR **3904176**
- M. Zabat, M. Vayer-Besançon, R. Harba, S. Bonnamy, and H. Van Damme,
*Surface topography and mechanical properties of smectite films*, Trends in Colloid and Interface Science XI, (1997), 96–102.

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Additional Information

**Hicham Chaouch**

Affiliation:
Faculty of Sciences and Technics, Sultan Mouly Slimane University, Campus Mghilla, BP 523, Beni-Mellal, Morocco

**Hamid El Maroufy**

Affiliation:
Faculty of Sciences and Technics, Sultan Mouly Slimane University, Campus Mghilla, BP 523, Beni-Mellal, Morocco

Email:
h.elmaroufy@usms.ma

**Mohamed El Omari**

Affiliation:
Polydisciplinary Faculty of Sidi Bennour, Chouaıb Doukkali University, B.P. 299, Jabrane Khalil Jabrane Street, 24000 El Jadida, Morocco

Email:
elomari.m@ucd.ac.ma

Keywords:
$n$-th order fractional Brownian motion,
maximum likelihood estimator,
least squares estimator,
consistency,
asymptotic normality

Received by editor(s):
July 25, 2022

Accepted for publication:
December 14, 2022

Published electronically:
May 2, 2023

Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv