Abstract
We construct the least-square estimator for the unknown drift parameter in the multifractional Ornstein–Uhlenbeck model and establish its strong consistency in the non-ergodic case. The proofs are based on the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of multifractional Brownian motion (mBm) and of some other functionals of mBm, including increments and fractional derivatives. As the auxiliary results having independent interest, we produce the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of the general Gaussian process and some functionals of it, in terms of the covariance function of its increments.
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References
Ayache A, Cohen S, Véhel JL (2000) The covariance structure of multifractional Brownian motion, with application to long range dependence. In: Proceedings 2000 IEEE international conference on acoustics, speech, and signal processing – ICASSP’00, vol 6, pp 3810–3813
Azmoodeh E, Morlanes JI (2015) Drift parameter estimation for fractional Ornstein-Uhlenbeck process of the second kind. Statistics 49(1):1–18
Belfadli R, Es-Sebaiy K, Ouknine Y (2011) Parameter estimation for fractional Ornstein-Uhlenbeck processes: non-ergodic case. Front Sci Eng 1(1):1–16
Benassi A, Jaffard S, Roux D (1997) Gaussian processes and pseudodifferential elliptic operators. Rev Math Iberoam 13(1):19–89
Buldygin V, Kozachenko Y (2000) Metric characterization of random variables and random processes. AMS, American Mathematical Society, Providence
El Machkouri M, Es-Sebaiy K, Ouknine Y (2016) Least squares estimator for non-ergodic Ornstein-Uhlenbeck processes driven by Gaussian processes. J Korean Stat Soc 45(3):329–341
Fernique X (1964) Continuité des processus Gaussiens. C R Acad Sci Paris 258:6058–6060
Fernique X (1975) Regularité des trajectoires des fonctions aléatoires gaussiennes. In: École d’Été de Probabilités de Saint-Flour, IV-1974 (Lecture notes in mathematics), vol 480. Springer, Berlin, pp 1–96
Hu Y, Nualart D (2010) Parameter estimation for fractional Ornstein-Uhlenbeck processes. Stat Probab Lett 80(11–12):1030–1038
Kleptsyna M, Le Breton A (2002) Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Stat Inference Stoch Process 5(3):229–248
Kozachenko Y, Melnikov A, Mishura Y (2015) On drift parameter estimation in models with fractional Brownian motion. Statistics 49(1):35–62
Kubilius K, Mishura Y, Ralchenko K, Seleznjev O (2015) Consistency of the drift parameter estimator for the discretized fractional Ornstein-Uhlenbeck process with Hurst index \(H\in (0,\frac{1}{2})\). Electron J Stat 9(2):1799–1825
Lifshits M (2012) Lectures on Gaussian processes. Springer briefs in mathematics. Springer, Heidelberg
Mishura Y, Ralchenko K (2014) On drift parameter estimation in models with fractional Brownian motion by discrete observations. Austrian J Stat 43(3):218–228
Mishura Y, Ralchenko K, Seleznev O, Shevchenko G (2014) Asymptotic properties of drift parameter estimator based on discrete observations of stochastic differential equation driven by fractional Brownian motion. In: Modern stochastics and applications. Springer optimization and its applications, vol 90. Springer, Cham, pp 303–318
Piterbarg VI (1996) Asymptotic methods in the theory of Gaussian processes and fields, Translations of Mathematical Monographs, vol 148. American Mathematical Society, Providence
Ralchenko KV (2011) Approximation of multifractional Brownian motion by absolutely continuous processes. Theory Probab Math Stat 82:115–127
Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives. Gordon and Breach Science, Yverdon
Stoev SA, Taqqu MS (2006) How rich is the class of multifractional Brownian motions? Stoch Process Appl 116(2):200–221
Talagrand M (2014) Upper and lower bounds for stochastic processes: modern methods and classical problems, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., A series of modern surveys in mathematicsSpringer, Heidelberg
Tudor CA, Viens FG (2007) Statistical aspects of the fractional stochastic calculus. Ann Stat 35(3):1183–1212
Zähle M (1998) Integration with respect to fractal functions and stochastic calculus. I. Probab Theory Relat Fields 111(3):333–374
Zähle M (1999) On the link between fractional and stochastic calculus. In: Stochastic dynamics (Bremen, 1997). Springer, New York, pp 305–325
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The authors are grateful to the anonymous referees for their useful remarks and suggestions which contributed to a substantial improvement of the text.
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Dozzi, M., Kozachenko, Y., Mishura, Y. et al. Asymptotic growth of trajectories of multifractional Brownian motion, with statistical applications to drift parameter estimation. Stat Inference Stoch Process 21, 21–52 (2018). https://doi.org/10.1007/s11203-016-9147-z
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DOI: https://doi.org/10.1007/s11203-016-9147-z
Keywords
- Gaussian process
- Multifractional Brownian motion
- Parameter estimation
- Consistency
- Strong consistency
- Stochastic differential equation