Modern Stochastics: Theory and Applications logo


  • Help
Login Register

  1. Home
  2. Issues
  3. Volume 8, Issue 3 (2021)
  4. Convergence rate of CLT for the drift es ...

Modern Stochastics: Theory and Applications

Submit your article Information Become a Peer-reviewer
  • Article info
  • Full article
  • Related articles
  • More
    Article info Full article Related articles

Convergence rate of CLT for the drift estimation of sub-fractional Ornstein–Uhlenbeck process of second kind
Volume 8, Issue 3 (2021), pp. 329–347
Maoudo Faramba Baldé   Khalifa Es-Sebaiy ORCID icon link to view author Khalifa Es-Sebaiy details  

Authors

 
Placeholder
https://doi.org/10.15559/21-VMSTA179
Pub. online: 20 May 2021      Type: Research Article      Open accessOpen Access

Received
8 October 2020
Revised
3 March 2021
Accepted
4 May 2021
Published
20 May 2021

Abstract

In this paper, we deal with an Ornstein–Uhlenbeck process driven by sub-fractional Brownian motion of the second kind with Hurst index $H\in (\frac{1}{2},1)$. We provide a least squares estimator (LSE) of the drift parameter based on continuous-time observations. The strong consistency and the upper bound $O(1/\sqrt{n})$ in Kolmogorov distance for central limit theorem of the LSE are obtained. We use a Malliavin–Stein approach for normal approximations.

References

[1] 
Alazemi, F., Alsenafi, A., Es-Sebaiy, K.: Parameter estimation for Gaussian mean-reverting Ornstein-Uhlenbeck processes of the second kind: non-ergodic case. Stoch. Dyn. 19(5), 2050011 (25 pages) (2020). MR4080159. https://doi.org/10.1142/S0219493720500112
[2] 
Azmoodeh, E., Morlanes, G.I.: Drift parameter estimation for fractional Ornstein-Uhlenbeck process of the second kind. Statistics 49(1), 1–18 (2013). MR3304364. https://doi.org/10.1080/02331888.2013.863888
[3] 
Azmoodeh, E., Viitasaari, L.: Parameter estimation based on discrete observations of fractional Ornstein-Uhlenbeck process of the second kind. Stat. Inference Stoch. Process. 18(3), 205–227 (2015). MR3395605. https://doi.org/10.1007/s11203-014-9111-8
[4] 
Balde, M.F., Belfadli, R., Es-Sebaiy, K.: Berry-Esséen bound for drift estimation of fractional Ornstein-Uhlenbeck process of second kind (2020). preprint. arXiv:2005.08397. MR4128741. https://doi.org/10.1142/S0219493720500239
[5] 
Bajja, S., Es-Sebaiy, K., Viitasaari, L.: Least squares estimator of fractional Ornstein-Uhlenbeck processes with periodic mean. J. Korean Stat. Soc. 46(4), 608–622 (2017). MR3718150. https://doi.org/10.1016/j.jkss.2017.06.002
[6] 
Chronopoulou, A., Viens, F.: Stochastic volatility and option pricing with long-memory in discrete and continuous time. Quant. Finance 12, 635–649 (2012). MR2909603. https://doi.org/10.1080/14697688.2012.664939
[7] 
Comte, F., Coutin, L., Renault, E.: Affine fractional stochastic volatility models. Ann. Finance 8, 337–378 (2012). MR2922801. https://doi.org/10.1007/s10436-010-0165-3
[8] 
Chen, Y., Kuang, N., Li, Y.: Berry-Esséeen bound for the parameter estimation of fractional Ornstein-Uhlenbeck processes. Stoch. Dyn. 20(1), 2050023 (2019). MR4128741. https://doi.org/10.1142/S0219493720500239
[9] 
Chen, Y., Li, Y.: Berry-Esséen bound for the parameter estimation of fractional Ornstein-Uhlenbeck processes with the hurst parameter $H\in (0,\frac{1}{2})$. Commun. Stat., Theory Methods, 1–18 (2019). https://doi.org/10.1080/03610926.2019.1704007
[10] 
Douissi, S., Es-Sebaiy, K., Viens, F.: Berry-Esséen bounds for parameter estimation of general Gaussian processes. ALEA Lat. Am. J. Probab. Math. Stat. 16, 633–664 (2019). MR3949273. https://doi.org/10.30757/alea.v16-23
[11] 
El Machkouri, M., Es-Sebaiy, K., Ouknine, Y.: Least squares estimator for non-ergodic Ornstein-Uhlenbeck processes driven by Gaussian processes. J. Korean Stat. Soc. 45(3), 329–341 (2016). MR3527650. https://doi.org/10.1016/j.jkss.2015.12.001
[12] 
El Onsy, B., Es-Sebaiy, K., Viens, F.: Parameter estimation for a partially observed Ornstein-Uhlenbeck process with long-memory noise. Stochastics 89(2), 431–468 (2017). MR3590429. https://doi.org/10.1080/17442508.2016.1248967
[13] 
Hu, Y., Nualart, D.: Parameter estimation for fractional Ornstein Uhlenbeck processes. Stat. Probab. Lett. 80(11–12), 1030–1038 (2010). MR2638974. https://doi.org/10.1016/j.spl.2010.02.018
[14] 
Kloeden, P., Neuenkirch, A.: The pathwise convergence of approximation schemes for stochastic differential equations. LMS J. Comput. Math. 10, 235–253 (2007). MR2320830. https://doi.org/10.1112/S1461157000001388
[15] 
Kaarakka, T., Salminen, P.: On Fractional Ornstein-Uhlenbeck process. Commun. Stoch. Anal. 5, 121–133 (2011). MR2808539. https://doi.org/10.31390/cosa.5.1.08
[16] 
Kim, Y.T., Park, H.S.: Optimal Berry-Esseen bound for statistical estimations and its application to SPDE. J. Multivar. Anal. 155, 284–304 (2017). MR3607896. https://doi.org/10.1016/j.jmva.2017.01.006
[17] 
Kleptsyna, M., Le Breton, A.: Statistical analysis of the fractional Ornstein-Uhlenbeck type process. In: Statistical Inference for Stochastic Processes, vol. 5, pp. 229–241 (2002). MR1943832. https://doi.org/10.1023/A:1021220818545
[18] 
Nourdin, I., Peccati, G.: Normal Approximations with Malliavin calculus: From Stein’s method to Universality. Cambridge University Press, (2012). MR2962301. https://doi.org/10.1017/CBO9781139084659
[19] 
Nualart, D.: The Malliavin calculus and related topics, 2nd edn. Springer, Berlin (2006). MR2200233
[20] 
Prakasa Rao, B.L.S.: Berry-Esseen type bound for fractional Ornstein-Uhlenbeck type process driven by sub-fractional Brownian motion. Theory Stoch. Process. 23(39), Issue 1, 82–92 (2018). MR3948508
[21] 
Tudor, C.: On the Wiener integral with respect to a sub-fractional Brownian motion on an interval. J. Math. Anal. Appl. 351, 456–468 (2009). MR2472957. https://doi.org/10.1016/j.jmaa.2008.10.041

Full article Related articles PDF XML
Full article Related articles PDF XML

Copyright
© 2021 The Author(s). Published by VTeX
by logo by logo
Open access article under the CC BY license.

Keywords
Sub-fractional Ornstein–Uhlenbeck process of second kind least squares estimator Berry–Esséen bound Malliavin–Stein approach for normal approximations

MSC2010
60G15 60G22 62F12 62M09 60H07

Funding
M. F. Baldé acknowledges support from NLAGA project of SIMONS foundation.

Metrics
since March 2018
621

Article info
views

570

Full article
views

392

PDF
downloads

138

XML
downloads

Export citation

Copy and paste formatted citation
Placeholder

Download citation in file


Share


RSS

MSTA

MSTA

  • Online ISSN: 2351-6054
  • Print ISSN: 2351-6046
  • Copyright © 2018 VTeX

About

  • About journal
  • Indexed in
  • Editors-in-Chief

For contributors

  • Submit
  • OA Policy
  • Become a Peer-reviewer

Contact us

  • ejournals-vmsta@vtex.lt
  • Mokslininkų 2A
  • LT-08412 Vilnius
  • Lithuania
Powered by PubliMill  •  Privacy policy