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Nonparametric estimation in fractional SDE

  • Fabienne Comte
  • Nicolas MarieEmail author
Article

Abstract

This paper deals with the consistency and a rate of convergence for a Nadaraya–Watson estimator of the drift function of a stochastic differential equation driven by an additive fractional noise. The results of this paper are obtained via both some long-time behavior properties of Hairer and some properties of the Skorokhod integral with respect to the fractional Brownian motion. These results are illustrated on the fractional Ornstein–Uhlenbeck process.

Keywords

Stochastic differential equations Fractional Brownian motion Nadaraya-Watson estimator Malliavin calculus Long-time behavior Fractional Ornstein-Uhlenbeck process 

Notes

References

  1. Bajja S, Es-Sebaiy K, Viitasaari L (2017) Least square estimator of fractional Ornstein–Uhlenbeck processes with periodic mean. J Korean Stat Soc 36(4):608–622MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bosq D (1996) Nonparametric statistics for stochastic processes: estimation and prediction. Springer, BerlinCrossRefzbMATHGoogle Scholar
  3. Cheridito P, Kawaguchi H, Maejima M (2003) Fractional Ornstein–Uhlenbeck processes. Electron J Probab 8(3):1–14MathSciNetzbMATHGoogle Scholar
  4. Chronopoulou A, Tindel S (2013) On inference for fractional differential equations. Stat Inference Stoch Process 16(1):29–61MathSciNetCrossRefzbMATHGoogle Scholar
  5. Friz P, Hairer M (2014) A course on rough paths. Springer, BerlinCrossRefzbMATHGoogle Scholar
  6. Friz P, Victoir N (2010) Multidimensional stochastic processes as rough paths: theory and applications. Cambridge studies in applied mathematics, vol 120. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  7. Hairer M (2005) Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann Probab 33(3):703–758MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hairer M, Ohashi A (2007) Ergodic theory for SDEs with extrinsic memory. Ann Probab 35(5):1950–1977MathSciNetCrossRefzbMATHGoogle Scholar
  9. Hairer M, Pillai NS (2013) Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. Annales de l’IHP 47(4):2544–2598zbMATHGoogle Scholar
  10. Hu Y, Nualart D (2010) Parameter estimation for fractional Ornstein–Uhlenbeck processes. Stat Probab Lett 80:1030–1038MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hu Y, Nualart D, Zhou H (2018) Drift parameter estimation for nonlinear stochastic differential equations driven by fractional Brownian motion. arXiv:1803.01032v1
  12. Kleptsyna ML, Le Breton A (2001) Some explicit statistical results about elementary fractional type models. Nonlinear Anal 47:4783–4794MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kubilius K, Skorniakov V (2016) On some estimators of the Hurst index of the solution of SDE driven by a fractional Brownian motion. Stat Probab Lett 109:159–167MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kutoyants Y (2004) Statistical inference for ergodic diffusion processes. Springer, BerlinCrossRefzbMATHGoogle Scholar
  15. Lejay A (2010) Controlled differential equations as Young integrals: a simple approach. J Differ Equ 249:1777–1798MathSciNetCrossRefzbMATHGoogle Scholar
  16. Lindgren G (2006) Lectures on stationary stochastic processes. Ph.D. course of Lund’s UniversityGoogle Scholar
  17. Mishra MN, Prakasa Rao BLS (2011) Nonparameteric estimation of trend for stochastic differential equations driven by fractional Brownian motion. Stat Inference Stoch Process 14(2):101–109MathSciNetCrossRefzbMATHGoogle Scholar
  18. Mishura Y, Ralchenko K (2014) On drift parameter estimation in models with fractional Brownian motion by discrete observations. Austrian J Stat 43(3–4):217–228Google Scholar
  19. Neuenkirch A, Tindel S (2014) A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise. Stat Inference Stoch Process 17(1):99–120MathSciNetCrossRefzbMATHGoogle Scholar
  20. Nualart D (2006) The Malliavin calculus and related topics. Springer, BerlinzbMATHGoogle Scholar
  21. Puig B, Poirion F, Soize C (2002) Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms. Probab Eng Mech 17:253–264CrossRefGoogle Scholar
  22. Revuz D, Yor M (1999) Continuous Martingales and Brownian motion. A series of comprehensive studies in mathematics, vol 293, 3rd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  23. Saussereau B (2014) Nonparametric inference for fractional diffusion. Bernoulli 20(2):878–918MathSciNetCrossRefzbMATHGoogle Scholar
  24. Taqqu MS (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z Warscheinlichkeitstheorie verw Gebiete 31:287–302MathSciNetCrossRefzbMATHGoogle Scholar
  25. Tsybakov AB (2009) Introduction to nonparametric estimation. Revised and extended from the 2004 French original. Translated by Vladimir Zaiats. Springer Series in Statistics. Springer, New YorkGoogle Scholar
  26. Tudor CA, Viens F (2007) Statistical aspects of the fractional stochastic calculus. Ann Stat 35(3):1183–1212MathSciNetCrossRefzbMATHGoogle Scholar
  27. Tudor CA, Viens F (2009) Variations and estimators for self-similarity parameters via Malliavin calculus. Ann Probab 37(6):2093–2134MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Laboratoire MAP5Université Paris DescartesParisFrance
  2. 2.Laboratoire Modal’XUniversité Paris NanterreNanterreFrance
  3. 3.ESME SudriaParisFrance

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