Asymptotic behavior of mixed power variations and statistical estimation in mixed models

Article
  • 158 Downloads

Abstract

We obtain results on both weak and almost sure asymptotic behaviour of power variations of a linear combination of independent Wiener process and fractional Brownian motion. These results are used to construct strongly consistent parameter estimators in mixed models.

Keywords

Power variation Fractional Brownian motion Hurst parameter Wiener process Consistent estimator 

Mathematics Subject Classification

60G22 62M09 60G15 62F25 

References

  1. Achard S, Coeurjolly J-F (2010) Discrete variations of the fractional Brownian motion in the presence of outliers and an additive noise. Stat Surv 4:117–147CrossRefMATHMathSciNetGoogle Scholar
  2. Androshchuk T, Mishura Y (2006) Mixed Brownian-fractional Brownian model: absence of arbitrage and related topics. Stochastics 78(5):281–300MATHMathSciNetGoogle Scholar
  3. Arcones MA (1994) Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann Probab 22(4):2242–2274CrossRefMATHMathSciNetGoogle Scholar
  4. Bardet J-M, Bertrand P (2007) Identification of the multiscale fractional Brownian motion with biomechanical applications. J Time Ser Anal 28(1):1–52CrossRefMATHMathSciNetGoogle Scholar
  5. Bardet J-M, Surgailis D (2013) Moment bounds and central limit theorems for Gaussian subordinated arrays. J Multivar Anal 114:457–473CrossRefMATHMathSciNetGoogle Scholar
  6. Begyn A (2007) Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. Bernoulli 13(3):712–753CrossRefMATHMathSciNetGoogle Scholar
  7. Benassi A, Cohen S, Istas J (1998) Identifying the multifractional function of a Gaussian process. Stat Probab Lett 39(4):337–345CrossRefMATHMathSciNetGoogle Scholar
  8. Benassi A., Deguy S (1999) Multi-scale fractional Brownian motion: definition and identification. Preprint LAICGoogle Scholar
  9. Breuer P, Major P (1983) Central limit theorems for nonlinear functionals of Gaussian fields. J Multivar Anal 13(3):425–441CrossRefMATHMathSciNetGoogle Scholar
  10. Cai, C, Chigansky P, Kleptsyna M (2012) ‘The maximum likelihood drift estimator for mixed fractional Brownian motion’. Preprint, available online at http://arxiv.org/abs/1208.6253
  11. Cheridito P (2001) Mixed fractional Brownian motion. Bernoulli 7(6):913–934CrossRefMATHMathSciNetGoogle Scholar
  12. Coeurjolly J-F (2000) Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study. J Stat Softw 5:1–53Google Scholar
  13. Coeurjolly J-F (2001) Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat Inference Stoch Process 4(2):199–227CrossRefMATHMathSciNetGoogle Scholar
  14. Coeurjolly J-F (2005) Identification of multifractional Brownian motion. Bernoulli 11(6):987–1008CrossRefMATHMathSciNetGoogle Scholar
  15. Dobrushin R, Major P (1979) Non-central limit theorems for nonlinear functionals of Gaussian fields. Z Wahrsch Verw Gebiete 50(1):27–52CrossRefMATHMathSciNetGoogle Scholar
  16. Filatova D (2008) Mixed fractional Brownian motion: some related questions for computer network traffic modeling. International conference on signals and electronic systems, Kraków 2008, pp 393–396Google Scholar
  17. Giraitis L, Robinson PM, Surgailis D (1999) Variance-type estimation of long memory. Stoch Process Appl 80(1):1–24CrossRefMATHMathSciNetGoogle Scholar
  18. Giraitis L, Surgailis D (1985) CLT and other limit theorems for functionals of Gaussian processes. Z Wahrsch Verw Gebiete 70(2):191–212CrossRefMATHMathSciNetGoogle Scholar
  19. Istas J, Lang G (1997) Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann Inst Henri Poincaré Probab Statist 33(4):407–436CrossRefMATHMathSciNetGoogle Scholar
  20. Janson S (1997) Gaussian Hilbert spaces, Vol. 129 of Cambridge tracts in mathematics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  21. Kent JT, Wood ATA (1997) Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J R Stat Soc Ser B 59(3):679–699MATHMathSciNetGoogle Scholar
  22. Kozachenko Y, Melnikov A, Mishura Y (2012) ‘On drift parameter estimation in models with fractional Brownian motion’. Statistics. To appear, available online at http://arxiv.org/abs/1112.2330
  23. Mishura Y, Shevchenko G (2012) Mixed stochastic differential equations with long-range dependence: existence, uniqueness and convergence of solutions. Comput Math Appl 64(10):3217–3227CrossRefMATHMathSciNetGoogle Scholar
  24. Mishura YS (2008) Stochastic calculus for fractional Brownian motion and related processes, Vol. (1929) of Lecture Notes in Mathematics. Springer-Verlag, BerlinGoogle Scholar
  25. Nourdin I (2008) Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion. Ann Probab 36(6):2159–2175CrossRefMATHMathSciNetGoogle Scholar
  26. Nourdin I, Nualart D, Tudor CA (2010) Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann Inst Henri Poincaré Probab Stat 46(4):1055–1079CrossRefMATHMathSciNetGoogle Scholar
  27. Taqqu MS (1979) Convergence of integrated processes of arbitrary Hermite rank. Z Wahrsch Verw Gebiete 50(1):53–83CrossRefMATHMathSciNetGoogle Scholar
  28. van Zanten H (2007) When is a linear combination of independent fBm’s equivalent to a single fBm? Stoch Process Appl 117(1):57–70CrossRefMATHGoogle Scholar
  29. Xiao W-L, Zhang W-G, Zhang X-L (2011) Maximum-likelihood estimators in the mixed fractional Brownian motion. Statistics 45(1):73–85CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Marco Dozzi
    • 1
  • Yuliya Mishura
    • 2
  • Georgiy Shevchenko
    • 2
  1. 1.Institut Elie CartanUniversité de LorraineVandoeuvre-les-Nancy CedexFrance
  2. 2.Department of Probability Theory, Statistics and Actuarial MathematicsTaras Shevchenko National University of KyivKyivUkraine

Personalised recommendations