Parameter Estimation for Gaussian Processes with Application to the Model with Two Independent Fractional Brownian Motions

  • Yuliya MishuraEmail author
  • Kostiantyn Ralchenko
  • Sergiy Shklyar
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 271)


The purpose of the article is twofold. Firstly, we review some recent results on the maximum likelihood estimation in the regression model of the form \(X_t = \theta G(t) + B_t\), where B is a Gaussian process, G(t) is a known function, and \(\theta \) is an unknown drift parameter. The estimation techniques for the cases of discrete-time and continuous-time observations are presented. As examples, models with fractional Brownian motion, mixed fractional Brownian motion, and sub-fractional Brownian motion are considered. Secondly, we study in detail the model with two independent fractional Brownian motions and apply the general results mentioned above to this model.


Discrete observations Continuous observations Maximum likelihood estimator Strong consistency Fractional Brownian motion Fredholm integral equation of the first kind 



The research of Yu. Mishura was funded (partially) by the Australian Government through the Australian Research Council (project number DP150102758). Yu. Mishura and K. Ralchenko acknowledge that the present research is carried through within the frame and support of the ToppForsk project nr. 274410 of the Research Council of Norway with title STORM: Stochastics for Time-Space Risk Models.


  1. 1.
    Belfadli, R., Es-Sebaiy, K., Ouknine, Y.: Parameter estimation for fractional Ornstein–Uhlenbeck processes: non-ergodic case. Front. Sci. Eng. 1(1), 1–16 (2011)Google Scholar
  2. 2.
    Benassi, A., Cohen, S., Istas, J.: Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett. 39(4), 337–345 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bercu, B., Coutin, L., Savy, N.: Sharp large deviations for the fractional Ornstein–Uhlenbeck process. Teor. Veroyatn. Primen. 55(4), 732–771 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Berger, J., Wolpert, R.: Estimating the mean function of a Gaussian process and the Stein effect. J. Multivar. Anal. 13(3), 401–424 (1983)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bertin, K., Torres, S., Tudor, C.A.: Maximum-likelihood estimators and random walks in long memory models. Statistics 45(4), 361–374 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bojdecki, T., Gorostiza, L.G., Talarczyk, A.: Sub-fractional Brownian motion and its relation to occupation times. Stat. Probab. Lett. 69(4), 405–419 (2004)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cai, C., Chigansky, P., Kleptsyna, M.: Mixed Gaussian processes: a filtering approach. Ann. Probab. 44(4), 3032–3075 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cénac, P., Es-Sebaiy, K.: Almost sure central limit theorems for random ratios and applications to LSE for fractional Ornstein–Uhlenbeck processes. Probab. Math. Stat. 35(2), 285–300 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cheridito, P.: Mixed fractional Brownian motion. Bernoulli 7(6), 913–934 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Es-Sebaiy, K.: Berry–Esséen bounds for the least squares estimator for discretely observed fractional Ornstein–Uhlenbeck processes. Stat. Probab. Lett. 83(10), 2372–2385 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Es-Sebaiy, K., Ndiaye, D.: On drift estimation for non-ergodic fractional Ornstein–Uhlenbeck process with discrete observations. Afr. Stat. 9(1), 615–625 (2014)Google Scholar
  13. 13.
    Es-Sebaiy, K., Ouassou, I., Ouknine, Y.: Estimation of the drift of fractional Brownian motion. Stat. Probab. Lett. 79(14), 1647–1653 (2009)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Houdré, C., Villa, J.: An example of infinite dimensional quasi-helix. In: Stochastic Models: Seventh Symposium on Probability and Stochastic Processes, 23–28 June 2002, Mexico City, Mexico. Selected papers, pp. 195–201. American Mathematical Society (AMS), Providence, RI (2003)Google Scholar
  15. 15.
    Hu, Y., Nualart, D.: Parameter estimation for fractional Ornstein–Uhlenbeck processes. Stat. Probab. Lett. 80(11–12), 1030–1038 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hu, Y., Song, J.: Parameter estimation for fractional Ornstein–Uhlenbeck processes with discrete observations. In: Malliavin calculus and stochastic analysis. A Festschrift in honor of David Nualart, pp. 427–442. Springer, New York (2013)Google Scholar
  17. 17.
    Hu, Y., Nualart, D., Xiao, W., Zhang, W.: Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation. Acta Math. Sci. Ser. B Engl. Ed. 31(5), 1851–1859 (2011)Google Scholar
  18. 18.
    Hu, Y., Nualart, D., Zhou, H.: Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter. arXiv preprint, arXiv:1703.09372 (2017)
  19. 19.
    Ibragimov, I.A., Rozanov, Y.A.: Gaussian Random Processes. Applications of Mathematics, vol. 9. Springer, New York (1978)Google Scholar
  20. 20.
    Kleptsyna, M.L., Le Breton, A.: Statistical analysis of the fractional Ornstein–Uhlenbeck type process. Stat. Inference Stoch. Process. 5, 229–248 (2002)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kozachenko, Y., Melnikov, A., Mishura, Y.: On drift parameter estimation in models with fractional Brownian motion. Statistics 49(1), 35–62 (2015)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kubilius, K., Mishura, Y.: The rate of convergence of Hurst index estimate for the stochastic differential equation. Stoch. Process. Appl. 122(11), 3718–3739 (2012)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kubilius, K., Mishura, Y., Ralchenko, K., Seleznjev, O.: 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 (2015)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kubilius, K.e., Mishura, Y., Ralchenko, K.: Parameter Estimation in Fractional Diffusion Models. Bocconi & Springer Series, vol. 8. Bocconi University Press, Milan; Springer, Cham (2017)zbMATHGoogle Scholar
  25. 25.
    Kukush, A., Mishura, Y., Valkeila, E.: Statistical inference with fractional Brownian motion. Stat. Inference Stoch. Process. 8(1), 71–93 (2005)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Le Breton, A.: Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion. Stat. Probab. Lett. 38(3), 263–274 (1998)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mishura, Y.: Stochastic Calculus for Fractional Brownian Motion and Related Processes, vol. 1929. Springer Science & Business Media (2008)Google Scholar
  28. 28.
    Mishura, Y.: Maximum likelihood drift estimation for the mixing of two fractional Brownian motions. In: Stochastic and Infinite Dimensional Analysis, pp. 263–280. Springer, Berlin (2016)Google Scholar
  29. 29.
    Mishura, Y., Ralchenko, K.: On drift parameter estimation in models with fractional Brownian motion by discrete observations. Austrian J. Stat. 43(3), 218–228 (2014)Google Scholar
  30. 30.
    Mishura, Y., Voronov, I.: Construction of maximum likelihood estimator in the mixed fractional-fractional Brownian motion model with double long-range dependence. Mod. Stoch. Theory Appl. 2(2), 147–164 (2015)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Mishura, Y., Ralchenko, K.: Drift parameter estimation in the models involving fractional Brownian motion. In: Panov, V. (ed.) Modern Problems of Stochastic Analysis and Statistics: Selected Contributions in Honor of Valentin Konakov, pp. 237–268. Springer International Publishing, Cham (2017)zbMATHGoogle Scholar
  32. 32.
    Mishura, Y., Ralchenko, K., Seleznev, O., Shevchenko, G.: 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, pp. 303–318. Springer, Cham (2014)Google Scholar
  33. 33.
    Mishura, Y., Ralchenko, K., Shklyar, S.: Maximum likelihood drift estimation for Gaussian process with stationary increments. Austrian J. Stat. 46(3–4), 67–78 (2017)Google Scholar
  34. 34.
    Mishura, Y., Ralchenko, K., Shklyar, S.: Maximum likelihood drift estimation for Gaussian process with stationary increments. Nonlinear Anal. Model. Control 23(1), 120–140 (2018)Google Scholar
  35. 35.
    Norros, I., Valkeila, E., Virtamo, J.: An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5(4), 571–587 (1999)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Peltier, R.F., Lévy Véhel, J.: Multifractional Brownian motion: definition and preliminary results. INRIA Research Report, vol. 2645 (1995)Google Scholar
  37. 37.
    Polyanin, A., Manzhirov, A.: Handbook of Integral Equations, 2nd edn. Chapman & Hall/CRC, Boca Raton (2008)Google Scholar
  38. 38.
    Prakasa Rao, B.L.S.: Statistical Inference for Fractional Diffusion Processes. Wiley, Chichester (2010)zbMATHGoogle Scholar
  39. 39.
    Privault, N., Réveillac, A.: Stein estimation for the drift of Gaussian processes using the Malliavin calculus. Ann. Stat. 36(5), 2531–2550 (2008)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Ralchenko, K.V., Shevchenko, G.M.: Paths properties of multifractal Brownian motion. Theory Probab. Math. Stat. 80, 119–130 (2010)Google Scholar
  41. 41.
    Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives. Taylor & Francis (1993)Google Scholar
  42. 42.
    Samuelson, P.A.: Rational theory of warrant pricing. Ind. Manag. Rev. 6(2), 13–32 (1965)Google Scholar
  43. 43.
    Shen, G., Yan, L.: Estimators for the drift of subfractional Brownian motion. Commun. Stat. Theory Methods 43(8), 1601–1612 (2014)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Tanaka, K.: Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process. Stat. Inference Stoch. Process. 16, 173–192 (2013)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Tanaka, K.: Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process. Stat. Inference Stoch. Process. 18(3), 315–332 (2015)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Tudor, C.: Some properties of the sub-fractional Brownian motion. Stochastics 79(5), 431–448 (2007)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Tudor, C.A., Viens, F.G.: Statistical aspects of the fractional stochastic calculus. Ann. Stat. 35(3), 1183–1212 (2007)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Xiao, W., Zhang, W., Xu, W.: Parameter estimation for fractional Ornstein–Uhlenbeck processes at discrete observation. Appl. Math. Model. 35, 4196–4207 (2011)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Xiao, W.L., Zhang, W.G., Zhang, X.L.: Maximum-likelihood estimators in the mixed fractional Brownian motion. Statistics 45, 73–85 (2011)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Zabreyko, P.P., Koshelev, A.I., Krasnosel’skii, M.A., Mikhlin, S.G., Rakovshchik, L.S., Stet’senko, V.Y.: Integral Equations: A Reference Text. Noordhoff, Leyden (1975)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Yuliya Mishura
    • 1
    Email author
  • Kostiantyn Ralchenko
    • 1
  • Sergiy Shklyar
    • 1
  1. 1.Department of Probability Theory, Statistics and Actuarial MathematicsTaras Shevchenko National University of KyivKyivUkraine

Personalised recommendations