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The Extended Orey Index for Gaussian Processes

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Parameter Estimation in Fractional Diffusion Models

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Abstract

Stationarity of the increments of fBm is a useful feature in certain applications. However, there are cases when this property is undesirable. In order to enlarge the variety of models to choose from, extensions of fBm have been introduced recently by Houdré and Villa [70] (bifractional Brownian motion) and Bojdecki et al. [27] (sub-fractional Brownian motion). These processes share with fBm such properties as self-similarity, Gaussian property and others, however they do not have stationary increments and possess some new features. Immediately the question arises about the estimation of the parameters of such processes. On tools for statistical estimation, we hope that the reader learned from the Chaps. 2–4 that one of the most important tools is quadratic variation. Except other applications, the asymptotic behavior of the quadratic variation leads to good results in estimation theory. As it was already mentioned in Chap. 2, the problem of the almost sure convergence of a quadratic variation has been solved for a wide class of processes by Baxter [10], Gladyshev [63], Klein and Giné [89], Bégyn [11], Malukas [114] etc.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. Appl. Math. Ser. 55, 62 (1966)

    Google Scholar 

  2. Achard, S., Coeurjolly, J.F.: Discrete variations of the fractional Brownian motion in the presence of outliers and an additive noise. Stat. Surv. 4, 117–147 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Androshchuk, T., Mishura, Y.: Mixed Brownian-fractional Brownian model: absence of arbitrage and related topics. Stochastics 78(5), 281–300 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arcones, M.A.: Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22(4), 2242–2274 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Athreya, K.B., Lahiri, S.N.: Measure Theory and Probability Theory. Springer Texts in Statistics. Springer, New York (2006)

    MATH  Google Scholar 

  6. Ayache, A., Cohen, S., Lévy Véhel, J.: The covariance structure of multifractional Brownian motion, with application to long range dependence. In: 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing – ICASSP’00. Proceedings, vol. 6, pp. 3810–3813 (2000)

    Google Scholar 

  7. Azmoodeh, E., Sottinen, T., Viitasaari, L., Yazigi, A.: Necessary and sufficient conditions for Hölder continuity of Gaussian processes. Stat. Probab. Lett. 94, 230–235 (2014)

    Article  MATH  Google Scholar 

  8. Bardet, J.M., Surgailis, D.: Measuring the roughness of random paths by increment ratios. Bernoulli 17(2), 749–780 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bardina, X., Es-Sebaiy, K.: An extension of bifractional Brownian motion. Commun. Stoch. Anal. 5(2), 333–340 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Baxter, G.: A strong limit theorem for Gaussian processes. Proc. Am. Math. Soc. 7, 522–527 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bégyn, A.: Quadratic variations along irregular subdivisions for Gaussian processes. Electron. J. Probab. 10, 691–717 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bégyn, A.: Generalized quadratic variations of Gaussian processes: limit theorems and applications to fractional processes. Ph.D. thesis, Toulouse 3 (2006)

    Google Scholar 

  13. Bégyn, A.: Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. Bernoulli 13(3), 712–753 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bel Hadj Khalifa, M., Mishura, Y., Zili, M.: Asymptotic properties of non-standard drift parameter estimators in the models involving fractional Brownian motion. Theory Probab. Math. Stat. 94, 73–84 (2016)

    Google Scholar 

  15. 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 

  16. Benassi, A., Jaffard, S., Roux, D.: Gaussian processes and pseudodifferential elliptic operators. Rev. Math. Iberoam. 13(1), 19–89 (1997)

    Article  MATH  Google Scholar 

  17. Benassi, A., Cohen, S., Istas, J.: Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett. 39(4), 337–345 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Benassi, A., Cohen, S., Istas, J., Jaffard, S.: Identification of filtered white noises. Stoch. Process. Appl. 75(1), 31–49 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Beran, J.: Statistics for Long-Memory Processes. Monographs on Statistics and Applied Probability, vol. 61. Chapman and Hall, New York (1994)

    Google Scholar 

  20. Beran, J., Feng, Y., Ghosh, S., Kulik, R.: Long-Memory Processes. Probabilistic Properties and Statistical Methods. Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  21. Bercu, B., Coutin, L., Savy, N.: Sharp large deviations for the fractional Ornstein–Uhlenbeck process. Teor. Veroyatn. Primen. 55(4), 732–771 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Berezansky, Y.M., Sheftel, Z.G., Us, G.F.: Functional Analysis, vol. 1. Birkhäuser, Basel (2012)

    MATH  Google Scholar 

  23. Bertin, K., Torres, S., Tudor, C.A.: Drift parameter estimation in fractional diffusions driven by perturbed random walks. Stat. Probab. Lett. 81(2), 243–249 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Berzin, C., Latour, A., León, J.R.: Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion. Lecture Notes in Statistics, vol. 216. Springer, Cham (2014)

    Google Scholar 

  25. Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London (2008)

    Book  MATH  Google Scholar 

  26. Bishwal, J.P.N.: Parameter Estimation in Stochastic Differential Equations. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  27. 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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Borodin, A., Salminen, P.: Handbook of Brownian Motion: Facts and Formulae, 2nd edn. Birkhäuser, Basel (2002)

    Book  MATH  Google Scholar 

  29. Boufoussi, B., Dozzi, M., Marty, R.: Local time and Tanaka formula for a Volterra-type multifractional Gaussian process. Bernoulli 16(4), 1294–1311 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Breton, J.C., Coeurjolly, J.F.: Confidence intervals for the Hurst parameter of a fractional Brownian motion based on finite sample size. Stat. Infer. Stoch. Process. 15(1), 1–26 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Breton, J.C., Nourdin, I., Peccati, G.: Exact confidence intervals for the Hurst parameter of a fractional Brownian motion. Electron. J. Stat. 3, 416–425 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Breuer, P., Major, P.: Central limit theorems for non-linear functionals of Gaussian fields. J. Multivar. Anal. 13, 425–441 (1983)

    Article  MATH  Google Scholar 

  33. Brouste, A., Kleptsyna, M.: Asymptotic properties of MLE for partially observed fractional diffusion system. Stat. Infer. Stoch. Process. 13(1), 1–13 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Buldygin, V., Kozachenko, Y.: Metric Characterization of Random Variables and Random Processes. Translated from the Russian by V. Zaiats. American Mathematical Society, Providence, RI (2000)

    MATH  Google Scholar 

  35. Cai, C., Chigansky, P., Kleptsyna, M.: Mixed Gaussian processes: a filtering approach. Ann. Probab. 44(4), 3032–3075 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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)

    MathSciNet  MATH  Google Scholar 

  37. Cheridito, P.: Mixed fractional Brownian motion. Bernoulli 7(6), 913–934 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Cheridito, P., Kawaguchi, H., Maejima, M.: Fractional Ornstein-Uhlenbeck processes. Electron. J. Probab. 8, 1–14 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Cherny, A.S., Engelbert, H.J.: Singular Stochastic Differential Equations. Lecture Notes in Mathematics, vol. 1858. Springer, Berlin (2005)

    Google Scholar 

  40. Coeurjolly, J.F.: Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inf. Stoch. Process. 4(2), 199–227 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Coeurjolly, J.F.: Identification of multifractional Brownian motion. Bernoulli 11(6), 987–1008 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Cramer, H., Leadbetter, M.: Stationary and Related Stochastic Processes. Sample Function Properties and Their Applications, vol. XII, 348 p. Wiley, New York-London-Sydney (1967)

    Google Scholar 

  43. Dellacherie, C., Meyer, P.A.: Probabilities and Potential. C. North-Holland Mathematics Studies, vol. 151. North-Holland, Amsterdam (1988)

    Google Scholar 

  44. Deza, M., Deza, E.: Encyclopedia of Distances, 3rd edn. Springer, Berlin (2014)

    MATH  Google Scholar 

  45. Dieudonné, J.: Foundations of Modern Analysis. Enlarged and Corrected Printing, vol. XV, 387 p. Academic, New York-London (1969)

    Google Scholar 

  46. Dobrić, V., Ojeda, F.M.: Fractional Brownian fields, duality, and martingales. In: High Dimensional Probability. IMS Lecture Notes Monographs Series, vol. 51, pp. 77–95. Institute of Mathematical Statistics, Beachwood, OH (2006)

    Google Scholar 

  47. Dobrushin, R., Major, P.: Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 27–52 (1979)

    Article  MATH  Google Scholar 

  48. Dozzi, M., Kozachenko, Y., Mishura, Y., Ralchenko, K.: Asymptotic growth of trajectories of multifractional Brownian motion, with statistical applications to drift parameter estimation. Stat. Inf. Stoch. Process. (2016)

    Google Scholar 

  49. Dozzi, M., Mishura, Y., Shevchenko, G.: Asymptotic behavior of mixed power variations and statistical estimation in mixed models. Stat. Infer. Stoch. Process. 18(2), 151–175 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. Dudley, R.: Real Analysis and Probability. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  51. Dudley, R.M., Norvaiša, R.: An Introduction to p-Variation and Young Integrals. MaPhySto Lecture Notes, vol. 1. Aarhus, Denmark (1998)

    Google Scholar 

  52. Dudley, R.M., Norvaiša, R.: Concrete Functional Calculus. Springer, New York (2011)

    Book  MATH  Google Scholar 

  53. El Machkouri, M., Es-Sebaiy, K., Ouknine, Y.: Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes. J. Korean Statist. Soc. 45(3), 329–341 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  54. Engelbert, H.J., Schmidt, W.: Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations, I, II, III. Math. Nachr. 143(1), 167–184 (1989); 144(1), 241–281 (1989); 151(1), 149–197 (1991)

    Google Scholar 

  55. 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)

    Article  MathSciNet  MATH  Google Scholar 

  56. 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)

    MathSciNet  MATH  Google Scholar 

  57. Filatova, D.: Mixed fractional Brownian motion: some related questions for computer network traffic modeling. In: International Conference on Signals and Electronic Systems, Kraków 2008, pp. 393–396 (2008)

    Google Scholar 

  58. Gasbarra, D., Sottinen, T., Valkeila, E.: Gaussian bridges. In: Stochastic Analysis and Applications. The Abel Symposium 2005. Proceedings of the Second Abel Symposium, Oslo, Norway, July 29–August 4, 2005, Held in Honor of Kiyosi Itô., pp. 361–382. Springer, Berlin (2007)

    Google Scholar 

  59. Gikhman, I.I., Skorokhod, A.V.: Introduction to the Theory of Random Processes. Dover, Mineola, NY (1996)

    MATH  Google Scholar 

  60. Gikhman, I.I., Skorokhod, A.V.: The Theory of Stochastic Processes. I. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  61. Giraitis, L., Surgailis, D.: CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 191–212 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  62. Giraitis, L., Robinson, P.M., Surgailis, D.: Variance-type estimation of long memory. Stoch. Process. Appl. 80(1), 1–24 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  63. Gladyshev, E.: A new limit theorem for stochastic processes with Gaussian increments. Theory Probab. Appl. 6, 52–61 (1962)

    Article  MATH  Google Scholar 

  64. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ontario (1980)

    Google Scholar 

  65. Grimm, C., Schlüchtermann, G.: IP-Traffic Theory and Performance. Springer Series on Signals and Communication Technology. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  66. Guerra, J., Nualart, D.: Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. Stoch. Anal. Appl. 26(5), 1053–1075 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  67. Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Application. Academic, New York–London (1980)

    MATH  Google Scholar 

  68. Hanson, D., Wright, F.: A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Stat. 42, 1079–1083 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  69. Heyde, C.: Quasi-Likelihood and Its Application: a General Approach to Optimal Parameter Estimation. Springer, New York (1997)

    Book  MATH  Google Scholar 

  70. Houdré, C., Villa, J.: An example of infinite dimensional quasi-helix. In: Stochastic Models. Seventh Symposium on Probability and Stochastic Processes, June 23–28, 2002, Mexico City, Mexico. Selected Papers, pp. 195–201. American Mathematical Society, Providence, RI (2003)

    Google Scholar 

  71. Hu, Y., Nualart, D.: Parameter estimation for fractional Ornstein–Uhlenbeck processes. Stat. Probab. Lett. 80(11–12), 1030–1038 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  72. 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 

  73. Hu, Y., Nualart, D., Xiao, W., Zhang, W.: Exact maximum likelihood estimators for drift fractional Brownian motion at discrete observation. Acta Math. Sci. Ser. B Engl. Ed. 31(5), 1851–1859 (2011)

    MathSciNet  MATH  Google Scholar 

  74. Hu, Y., Nualart, D., Zhou, H.: Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter (2017). arXiv preprint arXiv:1703.09372

    Google Scholar 

  75. Iacus, S.M.: Simulation and Inference for Stochastic Differential Equations. With R Examples. Springer, New York (2008)

    Book  MATH  Google Scholar 

  76. Ibe, O.: Elements of Random Walk and Diffusion Processes, 1st edn. Wiley Series in Operations Research and Management Science. Wiley, Hoboken, NJ (2013)

    Book  MATH  Google Scholar 

  77. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam (1989)

    MATH  Google Scholar 

  78. Istas, J., Lang, G.: Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré Probab. Stat. 33(4), 407–436 (1997)

    Article  MATH  Google Scholar 

  79. Itô, K.: On stochastic differential equations. Mem. Am. Math. Soc. 4, 1–51 (1951)

    Google Scholar 

  80. Itô, K., McKean Jr., H.P.: Diffusion Processes and Their Sample Paths, Second printing, Corrected. Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer, Berlin (1974)

    Google Scholar 

  81. Janson, S.: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  82. Jost, C.: Transformation formulas for fractional Brownian motion. Stoch. Process. Appl. 116(10), 1341–1357 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  83. Kahane, J.P.: Hélices et quasi-hélices. In: Mathematical Analysis and Applications, Part B. Advances in Mathematics Supplement Studies, vol. 7, pp. 417–433. Academic, New York-London (1981)

    Google Scholar 

  84. Karp, D., Sitnik, S.: Two-sided inequalities for generalized hypergeometric function. Res. Rep. Collect. 10(2) (2007)

    Google Scholar 

  85. Kawata, T.: Fourier Analysis in Probability Theory. Probability and Mathematical Statistics. A Series of Monographs and Textbooks. Academic, New York/London (1972)

    MATH  Google Scholar 

  86. Kent, J.T., Wood, A.T.: Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. R. Stat. Soc. Ser. B 59(3), 679–699 (1997)

    MathSciNet  MATH  Google Scholar 

  87. Kessler, M., Lindner, A., Sørensen, M. (eds.): Statistical Methods for Stochastic Differential Equations. Selected Papers Based on the Presentations at the 7th Séminaire Européen de Statistiques on “Statistics for stochastic differential equations models”, La Manga del Mar Menor, Cartagena, May 7–12, 2007. CRC, Boca Raton, FL (2012)

    Google Scholar 

  88. Khoshnevisan, D., Salminen, P., Yor, M.: A note on a.s. finiteness of perpetual integral functionals of diffusions. Electron. Commun. Probab. 11, 108–117 (electronic) (2006)

    Google Scholar 

  89. Klein, R., Gine, E.: On quadratic variations of processes with Gaussian increments. Ann. Probab. 3(4), 716–721 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  90. Kleptsyna, M.L., Le Breton, A.: Statistical analysis of the fractional Ornstein–Uhlenbeck type process. Stat. Infer. Stoch. Process. 5, 229–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  91. Kolmogoroff, A.N.: Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C. R. (Doklady) Acad. Sci. URSS (N.S.) 26, 115–118 (1940)

    Google Scholar 

  92. Kozachenko, Y., Melnikov, A., Mishura, Y.: On drift parameter estimation in models with fractional Brownian motion. Statistics 49(1), 35–62 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  93. Krylov, N.V.: Itô’s stochastic integral equations. Theor. Probab. Appl. 14(2), 330–336 (1969)

    Article  MATH  Google Scholar 

  94. Kubilius, K.: The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type. Stoch. Process. Appl. 98(2), 289–315 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  95. Kubilius, K.: On estimation of the extended Orey index for Gaussian processes. Stochastics 87(4), 562–591 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  96. Kubilius, K., Melichov, D.: On estimation of the Hurst index of solutions of stochastic integral equations. Liet. Mat. Rink. 48/49, 401–406 (2008)

    Google Scholar 

  97. Kubilius, K., Melichov, D.: Estimating the Hurst index of the solution of a stochastic integral equation. Liet. Mat. Rink. 50, 24–29 (2009)

    MATH  Google Scholar 

  98. Kubilius, K., Melichov, D.: Quadratic variations and estimation of the Hurst index of the solution of SDE driven by a fractional Brownian motion. Lith. Math. J. 50(4), 401–417 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  99. Kubilius, K., Melichov, D.: On comparison of the estimators of the Hurst index of the solutions of stochastic differential equations driven by the fractional Brownian motion. Informatica 22(1), 97–114 (2011)

    MathSciNet  MATH  Google Scholar 

  100. 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)

    Article  MathSciNet  MATH  Google Scholar 

  101. Kubilius, K., Skorniakov, V.: On some estimators of the Hurst index of the solution of SDE driven by a fractional Brownian motion. Statist. Probab. Lett. 109, 159–167 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  102. Kubilius, K., Skorniakov, V., Melichov, D.: Estimation of parameters of SDE driven by fractional Brownian motion with polynomial drift. J. Stat. Comput. Simul. 86(10), 1954–1969 (2016)

    Article  MathSciNet  Google Scholar 

  103. 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)

    Article  MathSciNet  MATH  Google Scholar 

  104. Kubilius, K., Skorniakov, V., Ralchenko, K.: The rate of convergence of the Hurst index estimate for a stochastic differential equation. Nonlinear Anal. Model. Control 22(2), 273–284 (2017)

    MathSciNet  Google Scholar 

  105. Kukush, A., Mishura, Y., Ralchenko, K.: Hypothesis testing of the drift parameter sign for fractional Ornstein–Uhlenbeck process. Electron. J. Stat. 11(1), 385–400 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  106. Kutoyants, Y.A.: Statistical Inference for Ergodic Diffusion Processes. Springer, London (2004)

    Book  MATH  Google Scholar 

  107. 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)

    Article  MathSciNet  MATH  Google Scholar 

  108. Lei, P., Nualart, D.: A decomposition of the bifractional Brownian motion and some applications. Stat. Probab. Lett. 79(5), 619–624 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  109. Lifshits, M., Volkova, K.: Bifractional Brownian motion: existence and border cases. ESAIM Probab. Stat. 19, 766–781 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  110. Liptser, R., Shiryayev, A.: Statistics of Random Processes. II. Applications. Translated by A. B. Aries. Applications of Mathematics, vol. 6, X, 339 p. Springer, New York, Heidelberg, Berlin (1978)

    Google Scholar 

  111. Liptser, R., Shiryayev, A.: Theory of Martingales. Kluwer Academic, Dordrecht (1989)

    Book  MATH  Google Scholar 

  112. Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes. II. Applications, Applications of Mathematics, vol. 6. Springer, Berlin (2001)

    Google Scholar 

  113. Liu, J., Yan, L., Peng, Z., Wang, D.: Remarks on confidence intervals for self-similarity parameter of a subfractional Brownian motion. Abstr. Appl. Anal. 2012, 14 (2012). Art. ID 804942

    Google Scholar 

  114. Malukas, R.: Limit theorems for a quadratic variation of Gaussian processes. Nonlinear Anal. Model. Control 16(4), 435–452 (2011)

    MathSciNet  MATH  Google Scholar 

  115. Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  116. Marcus, M.B.: Hölder conditions for Gaussian processes with stationary increments. Trans. Am. Math. Soc. 134, 29–52 (1968)

    MATH  Google Scholar 

  117. Marcus, M.B., Rosen, J.: Markov Processes, Gaussian Processes, and Local Times. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  118. Mémin, J., Mishura, Y., Valkeila, E.: Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Stat. Probab. Lett. 51(2), 197–206 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  119. Mijatović, A., Urusov, M.: Convergence of integral functionals of one-dimensional diffusions. Electron. Commun. Probab. 17, 1–13 (2012). Article 61

    Google Scholar 

  120. Mishura, Y.: Stochastic Calculus for Fractional Brownian Motion and Related Processes, vol. 1929. Springer Science & Business Media, Berlin (2008)

    MATH  Google Scholar 

  121. Mishura, Y.: Standard maximum likelihood drift parameter estimator in the homogeneous diffusion model is always strongly consistent. Stat. Probab. Lett. 86, 24–29 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  122. 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 

  123. 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)

    Article  Google Scholar 

  124. Mishura, Y.S., Shevchenko, G.M.: The rate of convergence for Euler approximations of solutions to of stochastic differential equations driven by fractional Brownian motion. Stochastics 809(5), 489–511 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  125. Mishura, Y.S., Shevchenko, G.M.: Existence and uniqueness of the solution of stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index H > 1∕2. Commun. Stat. Theory Methods 40(19–20), 3492–3508 (2011)

    Google Scholar 

  126. Mishura, Y., Shevchenko, G.: Mixed stochastic differential equations with long-range dependence: existence, uniqueness and convergence of solutions. Comput. Math. Appl. 64(10), 3217–3227 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  127. Mishura, Y., Shevchenko, G.: Theoretical and Statistical Aspects of Stochastic Processes. Elsevier, Amsterdam (2017)

    MATH  Google Scholar 

  128. 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)

    MathSciNet  MATH  Google Scholar 

  129. 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 Applications, vol. 90, pp. 303–318. Springer, Cham (2014)

    Google Scholar 

  130. Moers, M.: Hypothesis testing in a fractional Ornstein-Uhlenbeck model. Int. J. Stoch. Anal. 2012, 23 (2012). Art. ID 268568

    Google Scholar 

  131. 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)

    Article  MathSciNet  MATH  Google Scholar 

  132. Norvaiša, R.: A complement to Gladyshev’s theorem. Lith. Math. J. 51(1), 26–35 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  133. Norvaiša, R.: Gladyshev’s theorem for integrals with respect to a Gaussian process. Preprint (2011). arXiv:1105.1503v1

    Google Scholar 

  134. Norvaiša, R., Salopek, D.: Estimating the Orey index of a Gaussian stochastic process with stationary increments: An application to financial data set. In: Stochastic Models. Proceedings of the International Conference, In Honour of Prof. Donald A. Dawson, Ottawa, June 10–13, 1998, pp. 353–374. American Mathematical Society for the Canadian Mathematical Society, Providence, RI (2000)

    Google Scholar 

  135. Nourdin, I.: Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion. Ann. Probab. 36(6), 2159–2175 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  136. Nourdin, I.: Selected Aspects of Fractional Brownian Motion. Bocconi & Springer Series, vol. 4. Springer, Bocconi University Press, Milan (2012)

    Google Scholar 

  137. Nourdin, I., Viens, F.G.: Density formula and concentration inequalities with Malliavin calculus. Electron. J. Probab. 14, 2287–2309 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  138. Nourdin, I., Nualart, D., Tudor, C.A.: Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. Henri Poincaré, Probab. Stat. 46(4), 1055–1079 (2010)

    Google Scholar 

  139. Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Probability and Its Applications. Springer, Berlin (2006)

    MATH  Google Scholar 

  140. Nualart, D., Răşcanu, A.: Differential equations driven by fractional Brownian motion. Collect. Math. 53(1), 55–81 (2002)

    MathSciNet  MATH  Google Scholar 

  141. Orey, S.: Gaussian sample functions and the Hausdorff dimension of level crossings. Z. Wahrscheinlichkeitstheor. Verw. Geb. 15, 249–256 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  142. Peltier, R.F., Lévy Véhel, J.: Multifractional Brownian motion: definition and preliminary results. INRIA Research Report, vol. 2645 (1995)

    Google Scholar 

  143. Perestyuk, M., Mishura, Y., Shevchenko, G.: On the distribution of integral functionals of a homogeneous diffusion process. Mod. Stoch. Theory Appl. 1(2), 109–116 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  144. Pitman, J., Yor, M.: Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches. Bernoulli 9(1), 1–24 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  145. Prakasa Rao, B.L.S.: Asymptotic Theory of Statistical Inference. Wiley, New York (1987)

    MATH  Google Scholar 

  146. Prakasa Rao, B.L.S.: Statistical Inference for Fractional Diffusion Processes. Wiley, New York (2010)

    MATH  Google Scholar 

  147. R Core Team: A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna (2014). http://www.R-project.org/

  148. Ralchenko, K.V.: Approximation of multifractional Brownian motion by absolutely continuous processes. Theory Probab. Math. Stat. 82, 115–127 (2011)

    Article  MathSciNet  Google Scholar 

  149. Ralchenko, K.V., Shevchenko, G.M.: Paths properties of multifractal Brownian motion. Theory Probab. Math. Stat. 80, 119–130 (2010)

    Article  Google Scholar 

  150. Resnick, S.I.: Heavy-Tail Phenomena. Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. Springer, Berlin (2007)

    MATH  Google Scholar 

  151. Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales, vol. 2: Itô Calculus. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  152. Russo, F., Tudor, C.A.: On bifractional Brownian motion. Stoch. Process. Appl. 116(5), 830–856 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  153. Salminen, P., Yor, M.: Properties of perpetual integral functionals of Brownian motion with drift. Ann. Inst. H. Poincaré Probab. Stat. 41(3), 335–347 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  154. Salopek, D.: Tolerance to Arbitrage: Inclusion of fractional Brownian motion to model stock price fluctuations. Ph.D. thesis, Ottawa-Carleton Institute (1997)

    Google Scholar 

  155. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Transl. from the Russian. Gordon and Breach, New York, NY (1993)

    MATH  Google Scholar 

  156. Shevchenko, G.: Mixed stochastic delay differential equations. Theory Probab. Math. Stat. (89), 181–195 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  157. Shiryaev, A.: Essentials of Stochastic Finance: Facts, Models, Theory, 1st edn. Advanced Series on Statistical Science & Applied Probability, vol. 3. World Scientific, Singapore (1999)

    Google Scholar 

  158. Skorokhod, A.V.: Studies in the Theory of Random Processes. Addison-Wesley, Reading (1965)

    MATH  Google Scholar 

  159. Stoev, S.A., Taqqu, M.S.: How rich is the class of multifractional Brownian motions? Stoch. Process. Appl. 116(2), 200–221 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  160. Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with continuous coefficients. I, II. Commun. Pure Appl. Math. 22, 345–400, 479–530 (1969)

    Google Scholar 

  161. Tanaka, K.: Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein-Uhlenbeck process. Stat. Infer. Stoch. Process. 16, 173–192 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  162. Tanaka, K.: Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process. Stat. Infer. Stoch. Process. 18(3), 315–332 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  163. Taqqu, M.S.: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 287–302 (1975). https://doi.org/10.1007/BF00532868

    Article  MathSciNet  MATH  Google Scholar 

  164. Tricot, C.: Curves and Fractal Dimension. With a Foreword by Michel Mendès France. Transl. from the French. Springer, New York, NY (1995)

    Book  MATH  Google Scholar 

  165. Tudor, C.: Some properties of the sub-fractional Brownian motion. Stochastics 79(5), 431–448 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  166. Tudor, C.A., Viens, F.G.: Statistical aspects of the fractional stochastic calculus. Ann. Stat. 35(3), 1183–1212 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  167. van Zanten, H.: When is a linear combination of independent fBm’s equivalent to a single fBm? Stoch. Process. Appl. 117(1), 57–70 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  168. Wood, A.T.A., Chan, G.: Simulation of stationary Gaussian processes in [0, 1]d. J. Comput. Graph. Stat. 3(4), 409–432 (1994)

    MathSciNet  Google Scholar 

  169. Xiao, W.L., Zhang, W.G., Zhang, X.L.: Maximum-likelihood estimators in the mixed fractional Brownian motion. Statistics 45, 73–85 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  170. Xiao, W., Zhang, W., Xu, W.: Parameter estimation for fractional Ornstein-Uhlenbeck processes at discrete observation. Appl. Math. Model. 35, 4196–4207 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  171. Yamada, T., Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11(1), 155–167 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  172. Zähle, M.: Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Relat. Fields 111(3), 333–374 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  173. Zähle, M.: On the link between fractional and stochastic calculus. In: Stochastic Dynamics. Conference on Random Dynamical Systems, Bremen, April 28–May 2, 1997. Dedicated to Ludwig Arnold on the Occasion of his 60th birthday, pp. 305–325. Springer, New York, NY (1999)

    Google Scholar 

  174. Zähle, M.: Integration with respect to fractal functions and stochastic calculus. II. Math. Nachr. 225, 145–183 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  175. Zvonkin, A.K.: A transformation of the phase space of a diffusion process that will remove the drift. Math. USSR Sb. 93, 129–149 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Data Science and Digital Technologies, Vilnius University, Vilnius, Lithuania

    Kęstutis Kubilius

  2. Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Yuliya Mishura & Kostiantyn Ralchenko

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  1. Kęstutis Kubilius
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Kubilius, K., Mishura, Y., Ralchenko, K. (2017). The Extended Orey Index for Gaussian Processes. In: Parameter Estimation in Fractional Diffusion Models. Bocconi & Springer Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-71030-3_6

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