Abstract
In this chapter, we consider a problem of statistical estimation of an unknown drift parameter for a stochastic differential equation driven by fractional Brownian motion. Two estimators based on discrete observations of solution to the stochastic differential equations are constructed. It is proved that the estimators converge almost surely to the parameter value, as the observation interval expands and the distance between observations vanishes. A bound for the rate of convergence is given and numerical simulations are presented. As an auxilliary result of independent interest we establish global estimates for fractional derivative of fractional Brownian motion.
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References
Bertin, K., Torres, S., Tudor, C.A.: Drift parameter estimation in fractional diffusions driven by perturbed random walks. Statist. Probab. Lett. 81(2), 243–249 (2011)
Bishwal, J.P.N.: Parameter estimation in stochastic differential equations. Lecture Notes in Mathematics, vol. 1923. Springer, Berlin (2008)
Hu, Y., Nualart, D.: Differential equations driven by Hölder continuous functions of order greater than 1/2. In: Stochastic analysis and applications, Abel Symposium, vol. 2, pp. 399–413. Springer, Berlin (2007)
Hu, Y., Nualart, D.: Parameter estimation for fractional Ornstein-Uhlenbeck processes. Statist. Probab. Lett. 80(11–12), 1030–1038 (2010)
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)
Kleptsyna, M.L., Le Breton, A.: Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Stat. Inference Stoch. Process. 5(3), 229–248 (2002)
Kozachenko, Y., Melnikov, A., Mishura, Y.: On drift parameter estimation in models with fractional Brownian motion (2011). ArXiv:1112.2330
Kutoyants, Y.A.: Parameter estimation for stochastic processes. Research and Exposition in Mathematics, vol. 6. Heldermann Verlag, Berlin (1984)
Liptser, R.S., Shiryaev, A.N.: Statistics of random processes II: Applications. Applications of Mathematics, vol. 6. Springer, Berlin (2001)
Lyons, T.J.: Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14(2), 215–310 (1998)
Marcus, M.B.: Hölder conditions for Gaussian processes with stationary increments. Trans. Amer. Math. Soc. 134, 29–52 (1968)
Mishura, Y.S.: Stochastic calculus for fractional Brownian motion and related processes. Lecture Notes in Mathematics, vol. 1929. Springer, Berlin (2008)
Prakasa Rao, B.L.S.: Parametric estimation for linear stochastic differential equations driven by fractional Brownian motion. Random Oper. Stoch. Equ. 11(3), 229–242 (2003)
Prakasa Rao, B.L.S.: Statistical Inference for Fractional Diffusion Processes. Wiley Series in Probability and Statistics. Wiley, Chichester (2010)
Tudor, C.A., Viens, F.G.: Statistical aspects of the fractional stochastic calculus. Ann. Statist. 35(3), 1183–1212 (2007)
Xiao, W., Zhang, W., Xu, W.: Parameter estimation for fractional Ornstein-Uhlenbeck processes at discrete observation. Appl. Math. Model. 35(9), 4196–4207 (2011)
Xiao, W.L., Zhang, W.G., Zhang, X.L.: Maximum-likelihood estimators in the mixed fractional Brownian motion. Statistics 45(1), 73–85 (2011)
Zähle, M.: Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Relat. Fields 111(3), 333–374 (1998)
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Mishura, Y., Ral’chenko, K., Seleznev, O., Shevchenko, G. (2014). Asymptotic Properties of Drift Parameter Estimator Based on Discrete Observations of Stochastic Differential Equation Driven by Fractional Brownian Motion. In: Korolyuk, V., Limnios, N., Mishura, Y., Sakhno, L., Shevchenko, G. (eds) Modern Stochastics and Applications. Springer Optimization and Its Applications, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-03512-3_17
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DOI: https://doi.org/10.1007/978-3-319-03512-3_17
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