On some statistics of fractional Brownian motion
Keywords:fractional Brownian motion, parameter estimation, statistical hypothesis testing
Fractional Brownian motion as a method for estimating the parameters of a stochastic process by variance and one-step increment covariance is proposed and substantiated. The root-mean-square consistency of the constructed estimates has been proven. The obtained results complement and generalize the consequences of limit theorems for fractional Brownian motion, that have been proved in the number of articles. The necessity to estimate the variance is caused by the absence of a base unit of time and the estimation of the covariance allows one to determine the Hurst exponent. The established results let the known limit theorems to be used to construct goodness-of-fit criteria for the hypothesis “the observed time series is a transformation of fractional Brownian motion” and to estimate the error of optimal forecasting for time series.
Y. Mishura, “Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics 1929”, Springer, 393 p., 2008. doi: 10.1007/978-3-540-75873-0.
F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, “Stochastic Calculus for Fractional Brownian Motion and Applications”, Springer, 329 p., 2013. doi: 10.1007/978-1-84628-797-8.
R.F. Peltier and J. Levy Vehel, “A new method for estimating the parameter of fractional Brownian motion”, Rapport de recherché de l’INRIA, no. 2396, 27 p., 1994.
I. Nourdin, “Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion”, Ann. Probab., 36, no. 6, pp. 2159–2175, 2008. doi: 10.1214/07-AOP38.
I. Nourdin, “Noncentral convergence of multiple integrals”, Ann. Probab., vol. 37, no. 4, pp. 1412–1426, 2009. doi: 10.1214/08-AOP435.
M. Gradinaru and I. Nourdin, “Milstein's type schemes for fractional SDEs”, Ann. Inst H. Poincaré Probab. Statist, vol. 45, no. 4, pp. 1085–1098, 2009. doi: 10.1214/08-AIHP196.
I. Nourdin, D. Nualart, and C. Tudor, “Central and non-central limit theorems for weighted power variations of fractional Brownian motion”, Ann. Inst H. Poincaré Probab. Statist., vol. 46, no. 4, pp. 1055–1079, 2010. doi: 10.1214/09-AIHP342.
I. Nourdin, “Selected Aspects of fractional Brownian motion”, Springer, 124 p., 2012. doi: 10.1007/978-88-470-2823-4.
K. Kubilius, Yu. Mishura, and K.Ralchenko, “Parameter Estimation in Fractional Diffusion Models”, Bocconi & Springer Series, 380 p., 2017. doi: 10.1007/978-3-319-71030-3.
Y. Mishura, K. Ralchenko, and G. Shevchenko, “Existence and uniqueness of a mild solution to the stochastic heat equation with white and fractional noises”, Theor. Probability and Math. Statist., 98(2019), pp. 149–170. Available: https://doi.org/ 10.1090/ tpms/1068
O. Banna, Yu. Mishura, K. Ralchenko, and S. Shklyar, Fractional Brownian Motion. Approximations and Projections. Wiley-ISTE, 2019. 288 p. doi: 10.1002/9781119476771.