and Computational Methods

doi: 10.18698/2309-3684-2022-4-4862

The work summarizes the results obtained in the course of the implementation of Bachelor's final qualifying work and is devoted to the methods of simulating and applying the fractional Brownian motion in the problems of time series analysis. Software modules have been implemented to generate trajectories of fractal Brownian motion using the methods of stochastic representation, Cholesky decomposition and Davis-Hart. Algorithms vere compared in terms of their complexity and the quality of the resulting trajectories. The Hurst exponent was estimated by the Minkowski and R/S analysis methods. An approximation of time series by fractal Brownian motion using a power function is proposed and implemented for the subsequent application of a linear prediction algorithm based on the normal correlation theorem. It has been established that with the help of the presented approximation it is possible to achieve a satisfactory forecast of the exchange rate for several values ahead.

Morozov A.N., Skripkin A.V. Nemarkovskie fizicheskie processy [Non-Markov physical processes]. Moscow, Fizmatlit Publ., 2018, 288 p.

Shiryaev A.N. Osnovy stohasticheskoj finansovoj matematiki. Tom 1. Fakty. Modeli [Fundamentals of stochastic financial mathematics. Volume 1. Facts. Models]. Moscow, MCСME Publ., 2016, 440 p.

Yarygina I.Z., Gisin V.B., Putko B.A. Fractal asset pricing modelsfor financial risk management. Finance: Theory аnd Practice, 2019, vol. 23, no. 6 (114), pp. 117–130.

Mandelbrot B.B., Hudson R.L. The (mis) behavior of markets: a fractal view of risk, ruin and reward. London, Profile books, 2010, 352 c.

Dieker A.B., Mandjes M. On spectral simulation of fractional Brownian motion, Probability in the Engineering and Informational Sciences, 2003, vol. 17, iss. 3, pp. 417–434.

Hurst H.E. The long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers, 1951, vol. 116, pp. 770–799.

Hurst H.E., Black R.P., Simaika Y.M. Long-term storage: an experimental study. London, Constable, 1965, 145 p.

Geweke J., Porter-Hudak S. The estimation and application of long memory time series models. Journal of Time Series Analysis, 1983, vol. 4, iss. 4, pp. 221–238.

Kronover R.M. Fraktaly i haos v dinamicheskih sistemah. Osnovy teorii [Fractals and chaos in dynamical systems. Fundamentals of theory]. Moscow, Postmarket Publ., 2000, 352 c.

Coeurjolly J.-F. Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Statistical Inference for Stochastic Processes, 2001, vol. 4, iss. 2, pp. 199–227.

Porshnev S.V., Solomaha E.V., Ponomareva O.A. Peculiarities of estimating the Hurst exponent of classical Brownian motion, using the R/S analysis. International Journal of Open Information Technologies, 2020, vol. 8, no. 10, pp. 45–50.

Bulinsky A.V., Shiryaev A.N. Teoriya sluchajnyh processov [Theory of random processes]. Moscow, Fizmatlit Publ., 2005, 408 p.

Bondarenko V.V. The forecast of the time series by approximating the fractal Brownian motion. System research and information technologies, 2013, no. 4, pp. 80–88.

Korchagin S.A., Terin D.V. Klinaev Y.V. Simulating a fractal composite and studying its electrical characteristics. Маthematical Modeling and Coтputational Methods, 2017, no. 1, pp. 22–31.

Mandelbrot B.B., Van Ness J.W. Fractional Brownian motions, fractional noises and applications. SIAM Review, 1968, vol. 10, iss. 4, pp. 422–437.

Vyugin V.V. Matematicheskie osnovy teorii mashinnogo obucheniya i prognozirovaniya [Mathematical foundations of the theory of machine learning and forecasting]. Moscow, MCСME Publ., 2013, 387 p.

Облакова Т.В., Алексеев Д.С. Сравнительный анализ методов моделирования и прогнозирования временных рядов на основе теории фрактального броуновского движения. Математическое моделирование и численные методы, 2022, № 4, с. 48–62

Количество скачиваний: 191