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Nonparametric identification method of stochastic differential equation with fractal Brownian motion

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This Paper presents a methodology for estimating the parameters of stochastic differential equation (SDE) driven by froctional Brownian motion (fBm). The main idea is connected with simulated maximum likelihood.To develop the methodology, two iimportant questions, namely how to generote fBm sample paths with different values of the Hurst parameter and how to estimate Hurst parometer are studied. Aa Effectiveness of the methodology is analyzed through Monte Carlo simulations.
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  • [1] M. Taqqu, V. Teverovsky and W. Willinger, "Is network traffic self-similar of multifractal?", Fractals, 5, 1997, 63-73.
  • [2] A. Montanari, R. Rosso and M .S. Taqqu, "Fractionally differenced ARIMA models applied to hydrologic time series: Identification, estmation, and simulation", Water resources research, vol. 33 (7), 1997, pp. 1035-1044.
  • [3] G. Jumarie, "Stochastics of order n in biological systems applications to population dynamics, thermodynamics, nonequilibrium phase and complexity", Journal of Biological Systems, vol. 11 (3), 2003, pp. 113-137.
  • [4] G. Jumarie, "Further results on the modelling of complex fractals in finance, scaling observation and optimal portfolio selection", SAMS, vol. 42, 2002, pp. 1483-1498.
  • [5] B.B. Mandelbrot and J.W. van Ness, "Fractional Brownian motions, fractional noises and applications", SIAM Review, vol. 10, 1968, pp. 422-437.
  • [6] C. Kyung-Joon, W. Schucany, "Nonparametric kernel regression estimation near endpoints", Journal of statistical planning and inference, vol. 66, 1998, pp. 289-304.
  • [7] V. Katkovnik, I. Shmulevich, "Kernel density estimation with adaptive varying windows size", Pattern recognition letters, vol. 23, 2002, pp. 1641-1648.
  • [8] V. Pipiras and- M.S. Taqqu, "Integration questions related to fractional Brownian motion", Probab. Theory Relat. Fields, vol. 118, 2000, pp. 251-291.
  • [9] V. Pipiras and- M.S. Taqqu, "Deconvolution of fractional brownian motion", Journal of Time Series Analysis,vol. 23 (6), 2002, pp. 487-501.
  • [10]Y. Meyer, F. Sellan, and M. S. Taqqu, "Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion", The Journal of Fourier Analysis and Applications, vol. 5, 1999, pp.465-494.
  • [11]V. Pipiras, "Wavelet-based simulation of fractional Brownian motion revisited", Appl. Comput  Harmon. Anal., vol. 19, 2005, pp. 49-60.
  • [12]H. Hurst "Long-term storage capacity of reservoirs", Transaction of American Society of Civil Engineers, vol. 116, pp. 770-808, 1951.
  • [13]J. Nelder, R. Mead, "A simplex method for function minimization", The Computer Journal, vol. 7, 1965, pp.308-313.
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bwmeta1.element.baztech-article-BUJ5-0012-0013
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