Wavelet analysis in Wavelet Transform Modulus Maxima Approach
ISSN 1991-3087     НА ГЛАВНУЮ Анализ фрактального броуновского движения с помощью непрерывного вейвлет преобразования, с использованием вейвлет мексиканской шляпы   Пучков Андрей Александрович, магистр экономики, докторант Рижского технического университета.   Fractal Brownian motion analysis via ‘Mexican hat’ wavelets   Andrejs Puchkovs, Economist, Mg. oec., PhD student of Riga Technical University.   This article is dedicated for Fractal Brownian process analysis using Continuous Wavelet Transform (Direct and Inverse). Wavelet Analysis of stochastic processes is very important for financial time series analysis, risk estimation and financial time series forecasting. Wavelet Analysis is very precious for scalability analysis, because of its ability to analyze the signal (process) in scaling and shifting dimensions. In current research, Fractal Brownian motion is analyzed using Direct and Inverse Continuous Wavelet Transform, wavelet coefficients probability density function is estimated, wavelet coefficients lower and upper bounds are calculated using Mexican hat mother wavelet function. At the end estimation results are illustrated. Nowadays financial market requires a deeper and comprehensive understanding of financial risks. Needs new risk measurement approaches and methods. For many years economists, statisticians, and stock market players have been interested in developing and testing models of stock price behaviour The Hurst exponent, conceptually based on Benoit Mandelbrot Fractal theory and proposed by H. E. Hurst for use in fractal analysis, has been applied to many research fields. This theory was dedicated to extend view of classical Brownian motion using fractal geometry and fractal measure.   In Theory the Hurst exponent provides a measure of long-term memory and fractallity of a time series. One of the characteristic indicators of sustainability or persistence of the time series is a constant Hurst - statistic identifying the accumulation and inheritance of the past time series data, the fractal properties of the series. Hurst index (H) can take the values ​​0 ≤ H <1 (from zero to one), and: 1) the values ​​in the range 0 ≤ H <0,5 (from zero to one half) is commonly called pink noise, which has anti-persistent properties; 2) a value of H = 0,5 is called white noise associated with the Brownian motion - the observations are random and uncorrelated, consequently the present value of the time series does not affect the future; 3) the values ​​in the range 0,5
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