Movimento browniano fracionário: uma análise

Abstract

The irregular and erratic movement of small particles immersed in a solution, initially observed and described by English botanist Robert Brown, around 1828, became known as Brownian motion. From 1905 with the theoretical work of the German physicist Albert Einstein, and other later works, it was possible to connect the kinetic theory of fluids to the Brownian motion, explaining how the real cause of the phenomenon was the result of the mechanical interaction of the microparticles with the fluid. The modeling of this movement became since then one of the major themes of statistical mechanics. In 1968, Benoit Mandelbrot and Van Ness proposed a generalization of Brownian motion, called fractional Brownian motion, through a real exponent H (0 < H < 1), known as Hurst exponent. It was from Mandelbrot works that the fractional Brownian movement began to gain prominence and went on to found numerous applications in various areas of science. As an example of random fractal, fractional Brownian motion has fractal dimension given by Df = 2 – H. In this paper, we address some of the most important statistical properties of Brownian motion and fractional Brownian motion, showing some simple methods of simulation of a particle performing these types of motion in one dimension. Finally, we analyze the data generated in the simulations of fractional Brownian motion through some methods used to estimate the Hurst exponent.