Some topics in fractional Brownian motion

Abstract

Fractional Brownian motion denotes a family of Gaussian processes whose applicability has been demonstrated empirically for a wide variety of physical phenomena. For more than five decades, these processes have described risky outcomes or physical uncertainties and in particular for more than four decades these processes have been used to model fluctuations in economic data. These processes have a selfsimilarity or fractal property in probability law. They can provide a model for long range dependence, rare events and bursty behavior. Kolmogorov [15] initially defined the family of fractional Brownian motions motivated by his study of turbulence [16,17], though Schonberg [25] predated Kolmogorov’s construction by showing that the covariance functions for these processes are positive definite and thereby Gaussian processes can be constructed. Hurst [14] empirically identified an index parameter of these processes in his modeling of rainfall in the Nile River region. The rainfall in this region had a bursty behavior with often long periods of either excessive rainfall or meager rainfall. Mandelbrot [19] motivated by his study of fractals studied fractional Brownian motions to model economic data as well as turbulence [20]. He coined the name fractional Brownian motion and noted that Hurst empirically computed an index for these processes which is now called the Hurst index. Mandelbrot and van Ness [21] investigated some basic properties of fractional Brownian motion. The empirical evidence for models with a fractional Brownian motion has continued in hydrology, finance and turbulence. More recently the empirical evidence for the use of fractional Brownian motions (FBMs) as random models broadened to internet traffic [4,18], medicine [23] and cognition [3,12]. 2. Fractional Brownian motion