Benoît Mandelbrot and Fractional Brownian Motion

On the connection between Molchan-Golosov and Mandelbrot-Van Ness representations of fractional Brownian motion

We prove analytically a connection between the generalized Molchan-Golosov integral transform (see [4], Theorem 5.1) and the generalized Mandelbrot-VanNess integral transform (see [8], Theorem 1.1) of fractional Brownian motion (fBm). The former changes fBm of arbitrary Hurst index K into fBm of index H by integrating over [0, t], whereas the latter requires integration over (−∞, t] for t > 0. ...

Lacunary Fractional Brownian Motion

In this paper, a new class of Gaussian field is introduced called Lacunary Fractional Brownian Motion. Surprisingly we show that usually their tangent fields are not unique at every point. We also investigate the smoothness of the sample paths of Lacunary Fractional Brownian Motion using wavelet analysis.

Tempered fractional Brownian motion

Fractional Brownian motion and data

We analyze the fractal behavior of the high frequency part of the Fourier spectrum of fBm using multifractal analysis and show that it is not consistent with what is measured on real traac traces. We propose two extensions of fBm which come closer to actual traac traces multifractal properties.

Simulation of fractional Brownian motion

Preface In recent years, there has been great interest in the simulation of long-range dependent processes, in particular fractional Brownian motion. Motivated by applications in communications engineering, I wrote my master's thesis on the subject in 2002. Since many people turned out to be interested in various aspects of fractional Brownian motion, I decided to update my thesis and make it p...