Modelling and analysis of fractional Brownian motions

Abstract

This paper deals with the study of fractional Brownian motion (fbm) and fractional Brownian noises. First, we study the Barnes and Allan model, which is very close to the (fbm). Then, we propose an infinite dimension state model, where each state is the solution of a first order differential equation with time-varying parameter driving by a white noise. The simulation of (fbm) is thus transformed into the solution of the non-stationary model. The problem of discretization of the model is tricky and the solution proposed is an evolutive discretization from a very small initial time. Finally, the performances of this algorithm are shown on the basis of simulations with different fractal dimensions.