Prediction law of fractional Brownian motion

Power-law Polya’s urn and fractional Brownian motion

We introduce a natural family of random walks Sn on Z that scale to fractional Brownian motion. The increments Xn := Sn − Sn−1 ∈ {±1} have the property that given {Xk : k < n}, the conditional law of Xn is that of Xn−kn , where kn is sampled independently from a fixed law μ on the positive integers. When μ has a roughly power law decay (precisely, when μ lies in the domain of attraction of an α...

On Gaussian Processes Equivalent in Law to Fractional Brownian Motion

We consider Gaussian processes that are equivalent in law to the fractional Brownian motion and their canonical representations. We prove a Hitsuda type representation theorem for the fractional Brownian motion with Hurst index H [ 2 . For the case H> 2 we show that such a representation cannot hold. We also consider briefly the connection between Hitsuda and Girsanov representations. Using the...

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

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...