We investigate fractional Brownian motion with a microscopic random-matrix model and introduce a fractional Langevin equation. We use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath. We further compare fractional Brownian motion with the fractal time process. The respective mean-square displacements of these two forms of anomalous diffu...

We obtain a maximum principle for stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter H > 1/2). This maximum principle specifies a system of equations that the optimal control must satisfy (necessary condition for the optimal control). This system of equations consists of a backward stochastic differential eq...

We study the maximum likelihood estimator for stochastic equations with additive fractional Brownian sheet. We use the Girsanov transform for the twoparameter fractional Brownian motion, as well as the Malliavin calculus and Gaussian regularity theory. Mathematics Subject Classification (2000): 60G15, G0H07, 60G35, 62M40

We present a new framework for fractional Brownian motion in which processes with all indices can be considered under the same probability measure. Our results extend recent contributions by Hu, Øksendal, Duncan, Pasik-Duncan, and others. As an application we develop option pricing in a fractional Black-Scholes market with a noise process driven by a sum of fractional Brownian motions with vari...

Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractio...

In this paper, we propose a method to distinguish between mechanisms leading to single molecule subdiffusion in confinement. We show that the method of p-variation, introduced in the recent paper [M. Magdziarz, Phys. Rev. Lett. 103, 180602 (2009)], can be successfully applied also for confined systems. We propose a test which allows distinguishing between heavy-tailed continuous-time random wal...

In certain applications, for instance biomechanics, turbulence, finance, or Internet traffic, it seems suitable to model the data by a generalization of a fractional Brownian motion for which the Hurst parameter H is depending on the frequency as a piece-wise constant function. These processes are called multiscale fractional Brownian motions. In this contribution, we provide a statistical stud...

It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum Sn of dependent Gaussian random variables. In this paper we consider such a walk Zn that collects random rewards ξj for j ∈ Z, when the ceiling of the walk Sn is located at j. The random reward (or scenery) ξj is independent of the walk and with heavy tail. We show the convergence of the sum of i...

We study the stability of the solutions of stochastic differential equations driven by fractional Brownian motions with Hurst parameter greater than half. We prove that when the initial conditions, the drift, and the diffusion coefficients as well as the fractional Brownian motions converge in a suitable sense, then the sequence of the solutions of the corresponding equations converge in Hölder...