Packing Measure of the Sample Paths of Fractional Brownian Motion

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

The Packing Measure of the Trajectories of Multiparameter Fractional Brownian Motion

Let X = {X(t), t ∈ RN} be a multiparameter fractional Brownian motion of index α (0 < α < 1) in R. We prove that if N < αd , then there exist positive finite constants K1 and K2 such that with probability 1, K1 ≤ φ-p(X([0, 1] )) ≤ φ-p(GrX([0, 1] )) ≤ K2 where φ(s) = s/(log log 1/s), φ-p(E) is the φ-packing measure of E, X([0, 1] ) is the image and GrX([0, 1] ) = {(t,X(t)); t ∈ [0, 1]N} is the g...

Control of Some Stochastic Systems with a Fractional Brownian Motion

Some stochastic control systems that are described by stochastic differential equations with a fractional Brownian motion are considered. The solutions of these systems are defined by weak solutions. These weak solutions are obtained by the transformation of the measure for a fractional Brownian motion by a Radon-Nikodym derivative. This weak solution approach is used to solve a control problem...

Fractal Measures of the Sets Associated to Gaussian Random Fields

This paper summarizes recent results about the Hausdorff measure of the image, graph and level sets of Gaussian random fields, the packing dimension and packing measure of the image of fractional Brownian motion, the local times and multiple points of Gaussian random fields. Some open problems are also pointed out.

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.