Path Properties of a Generalized Fractional Brownian Motion (original) (raw)

On the Generalized Fractional Brownian Motion

Mathematical Models and Computer Simulations, 2018

We introduce a new Gaussian process, a generalization of both fractional and subfractional Brownian motions, which could serve as a good model for a larger class of natural phenomena. We study its main stochastic properties and some increments characteristics. As an application, we deduce the properties of nonsemimartingality, Hölder continuity, nondifferentiablity, and existence of a local time.

Generalized fractional Brownian motion

Modern Stochastics: Theory and Applications, 2017

On the mixed fractional Brownian motion

Journal of Applied Mathematics and Stochastic Analysis, 2006

The mixed fractional Brownian motion is used in mathematical finance, in the modelling of some arbitrage-free and complete markets. In this paper, we present some stochastic properties and characteristics of this process, and we study the α-differentiability of its sample paths.

A Dynamical Approach to Fractional Brownian Motion

Fractals, 1994

Herein we develop a dynamical foundation for fractional Brownian motion. A clear relation is established between the asymptotic behavior of the correlation function and diffusion in a dynamical system. Then, assuming that scaling is applicable, we establish a connection between diffusion (either standard or anomalous) and the dynamical indicator known as the Hurst coefficient. We argue on the basis of numerical simulations that although we have been able to prove scaling only for…

On Gaussian Processes Equivalent in Law to Fractional Brownian Motion

Journal of Theoretical Probability, 2004

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 [ 1 2. For the case H > 1 2 we show that such a representation cannot hold. We also consider briefly the connection between Hitsuda and Girsanov representations. Using the Hitsuda representation we consider a certain special kind of Gaussian stochastic equation with fractional Brownian motion as noise.

Fractional Brownian Motion

Elsevier eBooks, 2017

We consider the integral of fractional Brownian motion (IFBM) and its functionals ξT on the intervals (0, T) and (−T, T) of the following types: the maximum MT , the position of the maximum, the occupation time above zero etc. We show how the asymptotics of P (ξT < 1) = pT , T → ∞, is related to the Hausdorff dimension of Lagrangian regular points for the inviscid Burgers equation with FBM initial velocity. We produce computational evidence in favor of a power asymptotics for pT. The data do not reject the hypothesis that the exponent θ of the power law is related to the similarity parameter H of fractional Brownian motion as follows: θ = −(1 − H) for the interval (-T,T) and θ = −H(1 − H) for (0, T). The point 0 is special in that IFBM and its derivative both vanish there.

Some sample path properties of multifractional Brownian motion

Stochastic Processes and their Applications, 2015

The geometry of the multifractional Brownian motion (mBm) is known to present a complex and surprising form when the Hurst function is greatly irregular. Nevertheless, most of the literature devoted to the subject considers sufficiently smooth cases which lead to sample paths locally similar to a fractional Brownian motion (fBm). The main goal of this paper is therefore to extend these results to a more general frame and consider any type of continuous Hurst function. More specifically, we mainly focus on obtaining a complete characterization of the pointwise Hölder regularity of the sample paths, and the Box and Hausdorff dimensions of the graph. These results, which are somehow unusual for a Gaussian process, are illustrated by several examples, presenting in this way different aspects of the geometry of the mBm with irregular Hurst functions.

On the relation between the fractional Brownian motion and the fractional derivatives

Physics Letters A, 2008

The definition and simulation of fractional Brownian motion are considered from the point of view of a set of coherent fractional derivative definitions. To do it, two sets of fractional derivatives are considered: (a) the forward and backward and (b) the central derivatives, together with two representations: generalised difference and integral. It is shown that for these derivatives the corresponding autocorrelation functions have the same representations. The obtained results are used to define a fractional noise and, from it, the fractional Brownian motion. This is studied. The simulation problem is also considered.

Stochastic analysis of the fractional Brownian motion

Potential analysis, 1999

Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula, the Itô-Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations.

Fractional Brownian motion as a differentiable generalized Gaussian process

Institute of Mathematical Statistics Lecture Notes - Monograph Series, 2003

Brownian motion can be characterized as a generalized random process and, as such, has a generalized derivative whose covariance functional is the delta function. In a similar fashion, fractional Brownian motion can be interpreted as a generalized random process and shown to possess a generalized derivative. The resulting process is a generalized Gaussian process with mean functional zero and covariance functional that can be interpreted as a fractional integral or fractional derivative of the delta-function.

Path Properties of a Generalized Fractional Brownian Motion (original) (raw)

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