Related papers: Some Fractional and Multifractional Gaussian Proce…
The aim of this paper is to present the new results concerning some functionals of Brownian motion with drift and present their applications in financial mathematics. We find a probabilistic representation of the Laplace transform of…
Let $B=\{(B_{t}^{1},..., B_{t}^{d}), t\geq 0\}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H$ and let $R_{t}=% \sqrt{(B_{t}^{1})^{2}+... +(B_{t}^{d})^{2}}$ be the fractional Bessel process. It\^{o}'s formula for…
Brownian motion and fractional Brownian motion have been widely applied in statistical modeling in finance, telecommunication, network traffic, neuroscience, physics, and other fields. More realistic models for real time series data, such…
This paper addresses the problem of estimating drift parameter of the Ornstein - Uhlenbeck type process, driven by the sum of independent standard and fractional Brownian motions. The maximum likelihood estimator is shown to be consistent…
We introduce fractional Brownian motion processes (fBm) as an alternative model for the turbulent index of refraction. These processes allow to reconstruct most of the index properties, but they are not differentiable. We overcome the…
Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann--Liouville type derivatives. For these operators a Marchaud-type form is obtained…
A new extension of the sub-fractional Brownian motion, and thus of the Brownian motion, is introduced. It is a linear combination of a finite number of sub-fractional Brownian motions, that we have chosen to call the mixed sub-fractional…
This paper is devoted to establish an invariance principle where the limit process is a multifractional Gaussian process with a multifractional function which takes its values in $(1/2,1)$. Some properties, such as regularity and local…
The paper gives a new representation for the fractional Brownian motion that can be applied to simulate this self-similar random process in continuous time. Such a representation is based on the spectral form of mathematical description and…
This paper reviews and extends some recent results on the multivariate fractional Brownian motion (mfBm) and its increment process. A characterization of the mfBm through its covariance function is obtained. Similarly, the correlation and…
Transport phenomena are ubiquitous in nature and known to be important for various scientific domains. Examples can be found in physics, electrochemistry, heterogeneous catalysis, physiology, etc. To obtain new information about diffusive…
We study the problem of parameter estimation for the homogenization limit of multiscale systems involving fractional dynamics. In the case of stochastic multiscale systems driven by Brownian motion, it has been shown that in order for the…
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional…
Multifractional processes extend the concept of fractional Brownian motion by replacing the constant Hurst parameter with a time-varying Hurst function. This extension allows for modulation of the roughness of sample paths over time. The…
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…
The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power law shape function and…
Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Levy stable densities are discussed and used for…
We define a time dependent empirical process based on $n$ i.i.d.~fractional Brownian motions and establish Gaussian couplings and strong approximations to it by Gaussian processes. They lead to functional laws of the iterated logarithm for…
In this paper we discuss fractional generalizations of the filtering problem. The "fractional" nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the…
It is well known that martingale difference sequences are very useful in applications and theory. On the other hand, the operator fractional Brownian motion as an extension of the well-known fractional Brownian motion also plays important…