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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…
The $d$-dimensional fractional Brownian motion (FBM for short) $B_t=((B_t^{(1)},...,B_t^{(d)}),t\in\mathbb{R})$ with Hurst exponent $\alpha$, $\alpha\in(0,1)$, is a $d$-dimensional centered, self-similar Gaussian process with covariance…
It is proposed a class of statistical estimators $\hat H =(\hat H_1, \ldots, \hat H_d)$ for the Hurst parameters $H=(H_1, \ldots, H_d)$ of fractional Brownian field via multi-dimensional wavelet analysis and least squares, which are…
There is much confusion in the literature over Hurst exponents. Recently, we took a step in the direction of eliminating some of the confusion. One purpose of this paper is to illustrate the difference between fBm on the one hand and…
Brownian motion is the only random process which is Gaussian, stationary and Markovian. Dropping the Markovian property, i.e. allowing for memory, one obtains a class of processes called fractional Brownian motion, indexed by the Hurst…
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…
The goal of this paper is to establish a relation between characteristic polynomials of $N\times N$ GUE random matrices $\mathcal{H}$ as $N\to\infty$, and Gaussian processes with logarithmic correlations. We introduce a regularized version…
We consider fractional Brownian motion with the Hurst parameters from (1/2,1). We found that the increment of a fractional Brownian motion can be represented as the sum of a two independent Gaussian processes one of which is smooth in the…
We develop a GMM approach for estimation of log-normal stochastic volatility models driven by a fractional Brownian motion with unrestricted Hurst exponent. We show that a parameter estimator based on the integrated variance is consistent…
The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due…
We estimate the Hurst parameter $H$ of a fractional Brownian motion from discrete noisy data observed along a high frequency sampling scheme. The presence of systematic experimental noise makes recovery of $H$ more difficult since relevant…
In this paper we will consider the LAN property for both the Hurst parameter $H>3/4$ and the variance of the fractional Brownian motion plus an independent standard Brownian motion (called mixed fractional Brownian motion) with…
We study how to construct a stochastic process on a finite interval with given `roughness' and finite joint moments of marginal distributions. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a…
We investigate the process of eigenvalues of a symmetric matrix-valued process which upper diagonal entries are independent one-dimensional H\"older continuous Gaussian processes of order gamma in (1/2,1). Using the stochastic calculus with…
In Chen and Zhou 2021, they consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function…
The characterization of intermittency in turbulence has its roots in the K62 theory, and if no proper definition is to be found in the literature, statistical properties of intermittency were studied and models were developed in attempt to…
Fractional Brownian motion is a self-affine, non-Markovian and translationally invariant generalization of Brownian motion, depending on the Hurst exponent $H$. Here we investigate fractional Brownian motion where both the starting and the…
We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter $\hat{\cal H} \in ( 1/2, 1)$. Using fixed point techniques, a…
Graph signal processing (GSP) leverages the inherent signal structure within graphs to extract high-dimensional data without relying on translation invariance. It has emerged as a crucial tool across multiple fields, including learning and…
This note is devoted to show how to push forward the algebraic integration setting in order to treat differential systems driven by a noisy input with H\"older regularity greater than 1/4. After recalling how to treat the case of ordinary…