Related papers: Integral Representation of Generalized Grey Browni…

We analyze {\em the Rosenblatt process} which is a selfsimilar process with stationary increments and which appears as limit in the so-called {\em Non Central Limit Theorem} (Dobrushin and Major (1979), Taqqu (1979)). This process is…

In this paper we suggest a consistent approach to derivation of generalized Fokker-Planck equation (GFPE) for Gaussian non-Markovian processes with stationary increments. This approach allows us to construct the probability density function…

We consider a linear stochastic differential equation with stochastic drift. We study the problem of approximating the solution of such equation through an Ornstein-Uhlenbeck type process, by using direct methods of calculus of variations.…

Gaussian processes (GPs) are nonparametric Bayesian models that have been applied to regression and classification problems. One of the approaches to alleviate their cubic training cost is the use of local GP experts trained on subsets of…

In this paper, we study the existence and uniqueness of a class of stochastic differential equations driven by fractional Brownian motions with arbitrary Hurst parameter $H\in (0,1)$. In particular, the stochastic integrals appearing in the…

We generalise the Langevin equation with Gaussian white noise by replacing the velocity term by a local fractional derivative. The solution of this equation is a Levy process. We further consider the Brownian motion of a fractal particle,…

In this paper, we construct operator fractional L\'evy motion (ofLm), a broad class of non-Gaussian stochastic processes that are covariance operator self-similar, have wide-sense stationary increments and display infinitely divisible…

In this article we explore the phenomena of nonequilibrium stochastic process starting from the phenomenological Brownian motion. The essential points are described in terms of Einstein's theory of Brownian motion and then the theory…

We present the non-Markovian generalization of the widely used stochastic Schrodinger equation. Our result allows to describe open quantum systems in terms of stochastic state vectors rather than density operators, without approximation.…

Gaussian processes (GPs) are versatile tools that have been successfully employed to solve nonlinear estimation problems in machine learning, but that are rarely used in signal processing. In this tutorial, we present GPs for regression as…

We consider an ensemble of Ornstein-Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass like variable…

We derive fractional Brownian motion and stochastic processes with multifractal properties using a framework of network of Gaussian conditional probabilities. This leads to the derivation of new representations of fractional Brownian…

A short review of the classical theory of Brownian motion is presented. A new method is proposed for derivation of the Fokker-Planck equations, describing the probability density evolution, from stochastic differential equations. It is also…

The analytic inference, e.g. predictive distribution being in closed form, may be an appealing benefit for machine learning practitioners when they treat wide neural networks as Gaussian process in Bayesian setting. The realistic widths,…

An innovative extension of Geometric Brownian Motion model is developed by incorporating a weighting factor and a stochastic function modelled as a mixture of power and trigonometric functions. Simulations based on this Modified Brownian…

This paper presents a unified geometric framework for Brownian motion on manifolds, encompassing intrinsic Riemannian manifolds, embedded submanifolds, and Lie groups. The approach constructs the stochastic differential equation by…

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…

We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes,…

Gaussian processes models are widely adopted for nonparameteric/semi-parametric modeling. Identifiability issues occur when the mean model contains polynomials with unknown coefficients. Though resulting prediction is unaffected, this leads…

We present new exact expressions for a class of moments for the geometric Brownian motion, in terms of determinants, obtained using a recurrence relation and combinatorial arguments for the case of a Ito's Wiener process. We then apply the…