Elliptic gaussian random processes

2012

A recurrent theme in functional analysis is the interplay between the theory of positive definite functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. This central theme is motivated by a host of applications, e.g., in mathematical physics, and in stochastic differential equations, and their use in financial models. In this paper, we show that, for three classes of cases in the correspondence, it is possible to obtain explicit formulas which are amenable to computations of the respective Gaussian stochastic processes. For achieving this, we first develop two functional analytic tools. They are: (i) an identification of a universal sample space Ω where we may realize the particular Gaussian processes in the correspondence; and (ii) a procedure for discretizing computations in Ω. The three classes of processes we study are as follows: Processes associated with: (a) arbitrarily given sigma finite regular measures on a fixed Borel m...

Complex Analysis and Operator Theory, 2012

Given a fractional Brownian motion (fBm) with Hurst index H ∈ (0, 1), we associate with this a special family of representions of Cuntz algebras related to frequency domains and wavelets. Vice versa, we consider a pair of Haar wavelets satisfying some compatibility conditions, and we construct the covariance functions of fBm with a fixed Hurst index. The Cuntz algebra representations enter the picture as filters of the associated wavelets. Extensions to q− dependent covariance functions leading to a corresponding fBm process will also be discussed.

Random Operators and Stochastic Equations, 2006

In the paper we present conditions for uniform convergence with probability one of wavelet expansions of ϕ-sub-Gaussian (in particular, Gaussian) random processes defined on the space R. It is shown that upon certain conditions for the bases of wavelets the wavelet expansions of stationary almost sure continuous Gaussian processes and wavelet expansions of fractional Brownian motion converge uniformly with probability one on any finite interval.

2010

In this mini-course we explain how one can construct an orthonormal wavelet basis starting from a multiresolusition analysis. Also, we present one of the most standard random wavelet series representation of Fractional Brownian Motion and use it in order to solve a problem concerning the pointwise Hölder regularity of the trajectories of the latter process

arXiv (Cornell University), 2023

The generalization of fractional Brownian motion in infinite-dimensional white and grey noise spaces has been recently carried over, following the Mandelbrot-Van Ness representation, through Riemann-Liouville type fractional operators. Our aim is to extend this construction by means of general fractional derivatives and integrals, which we define through Bernstein functions. According to the conditions satisfied by the latter, some properties of these processes (such as continuity, local times, variance asymptotics and persistence) are derived. On the other hand, they are proved to display short-or long-range dependence, if obtained by means of a derivative or an integral, respectively, regardless of the Bernstein function chosen. Moreover, this kind of construction allows us to define the corresponding noise and to derive an Ornstein-Uhlenbeck type process, as solution of an integral equation.

Revista Matemática Iberoamericana, 2000

We construct Generalized Multifractional Processes with Random Exponent (GMPREs). These processes, defined through a wavelet representation, are obtained by replacing the Hurst parameter of Fractional Brownian Motion by a sequence of continuous random processes. We show that these GMPREs can have the most general pointwise Hölder exponent function possible, namely, a random Hölder exponent which is a function of time and which can be expressed in the strong sense (almost surely for all t), as a lim inf of an arbitrary sequence of continuous processes with values in [0, 1].

2012

A recurrent theme in functional analysis is the interplay between the theory of positive definite functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. This central theme is motivated by a host of applications, e.g., in mathematical physics, and in stochastic differential equations, and their use in financial models. In this paper, we show that, for three classes of cases in the correspondence, it is possible to obtain explicit formulas which are amenable to computations of the respective Gaussian stochastic processes. For achieving this, we first develop two functional analytic tools. They are: $(i)$ an identification of a universal sample space $\Omega$ where we may realize the particular Gaussian processes in the correspondence; and (ii) a procedure for discretizing computations in $\Omega$. The three classes of processes we study are as follows: Processes associated with: (a) arbitrarily given sigma finite regular measures on a fixed Borel measure space; (b) with Hilbert spaces of sigma-functions; and (c) with systems of self-similar measures arising in the theory of iterated function systems. Even our results in (a) go beyond what has been obtained previously, in that earlier studies have focused on more narrow classes of measures, typically Borel measures on $\mathbb R^n$. In our last theorem (section 10), starting with a non-degenerate positive definite function $K$ on some fixed set $T$, we show that there is a choice of a universal sample space $\Omega$, which can be realized as a "boundary" of $(T, K)$. Its boundary-theoretic properties are analyzed, and we point out their relevance to the study of electrical networks on countable infinite graphs.

Journal of Theoretical Probability, 2008

We investigate the approximation rate for certain centered Gaussian fields by a general approach. Upper estimates are proved in the context of so-called Hölder operators and lower estimates follow from the eigenvalue behavior of some related self-adjoint integral operator in a suitable L 2 (µ)-space. In particular, we determine the approximation rate for the Lévy fractional Brownian motion X H with Hurst parameter H ∈ (0, 1), indexed by a self-similar set T ⊂ R N of Hausdorff dimension D. This rate turns out to be of order n −H/D (log n) 1/2 . In the case T = [0, 1] N we present a concrete wavelet representation of X H leading to an approximation of X H with the optimal rate n −H/N (log n) 1/2 .

Operator fractional Brownian motion (OFBM) is the natural vector-valued extension of the univariate fractional Brownian motion. Instead of a scalar parameter, the law of an OFBM scales according to a Hurst matrix that affects every component of the process. In this paper, we develop the wavelet analysis of OFBM, as well as a new estimator for the Hurst matrix of bivariate OFBM. For OFBM, the univariate-inspired approach of analyzing the entry-wise behavior of the wavelet spectrum as a function of the (wavelet) scales is fraught with difficulties stemming from mixtures of power laws. The proposed approach consists of considering the evolution along scales of the eigenstructure of the wavelet spectrum. This is shown to yield consistent and asymptotically normal estimators of the Hurst eigenvalues, and also of the coordinate system itself under assumptions. A simulation study is included to demonstrate the good performance of the estimators under finite sample sizes.

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.