Existence of the linear prediction for Banach space valued Gaussian processes (original) (raw)

Trajectories of Gaussian processes and interpolation of Banach spaces

Stochastic Processes and Their Applications, 1987

We investigate a condition for a Gaussian process with trajectories in a Banach space E to actually have trajectories in a "more regular" Banach space E o (0< 0 < 1). "More regular" here means that Ee is an interpolation space between E and some sub-space of the reproducing kernel Hilbert space of the process. As an illustration we give an interpolation-theoretic proof of H61der property for trajectories of some Gaussian processes related to a Wiener process.

The covariation for Banach space valued processes and applications

Metrika, 2014

This article focuses on a recent concept of covariation for processes taking values in a separable Banach space B and a corresponding quadratic variation. The latter is more general than the classical one of Métivier and Pellaumail. Those notions are associated with some subspace χ of the dual of the projective tensor product of B with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the Itô process and the concept ofν 0 -semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calculus via regularization in Banach spaces. Two main applications are mentioned: one related to Clark-Ocone formula for finite quadratic variation processes; the second one concerns the probabilistic representation of a Hilbert valued partial differential equation of Kolmogorov type.

Stochastic Analysis of Gaussian Processes via Fredholm Representation

We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. In particular, we prove an Itô formula that is, as far as we know, the most general Malliavin-type Itô formula for Gaussian processes so far. Finally, we give applications to equivalence in law and series expansions of Gaussian processes.

On linear prediction of random processes under conditions of uncertainty

Theory of Probability and Mathematical Statistics, No. 45,1992, 1992

A b s t r a c t . This article contains a treatment of the problem of linear estimation of the transform roc A£ = a(t)((t)dt Jo of a stationary random process { ( t) from observations of t (s ) for 5 < 0 . Least favorable spectral densities /o(A) € 3 ! are found along with the minimax (robust) spectral characteristics of an optimal estimator of At, for different classes 3 of densities.

Gauss-Markov processes on Hilbert spaces

Transactions of the American Mathematical Society, 2015

K. Itô characterised in 1984 zero-mean stationary Gauss–Markov processes evolving on a class of infinite-dimensional spaces. In this work we extend the work of Itô in the case of Hilbert spaces: Gauss–Markov families that are time-homogenous are identified as solutions to linear stochastic differential equations with singular coefficients. Choosing an appropriate locally convex topology on the space of weakly sequentially continuous functions we also characterize the transition semigroup, the generator and its core, thus providing an infinite-dimensional extension of the classical result of Courrège in the case of Gauss–Markov semigroups.

Regularity of Gaussian Processes on Dirichlet Spaces

Constructive Approximation

We are interested in the regularity of centered Gaussian processes (Z x (ω)) x∈M indexed by compact metric spaces (M, ρ). It is shown that the almost everywhere Besov space regularity of such a process is (almost) equivalent to the Besov regularity of the covariance K(x, y) = E(Z x Z y) under the assumption that (i) there is an underlying Dirichlet structure on M which determines the Besov space regularity, and (ii) the operator K with kernel K(x, y) and the underlying operator A of the Dirichlet structure commute. As an application of this result we establish the Besov regularity of Gaussian processes indexed by compact homogeneous spaces and, in particular, by the sphere. Heat kernel, Gaussian processes, Besov spaces. MSC 58J35 MSC 46E35MSC 42C15MSC 43A85 Clearly, K(x, y) determines the law of all finite dimensional random variables (Z x 1 ,. .. , Z xn). Conversely, if K(x, y) is a real valued, symmetric, and positive definite function on M × M , there exists a unique Hilbert space H of functions on M (the associated RKHS), for which K is a reproducing kernel, i.e. f (x) = f, K(x, •) H , ∀f ∈ H, ∀x ∈ M (see [5], [37], [15]). Further, if (u i) i∈I is an orthonormal basis for H, then the following representation in H holds: K(x, y) = i∈I u i (x)u i (y), ∀x, y ∈ M. Therefore, if (B i (ω)) i∈I is a family of independent N (0, 1) variables, then Z x (ω) := i∈I u i (x)B i (ω) is a centered Gaussian process with covariance K(x, y). Thus, this is a version of the previous process Z x (ω). 2.2 Gaussian processes with a zest of topology We now consider the following more specific setting. Let M be a compact space and let µ be a Radon measure on (M, B) with support M and B being the Borel sigma algebra on M. Assuming that (Ω, A, P) is a probability space we let Z : (M, B) ⊗ (Ω, A) → Z x (ω) ∈ R, be a measurable map such that (Z x) x∈M is a Gaussian process. In addition, we suppose that K(x, y) is a symmetric, continuous, and positive definite function on M × M. Then obviously the operator K defined by Kf (x) := M K(x, y)f (y)dµ(y), f ∈ L 2 (M, µ), is a self-adjoint compact positive operator (even trace-class) on L 2 (M, µ). Moreover, K(L 2) ⊂ C(M), the Banach space of continuous functions on M. Let ν 1 ≥ ν 2 ≥ • • • > 0 be the sequence of eigenvalues of K repeated according to their multiplicities and let (u k) k≥1 be the sequence of respective normalized eigenfunctions: M K(x, y)u k (y)dµ(y) = ν k u k (x). The functions u k are continuous real-valued functions and the sequence (u k) k≥1 is an orthonormal basis for L 2 (M, µ). By Mercer Theorem we have the following representation: K(x, y) = k ν k u k (x)u k (y), where the convergence is uniform. Let H ⊂ L 2 (Ω, P) be the closed Gaussian space spanned by finite linear combinations of (Z x) x∈M. Clearly, interpreting the following integral as Bochner integral with value in the Hilbert space H, we have B k (ω) = 1 √ ν k M Z x (ω)u k (x)dµ(x) ∈ H.

The Modeling of Gaussian Stationary Random Processes with a Certain Accuracy and Reliability

2017

In this chapter, the accuracy and reliability of the models of stationary Gaussian random processes are studied in spaces L p ([0, T ]), p ≥ 1; in Orlicz spaces and in the space of continuous functions C ([0, T ]). The properties of models of stationary Gaussian processes in a uniform metric, applying the theory of Sub φ (Ω) spaces, are investigated. A generalized model of Gaussian stationary processes is also considered.

Hilbert Space Valued Generalized Random Processes – Part I

2007

Generalized random processes by various types of continuity are considered and classified as generalized random processes (GRPs) of type (I) and (II). Structure theorems for Hilbert space valued generalized random processes are obtained: Series expansion theorems for GRPs (I) considered as elements of the spaces L(A, S(H)−1) are derived, and structure representation theorems for GRPs (II) on K{M p }(H) on a set with arbitrary large probability are given.

State Space Models for Gaussian Stochastic Processes

Stochastic Systems: The Mathematics of Filtering and Identification and Applications, 1981

ABSTP CT:K comprehensive theory of stochastic realization for multivariate stationary Gaussian processes is presented. It is coordinate-free in nature, starting out with an abstract state space theory in Hilbert space, based on the concept of splitting subspace. These results are then carried over to the spectral domain and described in terms of Hardy functions. Each state space is uniquely characterized by its structural function, an inner function which contains all the systems theoretical characteristics of the corresponding realizations. Finally coordinates are introduced and concrete differential-equation-type representations are obtained. This paper is an abridged version of a forthcoming paper, which in turn summarizes and considerably extends results which have previously been presented in a series of preliminary conference papers. ' ... V :-, t: reviewed and is. .' , AY" ApR 1L-12 (7b).

Prediction Error for Continuous-Time Stationary Processes with Singular Spectral Densities

Acta Applicandae Mathematicae, 2008

The paper considers the mean square linear prediction problem for some classes of continuous-time stationary Gaussian processes with spectral densities possessing singularities. Specifically, we are interested in estimating the rate of decrease to zero of the relative prediction error of a future value of the process using the finite past, compared with the whole past, provided that the underlying process is nondeterministic and is "close" to white noise. We obtain explicit expressions and asymptotic formulae for relative prediction error in the cases where the spectral density possess either zeros (the underlying model is an anti-persistent process), or poles (the model is a long memory processes). Our approach to the problem is based on the Krein's theory of continual analogs of orthogonal polynomials and the continual analogs of Szegö theorem on Toeplitz determinants. A key fact is that the relative prediction error can be represented explicitly by means of the so-called "parameter function" which is a continual analog of the Verblunsky coefficients (or reflection parameters) associated with orthogonal polynomials on the unit circle. To this end first we discuss some properties of Krein's functions, state continual analogs of Szegö "weak" theorem, and obtain formulae for the resolvents and Fredholm determinants of the corresponding Wiener-Hopf truncated operators.

Representations and regularity of Gaussian processes

2015

Julkaisija Julkaisupäivämäärä Vaasan yliopisto Elokuu 2015 Tekijä(t) Julkaisun tyyppi Adil Yazigi Artikkelikokoelma Julkaisusarjan nimi, osan numero Acta Wasaensia, 329 Yhteystiedot ISBN Vaasan Yliopisto Teknillinen tiedekunta Matemaattisten tieteiden yksikkö PL 700 64101 Vaasa 978-952-476-628-9 (painettu) 978-952-476-629-6 (verkkojulkaisu) ISSN 0355-2667 (Acta Wasaensia 329, painettu) 2323-9123 (Acta Wasaensia 329, verkkojulkaisu) 1235-7928 (Acta Wasaensia. Mathematics 12, painettu) 2342-9607 (Acta Wasaensia. Mathematics 12, verkkojulkaisu) Sivumäärä Kieli 80 Englanti Julkaisun nimike Gaussisten prosessien esityslauseet ja säännöllisyys Tiivistelmä Tämä työ käsittelee gaussisia prosesseja, niiden esityslauseita ja säännöllisyyttä. Aluksi tarkastelemme gaussisten prosessien polkujen säännöllisyyttä ja annamme riittävät ja välttämättömät ehdot niiden Hölder-jatkuvuudelle. Sitten tarkastelemme itsesimilaaristen gaussisten prosessien kanonista Volterra-esitystä, joka perustuu Lamperti-muunnokselle ja annamme riittävät ja välttämättömät ehdot esityksen olemassaololle käyttäen ns. täyttä epädeterminismiä. Täysi epädeterminismi on luonnollinen ehto stationaarisille prosesseille kuten myös itsesimilaarisille prosesseille. Sitten sovellamme tulosta ja määrittelemme luokan, jolla on sama itsesimilaarisuusindeksi ja jotka ovat ekvivalentteja jakaumamielessä. Lopuksi tarkastelemme gaussisia siltoja yleistetyssä muodossa, jossa gaussiset prosessit on ehdollistettu monella päätearvolla. Yleistetylle sillalle annamme ortogonaalisen ja kanonisen esitysmuodon sekä martingaali-että ei-martingaalitapauksessa. Johdamme kanonisen esityslauseen myös kääntyville gaussisille Volterraprosesseille.

Generalized covariation for Banach space valued processes, It\^o formula and applications

OSAKA JOURNAL OF MATHEMATICS

This paper concerns the notion of quadratic variation and covariation for Banach valued processes and related Itô formula. If X and Y take respectively values in Banach spaces B1 and B2 (denoted by (B1⊗ π B2) * ) and χ is a suitable subspace of the dual of the projective tensor product of B1 and B2 we define the so-called χ-covariation of X and Y. If X = Y the χ-covariation is called χ-quadratic variation. The notion of χ-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if χ is the whole space (B1⊗ π B1) * then the χ-quadratic variation coincides with the quadratic variation of a B1-valued semimartingale. We evaluate the χ-covariation of various processes for several examples of χ with a particular attention to the case B1 = B2 = C([−τ, 0]) for some τ > 0 and X and Y being window processes. If X is a real process, we call window process associated with X the C([−τ, 0])-valued process X := X(·) defined by Xt(y) = Xt+y, where y ∈ [−τ, 0].

Realization Theory for Multivariate Stationary Gaussian Processes

SIAM Journal on Control and Optimization, 1985

This paper collects in one place a comprehensive theory of stochastic realization for continuous-time stationary Gaussian vector processes which in various pieces has appeared in a number of our earlier papers. It begins with an abstract state space theory, based on the concept of splitting subspace. These results are then carried over to the spectral domain and described in terms of Hardy functions. Finally, differential-equations type stochastic realizations are constructed. The theory is coordinate-free, and it accommodates infinite-dimensional representations, minimality and other systems-theoretical concepts being defined by subspace inclusion rather than by dimension. We have strived for conceptual completeness rather than generality, and the same framework can be used for other types of stochastic realization problems. CONTENTS 1. Introduction 809 2. Perpendicular intersection 813 3. The geometry of splitting subspaces 815 4. Observability, constructibility, and minimality 819 5. Reconciliation with systems theory 823 6. Generating processes 826 7. Hardy space representation of Markovian splitting subspaces 831 8. Stochastic realizations: the finite-dimensional case 838 9. Stochastic realizations: the general case 844 10. State space isomorphism 850 References 855 *