scholarly journals A Version of Fundamental Theorem for the Ito-McShane Integral of an Operator-Valued Stochastic Process

2019 ◽  
Vol 12 (1) ◽  
pp. 101-117
Author(s):  
Jeffer Dave Cagubcob ◽  
Mhelmar Avila Labendia
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Wiener Process ◽  
Absolute Continuity ◽  
Fundamental Theorem ◽  
Mcshane Integral ◽  
Descriptive Definition

In this paper, we formulate a descriptive definition or a version of fundamental theorem for the Ito-McShane integral of an operator-valued stochastic process with respect to a Hilbert space-valued Wiener process. For this reason, we introduce the concept of belated Mcshane dierentiability and a version of absolute continuity of a Hilbert space-valued stochastic process.

scholarly journals Double Lusin Condition for the Ito-Henstock Integrable Operator-Valued Stochastic Process

2018 ◽  
Vol 11 (4) ◽  
pp. 1003-1013
Author(s):  
Mhelmar Avila Labendia ◽  
Jayrold Arcede
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Wiener Process ◽  
Henstock Integral ◽  
Integrable Operator

In this paper, using double Lusin condition, we give an equivalent denition of the Ito-Henstock integral of an operator-valued stochastic process with respect to a Hilbert space-valued Wiener process.

Stochastic barriers for the Wiener process

1983 ◽  
Vol 20 (2) ◽  
pp. 338-348 ◽  
Author(s):  
C. Park ◽  
J. A. Beekman
Keyword(s):  
Stochastic Process ◽  
Wiener Process ◽  
Poisson Processes ◽  
Standard Wiener Process ◽  
Compound Poisson ◽  
Compound Poisson Processes ◽  
Deterministic Function

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The probabilities of the type P[sup0≦t ≦ TW(t) − f(t) ≧ 0] have been extensively studied when f(t) is a deterministic function. This paper discusses the probabilities of the type P{sup0≦t ≦ TW(t) − [f(t) + X(t)] ≧ 0} when X(t) is a stochastic process. By taking compound Poisson processes as X(t), the paper gives procedures for finding such probabilities.

Convergence Theorems in the Sense of Vitali and Lusin for the Itô–Henstock Integral

2021 ◽  
Vol 15 (01) ◽  
pp. 23-34
Author(s):  
Mhelmar A. Labendia ◽  
Jayrold P. Arcede
Keyword(s):  
Stochastic Process ◽  
Wiener Process ◽  
Convergence Theorem ◽  
Convergence Theorems ◽  
Henstock Integral

In this paper, we formulate a version of convergence theorem using double Lusin condition and a version of Vitali convergence theorem for the Itô–Henstock integral of an operator-valued stochastic process with respect to a [Formula: see text]-Wiener process.

scholarly journals A SIMPLE PROOF OF THE FUNDAMENTAL THEOREM ABOUT ARVESON SYSTEMS

2006 ◽  
Vol 09 (02) ◽  
pp. 305-314 ◽  
Author(s):  
MICHAEL SKEIDE
Keyword(s):  
Hilbert Space ◽  
Simple Proof ◽  
Fundamental Theorem ◽  
Hilbert Module ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Product Systems ◽  
Dimensional Hilbert Space ◽  
The One

With every E0-semigroup (acting on the algebra of of bounded operators on a separable infinite-dimensional Hilbert space) there is an associated Arveson system. One of the most important results about Arveson systems is that every Arveson system is the one associated with an E0-semigroup. In these notes we give a new proof of this result that is considerably simpler than the existing ones and allows for a generalization to product systems of Hilbert module (to be published elsewhere).

A SIMILARITY BETWEEN THE GROSS LAPLACIAN AND THE LÉVY LAPLACIAN

2007 ◽  
Vol 10 (02) ◽  
pp. 261-276 ◽  
Author(s):  
UN CIG JI ◽  
KIMIAKI SAITÔ
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Infinitesimal Generator ◽  
Separable Hilbert Space ◽  
Fock Spaces ◽  
Parameter Group ◽  
Infinite Dimensional ◽  
Lévy Laplacian

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

scholarly journals On composite formulas for mathematical expectation of functionals of solution of the Ito equation in Hilbert space

2019 ◽  
Vol 55 (2) ◽  
pp. 158-168
Author(s):  
A. D. Egorov
Keyword(s):  
Hilbert Space ◽  
Random Process ◽  
Wiener Process ◽  
Mathematical Expectation ◽  
Quadrature Formulas ◽  
Third Order ◽  
One Dimensional ◽  
Temporal Variables ◽  
Nonlinear Functionals ◽  
Ito Equation

This article is devoted to constructing composite approximate formulas for calculation of mathematical expectation of nonlinear functionals of solution of the linear Ito equation in Hilbert space with additive noise. As the leading process, the Wiener process taking values in Hilbert space is examined. The formulas are a sum of the approximations of the nonlinear functionals obtained by expanding the leading random process into a series of independent Gaussian random variables and correcting approximating functional quadrature formulas that ensure an approximate accuracy of compound formulas for third-order polynomials. As a test example, the application of the obtained formulas to the case of a one-dimensional wave equation with a leading Wiener process indexed by spatial and temporal variables is considered. This article continues the research begun in [1].The problem is motivated by the necessity to calculate the nonlinear functionals of solution of stochastic partial differential equations. Approximate evaluation of mathematical expectation of stochastic equations with a leading random process indexed only by the time variable is considered in [2–11]. Stochastic partial equations in various interpretations are considered [12–16]. The present article uses the approach given in [12].

scholarly journals Spectral theory of discrete processes

Open Physics ◽  
2010 ◽  
Vol 8 (3) ◽  
Author(s):  
Palle Jorgensen ◽  
Myung-Sin Song
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Spectral Analysis ◽  
Stochastic Processes ◽  
Spectral Theory ◽  
Transfer Operator ◽  
Infinite Graphs ◽  
Spectral Theorem ◽  
Transfer Operators ◽  
Wide Range

AbstractWe offer a spectral analysis for a class of transfer operators. These transfer operators arise for a wide range of stochastic processes, ranging from random walks on infinite graphs to the processes that govern signals and recursive wavelet algorithms; even spectral theory for fractal measures. In each case, there is an associated class of harmonic functions which we study. And in addition, we study three questions in depthIn specific applications, and for a specific stochastic process, how do we realize the transfer operator T as an operator in a suitable Hilbert space? And how to spectral analyze T once the right Hilbert space H has been selected? Finally we characterize the stochastic processes that are governed by a single transfer operator.In our applications, the particular stochastic process will live on an infinite path-space which is realized in turn on a state space S. In the case of random walk on graphs G, S will be the set of vertices of G. The Hilbert space H on which the transfer operator T acts will then be an L 2 space on S, or a Hilbert space defined from an energy-quadratic form.This circle of problems is both interesting and non-trivial as it turns out that T may often be an unbounded linear operator in H; but even if it is bounded, it is a non-normal operator, so its spectral theory is not amenable to an analysis with the use of von Neumann’s spectral theorem. While we offer a number of applications, we believe that our spectral analysis will have intrinsic interest for the theory of operators in Hilbert space.

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