White noise approach to stochastic integration
type: Item
instance of: scholarly article
title: White noise approach to stochastic integration
zbMATH Open document ID: 0636.60053
DOI: 10.1016/0047-259X(88)90037-1
author: Hui-Hsiung Kuo
author: Andrezej Russek
published in: Journal of Multivariate Analysis
publication date: 1988-01-01T00:00:00Z
review text: The purpose of this paper is to extend the well-known Ito integral \(\int^{b}_{a}\Phi (t,\omega)dB(t,\omega)\) with respect to a Brownian motion to not necessarily nonanticipating stochastic processes \(\Phi\) by using the white noise calculus. Starting from the formal equality \[ \int^{b}_{a}\Phi (t,\omega)dB(t,\omega)=\int^{b}_{a}\dot B(t,\omega)\Phi (t,\omega)dt \] the authors define two kinds of extended stochastic integrals, where they use two kinds of white noise multiplications introduced in the paper. In the nonanticipating case one of these integrals coincides with the usual Ito integral. After proving the existence and some basic properties of the introduced integrals the authors solve the following problem concerning the series expansion of the Ito integral: Let \((e_ k)_{k=1,2,...}\) be an orthonormal basis in \(L_ 2(a,b)\). If \(\int^{b}_{a}| \Phi (t,\omega)|\) 2 dt\(<\infty\omega\)-almost surely, then \(\Phi\) can be expanded by \(\Phi (t,\omega)=\sum^{\infty}_{k=1}a_ k(\omega)e_ k(t)\). Under which conditions does the series \(\sum^{\infty}_{k=1}a_ k(\omega)\int^{b}_{a}e_ k(t)dB(t,\omega)\) converge and, if it does, is the limit just the Ito integral? The authors give conditions on \(\Phi\) generating the convergence of the series and show that the limit is an extended Stratonovich integral defined in the paper. Note that the authors do not compare their integral with other extensions of the Ito integral known from the literature.
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60G20
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4036835
zbMATH Keywords: white noise calculus
zbMATH Keywords: extended Stratonovich integral
zbMATH Keywords: extensions of the Ito integral
MaRDI profile type: Publication
cites work: Q4073917
cites work: An extension of the stochastic integral
cites work: ESTIMATES OF SINGULAR NUMBERS OF INTEGRAL OPERATORS
cites work: Q5471003
cites work: Generalized multiple Wiener integrals
cites work: Quadratic functionals of Brownian motion
cites work: Q4174026
cites work: Multiple Wiener integral
cites work: Q4197141
cites work: Q3311431
cites work: Calculus on Gaussian white noise. II
cites work: Calculus on Gaussian white noise. III
cites work: On Fourier transform of generalized Brownian functionals
cites work: Brownian functionals and applications
cites work: Q5628582
cites work: On a Generalization of a Stochastic Integral
full work available at URL: 0047-259x(88)90037-1
OpenAlex ID: W2042847706
description: scientific article; zbMATH DE number 4036835