On stochastic integration by series of Wiener integrals
type: Item
instance of: scholarly article
title: On stochastic integration by series of Wiener integrals
zbMATH Open document ID: 0661.60065
DOI: 10.1007/BF01448196
published in: Applied Mathematics and Optimization
publication date: 1989-01-01T00:00:00Z
review text: \textit{D. E. Varberg} [Ann. Math. Stat. 37, 567-576 (1966; Zbl 0143.206)] defined a certain stochastic integral as the limit of suitable orthogonal series of random variables which are closely related to the martingale differences, and noted that the resulting integral is more general than the Wiener integral in some respects. The present paper abstracts that construction and obtains the corresponding results in a somewhat more general context. Thus one has: ``Stochastic integrals of random functions with respect to a white noise random measure are defined in terms of random series of the usual Wiener integrals. Conditions for the existence of such integrals are obtained in terms of the nuclearity of certain operators on \(L^ 2\)- spaces. The relation with the Fisk-Stratonovich symmetric integral is also discussed.'' The paper contains several examples to illustrate the significance of the thus obtained generalization. In particular neither integral includes the other although they have a ``large'' intersection.
reviewed by: Malempati M. Rao
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 4080539
zbMATH Keywords: multiple stochastic integrals
zbMATH Keywords: white noise random measures
zbMATH Keywords: martingale differences
zbMATH Keywords: Fisk-Stratonovich symmetric integral
author: Jan Rosiński
MaRDI profile type: Publication
cites work: Q4099504
cites work: Q5336643
cites work: Q3959169
cites work: Multiple Wiener integral
cites work: Gaussian measures in Banach spaces
cites work: White noise approach to stochastic integration
cites work: Une remarque sur l'approximation de l'integrale stochastique du type noncausal par une suite des integrales de Stieltjes
cites work: A Sufficient Condition for an Integral Operator to Have a Trace.
cites work: Convergence of Quadratic Forms in Independent Random Variables
description: scientific article; zbMATH DE number 4080539