A weak stochastic integral in Banach space with application to a linear stochastic differential equation
type: Item
instance of: scholarly article
title: A weak stochastic integral in Banach space with application to a linear stochastic differential equation
zbMATH Open document ID: 0534.60048
DOI: 10.1007/BF01448381
author: Nadav Berman
author: William L. Root
published in: Applied Mathematics and Optimization
publication date: 1983-01-01T00:00:00Z
review text: In this readable paper the authors study in detail cylindrical stochastic processes in separable Banach spaces. They start with the notion of cylindrical Wiener process \(W_ t\), which may be represented as a weakly convergent sum of independent, real Wiener processes. Similar constructions may be found in \textit{A. Bensoussan}, Filtrage optimal des systèmes linéaires (1971; Zbl 0231.93022), \textit{A. V. Balakrishnan}, Applied functional analysis (1976; Zbl 0333.93051) and \textit{L. Gross}, Proc. 5th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1965/1966, 2, part 1, 31-42 (1967; Zbl 0187.409). Since the covariance may be a general bounded symmetric operator, the distribution of \(W_ t\) is only a cylinder set measure. They then define an adequate notion of stochastic integral with respect to \(W_ t\) and give a theorem on existence and uniqueness for certain stochastic differential equations with linear drift and state-independent diffusion.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 3848338
zbMATH Keywords: cylindrical stochastic process in Banach space
zbMATH Keywords: cylindrical Wiener process in Banach space
zbMATH Keywords: linear stochastic differential equations in Banach space
Wikidata QID: Q115393508
reviewed by: Gunter Ritter
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01448381
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description: scientific article; zbMATH DE number 3848338