Malliavin calculus in abstract Wiener space using infinitesimals
type: Item
instance of: scholarly article
title: Malliavin calculus in abstract Wiener space using infinitesimals
zbMATH Open document ID: 1018.60058
DOI: 10.1016/S0001-8708(02)00033-6
author: Horst Osswald
published in: Advances in Mathematics
publication date: 2003-05-25T00:00:00Z
review text: The author constructs a Malliavin calculus on \(C([0,1],{\mathbb B})\), where \({\mathbb B}\) is an abstract Wiener space with Cameron-Martin space \({\mathbb H}\), in a non-standard framework. Here, \([0,1]\) is replaced by \(T=\{1,\ldots ,H\}\) where \(H\) is a hyperfinite integer, and \({\mathbb H}\) is replaced with a hyperfinite dimensional space \({\mathbb F}\) with dimension \(\omega\). The Lebesgue measure on \([0,1]\) is replaced with the Loeb measure over the uniform probability on \(T\), and the basic probability space is now \({\mathbb F}^H\), endowed with the Loeb measure over the \(\omega \cdot H\)-dimensional standard Gaussian distribution of variance \(1/H\). In this framework a construction of Brownian motion in abstract Wiener spaces, iterated Itô integrals, chaos decompositions, Malliavin derivative, and Skorokhod integrals, is provided. The absolute continuity of certain anticipating shifts (called adequate shifts) of \({\mathbb B}\)-valued Brownian motion is studied, and as an application a Girsanov theorem for adapted shifts of Brownian motion by Bochner integrals is obtained.
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 03H05
zbMATH DE Number: 1915629
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: abstract Wiener spaces
zbMATH Keywords: nonstandard analysis
MaRDI profile type: Publication
cites work: Q3742397
cites work: Dirichlet forms and analysis on Wiener space
cites work: Infinitesimals in Action
cites work: Q4839666
cites work: Measurable Functions on Hilbert Space
cites work: An infinitesimal approach to stochastic analysis
cites work: Gaussian measures in Banach spaces
cites work: Characterization of the law of the iterated logarithm in Banach spaces
cites work: Nonstandard integration theory in topological vector lattices
cites work: Q4509263
cites work: Infinitesimals in abstract Wiener spaces
cites work: On the Clark Ocone formula for the abstract Wiener space
cites work: Embedding the abstract Wiener space in a probability space
full work available at URL: s0001-8708(02)00033-6
OpenAlex ID: W2048518870
description: scientific article; zbMATH DE number 1915629