Nonlinear transformations on the Wiener space
type: Item
instance of: scholarly article
title: Nonlinear transformations on the Wiener space
zbMATH Open document ID: 0803.60048
author: Andrew Lyasoff
published in: The Annals of Probability
publication date: 1995-01-03T00:00:00Z
review text: The author studies transformations on an abstract Wiener space \((E, H, \mu)\) which have the form \[ E\ni w\to S(w)= w- \int^ T_ 0 \varphi_ t (w) Z(dt)\in E, \] where \(\varphi_ t(w)\) is a scalar function on \([0,T]\times E\) and \(Z\) is an orthogonal \(H\)-valued measure. Assuming that \(\varphi\in \mathbb{L}^{1,2}\) and that the Malliavin derivative \(D\varphi\) of \(\varphi\) is such that \(| D\varphi (w)|_{H,S} <1\), \(\mu(dw)\)-a.e., the author applies a special factorization technique developed by Gokhberg and Krejn (1970) to express explicitly the Carleman-Fredholm determinant which occurs in Kusuoka's formula (1982) of the density \(R_ \varphi\) of the image of \(\mu\) under \(S\) w.r.t. \(\mu\) as exponential of the product of two integral kernels. This approach works also in cases where \(S\) is not invertible. The main result, stated under the assumptions presented above, constitutes an anticipative analogon of the Cameron-Martin-Girsanov formula: if \(R_ \varphi\) has the expectation 1 under \(\mu\), then \((R_ \varphi\mu)\cdot S^{-1} =\mu\). This generalizes also a paper of the reviewer (1991) who has determined explicitly the integral kernels from the factorization of the Carleman-Fredholm determinant under the assumption \(\text{ess sup}_{w\in E} | D_ \varphi (w)|_{H\cdot S} <1\).
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 46G12
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A68
zbMATH DE Number: 528636
zbMATH Keywords: Skorokhod integral
zbMATH Keywords: Gokhberg-Krejn factorization
zbMATH Keywords: transformations of Wiener measure
zbMATH Keywords: abstract Wiener space
zbMATH Keywords: Malliavin derivative
zbMATH Keywords: factorization technique
zbMATH Keywords: Carleman-Fredholm determinant
zbMATH Keywords: Cameron-Martin- Girsanov formula
zbMATH Keywords: integral kernels
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989015
OpenAlex ID: W2104603361
DOI: 10.1214/AOP/1176989015
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description: scientific article; zbMATH DE number 528636