Integration by parts on Wiener space and the existence of occupation densities
type: Item
instance of: scholarly article
title: Integration by parts on Wiener space and the existence of occupation densities
zbMATH Open document ID: 0803.60049
author: P. Imkeller
author: David Nualart
published in: The Annals of Probability
publication date: 1995-01-03T00:00:00Z
review text: Motivated by recent developments in the anticipative stochastic calculus [see \textit{D. Nualart} and \textit{E. Pardoux}, Stochastic differential systems, Proc. IFIP-WG 7/1 Work Conf., Eisenach/GDR 1986, Lect. Notes Control Inf. Sci. 96, 363-372 (1987; Zbl 0633.60075)] and, in particular, in the theory of Skorokhod's stochastic integral (introduced by Skorokhod, 1975) which allows to integrate also non-adapted processes, the authors investigate conditions under which Skorokhod integral processes possess an occupation density or a local time. The paper generalizes earlier results of Imkeller about Skorokhod integral processes belonging to the second Wiener chaos, and it uses for this purpose a new general approach: the authors combine a sufficient criterion for the existence of an occupation density by \textit{D. Geman} and \textit{J. Horowitz} [Ann. Probab. 8, 1-67 (1980; Zbl 0499.60081)] with the technique of integration by parts on the Wiener space and deduce several sufficient criteria for the existence of occupation densities of Skorokhod integral processes. In particular, the authors prove that, if \((W_ t)\) denotes the Wiener process and \(u\) a possibly anticipating process which belongs w.r.t. \((W_ t)\) to \(\mathbb{L}^{2,2}\), then the process \((W_ t+ \int_ 0^ t u_ s dW_ s)\) has a square integrable occupation density. If, moreover, for some suitable \(\gamma>0\), \(E[\int^ 1_ 0 | u_ t|^{-\gamma} dt] <+\infty\), then, under some additional smoothness assumptions and boundedness assumptions on the Malliavin derivatives of \(u\), also the process \((\int^ t_ 0 u_ s dW_ s)\) possesses a square integrable occupation density.
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 567195
zbMATH Keywords: Tanaka's formula
zbMATH Keywords: Skorokhod integral processes
zbMATH Keywords: occupation density
zbMATH Keywords: local time
zbMATH Keywords: second Wiener chaos
zbMATH Keywords: integration by parts
zbMATH Keywords: Wiener space
zbMATH Keywords: anticipating process
zbMATH Keywords: Malliavin derivatives
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988869
OpenAlex ID: W1994216870
DOI: 10.1214/AOP/1176988869
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description: scientific article; zbMATH DE number 567195