Anticipating integrals for a class of martingales
type: Item
instance of: scholarly article
title: Anticipating integrals for a class of martingales
zbMATH Open document ID: 0897.60058
DOI: 10.2307/3318533
author: Jin Ma
author: Philip Protter
author: Jaime San Martín
published in: Bernoulli
publication date: 1998-05-25T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bj/1175865491
review text: Let \(M=(M_t)\) be a martingale with \(\langle M,M \rangle_t =t\), \(t\geq 0\), which possesses the additional property of chaos representation. For such a martingale \(M\) the authors define an anticipating integral by the method of chaos expansion of the possibly nonadapted integrand. This approach made in the spirit of the work by \textit{D. Nualart} and \textit{E. Pardoux} [Probab. Theory Relat. Fields 78, No. 4, 535-581 (1988; Zbl 0629.60061)] generalizes the notion of the Skorokhod integral (which is defined with respect to a Wiener process \(W)\). So it is not very surprising that there are a lot of similarities between this martingale anticipating integral and the Skorokhod integral. But there are also quite new effects which come from the fact that for the Wiener case \([W,W]_t=\langle W,W\rangle_t=t\), while in the general case only \(\langle M,M \rangle_t =t\), and \([M,M]_t\) can be random. In particular and besides other subtle differences, this leads to different Malliavin derivatives which enter together with \([M,M]_t\) in the author's integration by parts formula. The very interesting paper about this new martingale anticipating integral and its properties is completed by some preliminary results on associated linear stochastic differential equations with anticipation; the results are obtained by the method of chaos expansion.
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 1143813
zbMATH Keywords: anticipating stochastic differential equation
zbMATH Keywords: anticipating stochastic integral
zbMATH Keywords: chaos decomposition
zbMATH Keywords: integration by parts formula
zbMATH Keywords: normal martingale
zbMATH Keywords: structure equation
zbMATH Keywords: Malliavin derivative
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
OpenAlex ID: W2089724057
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description: scientific article; zbMATH DE number 1143813