Anticipating integrals for a class of martingales
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.
anticipating stochastic differential equation
scientific article; zbMATH DE number 1143813