Stochastic integrals for nonprevisible, multiparameter processes (Q687076)

The authors develop a theory of stochastic integrals of generalized multiparameter stochastic processes in the framework of the white noise analysis. Basic ingredients of this theory are the weak derivatives \({\partial X\over\partial t_ i}\) and the white-noise integrals \(\int X(t)\hat\otimes {\partial^ rW^ j\over\partial t_{i_ 1}\cdots\partial t_{i_ r}} (t)dt\) of a given generalized process \(X: {\mathbb{R}}^ N\to({\mathcal D}^*)\). Using these notions the authors establish a derivative version of Itô's formula for a process of the form \(\{F(W^ 1(t),\ldots,W^ M(t)), t\in{\mathbb{R}}^ N\}\), where \(W\) is an \(M\)-dimensional \(N\)-parameter Wiener process. On the other hand, the above white-noise integrals can be expressed in terms of multiple Skorokhod integrals which coincide with the nonmixed stochastic integrals introduced by \textit{P. Imkeller} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 65, 535-562 (1984; Zbl 0557.60040)] when the integrand is previsible.