Anticipating Hilbert integrals with respect to a cylindrical Wiener process and associated stochastic calculus
This paper is devoted to extend the results on the anticipating stochastic calculus established by \textit{D. Nualart} and \textit{E. Pardoux} [Probab. Theory Relat. Fields 78, No. 4, 535-581 (1988; Zbl 0629.60061)] to the case of Hilbert-valued processes. Given a cylindrical Wiener process \(W=\{W_ t(h), t\geq 0, h\in H\}\) on a Hilbert space \(H\), the authors first define the Skorokhod integral of an operator-valued stochastic process with respect to \(W\), as the adjoint of the derivation operator. This stochastic integral generalizes the Itô integral of operator-valued square integrable adapted processes. Some properties of this integral are presented (isometry, approximation by Riemann sums, \(L^ p\)-estimates, continuity and quadratic variation of the indefinite integral) which extend similar finite-dimensional properties. A Hilbert- valued version of the Stratonovich integral is also introduced, and Itô formulas for both the Skorokhod and Stratonovich integrals are established, following the ideas of the finite-dimensional case.
scientific article; zbMATH DE number 36183