Green formulas in anticipating stochastic calculus (Q1890717)

Let \(T = [0,1]^ 2\) and \(W = \{W_ z, z\in T\}\) be a Brownian sheet. Consider two stochastic processes \(u = \{u_ z, z\in T\}\), \(v = \{v_ z, z\in T\}\) which are not necessarily adapted with respect to \(W\) but satisfy some smoothness assumptions in the sense of the Malliavin calculus. Set \[ u_ z = \int_{R_ z} u_ r \circ dw_ r + \int_{R_ z} v_ r dr, \qquad z \in T,\;R_ z := (0,z), \] where the stochastic integral is defined in the Skorokhod-Stratonovich sense. The paper establishes a Green-Stratonovich formula for anticipating processes with respect to \(U\), i.e., a formula which relates ``line'' and ``surface'' stochastic integrals. Also a Skorokhod counterpart of this formula is provided. As a useful tool for deriving Green's formula a Fubini theorem for Stratonovich integrals with respect to \(U\) is proved by studying trace terms relating Skorokhod and Stratonovich integrals. The paper generalizes earlier work of \textit{R. Cairoli} and \textit{J. B. Walsh} [Acta Math. 134, 111-183 (1975; Zbl 0334.60026)] who considered adapted integrands and \(u = 1\), \(v = 0\), and that of \textit{J. L. Solé} and \textit{F. Utzet} [Stochastic Processes Appl. 39, No. 2, 239-262 (1991; Zbl 0744.60059) and in: Séminaire de probabilités, Lect. Notes Math. 1485, 270-283 (1991; Zbl 0744.60060)] for the same situation but for integrands which can be nonadapted.