On certain relations between the path integrals and the translation operator and its dual in canonical Poisson space (Q1580330)

The present paper is devoted to a ``Malliavin-type'' stochastic calculus for Poisson processes. After recalling the concept of the translation operator \(\Psi\) and its \(L^2\)-dual \(\Phi\) introduced by \textit{D. Nualart} and \textit{J. Vives} [C. R. Acad. Sci., Paris, Sér. I 307, No. 7, 349-354 (1988; Zbl 0651.60066) and in: Seminar on stochastic analysis, random fields and applications. Prog. Probab. 36, 205-213 (1995; Zbl 0856.60057)] on the canonical Poisson space, the authors study the relation between both operators, the pathwise integral and the Skorokhod integral (defined via the Poisson chaos expansion) and present a small new contribution. In the last section they consider the notions of forward and backward integrals introduced by \textit{F. Russo} and \textit{P. Vallois} [C. R. Acad. Sci., Paris, Sér. I 312, No. 8, 615-618 (1991; Zbl 0723.60058)] and translate them into their setting. This allows to the authors to present, in the special case of a càdlàg integrand of the pathwise integral, another approach to characterize the relation between pathwise integral, \(\Phi\) and \(\Psi\).