Ito stochastic integral in the dual of a nuclear space

Ito's definition of the stochastic integral with respect to a Wiener process in the dual of a nuclear space is simplified and slightly generalized. This definition yields a completely intrinsic description of the class of random, operator-valued integrands. For a large class of spaces (e.g. for Schwartz distribution spaces) any time-inhomogeneous Wiener process is proved to have a representation as the stochastic integral with respect to a homogeneous (standard) Wiener process. A relation between this definition of stochastic integral and the notion of isometric integral in Hilbert spaces, defined by Metivier, is established.