In this paper a generalized random process is modeled through the randomization of a bilinear form between the space D of test functions and the Colombeau generalized functions. This results in a theory akin to Gelfand–Vilankin's random Schwartz distributions. An extension theorem in Bochner–Badrikian style is proved under some continuity assumptions. An important application is a natural representation of nonlinear functionals of white noise.