The main notions and tools from white noise analysis are set up on the basis of the calculus of Gaussian random variables and the S-transform. A new proof of the formula for the S-transform of Itô integrals is given. Moreover, measurability and the martingale property with respect to the Brownian filtration are characterized in terms of the S-transform. This allows to extend these notions to random variables and processes, respectively, in the space of Hida distributions.
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