The Gross derivative and generalized random variables

We extend the Gross derivative to a space of generalized random variables w hich have a (formal) chaos expansion with kernels from the space of tempere d Schwartz, distributions. The extended derivative, which we call the Hida derivative, has to be interpreted in the sense of distributions. Many of th e properties of the Gross derivative are proved to hold for the extension a s well. In addition, we derive a representation formula for the Hida deriva tive involving the Wick product and a centered Gaussian random variable. We apply our results to calculate the Hida derivative of a class of stochasti c differential equations of Wick type.