We investigate the properties of the Wick square of Gaussian white noises
through a new method to perform non linear operations on Hida distributions.
This method lays in between the Wick product interpretation and the usual
definition of nonlinear functions. We prove on Ito-type formula and solve
stochastic differential equations driven by the renormalized square of the
Gaussian white noise. Our approach works with standard assumptions on the
coefficients of the equations, Lipschitz continuity and linear growth
condition, and produces existence and uniqueness results in the space where the
noise lives. The linear case is studied in details and positivity of the
solution is proved.Comment: 23 page
