We consider linear SDEs with the generalized positive noise process standing for the noisy term. Under certain conditions, the solution, a Colombeau generalized stochastic process, is proved to exist. Due to the blowing-up of the variance of the solution, we introduce a ``new" positive noise process, a renormalization of the usual one. When we consider the same equation but now with the renormalized positive noise, we obtain a solution in the space of Colombeau generalized stochastic processes with both, the first and the second moment, converging to a finite limit.