The Wiener process has provided lots of practically useful mathematical tools to model stochastic noise in many applications. However, this framework is not enough for modeling extremal events since many statistical properties of dynamical systems driven by Wiener processes are inevitably Gaussian. The goal of this work is to develop a framework that can represent heavy tailed distribution without losing the advantages of the Wiener process. To this end, we investigate models based on stable processes, and propose a method for stochastic linearization. It is applied to renewable energy assessment to show the effectiveness.