The Wiener process has provided a lot 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 the Wiener process are inevitably Gaussian. The goal of this work is to develop a framework that can represent a heavy-tailed distribution without losing the advantages of the Wiener process. To this end, we investigate models based on stable processes (this term "stable" has nothing to do with "dynamical stability") and clarify their fundamental properties. In addition, we propose a method for stochastic linearization, which enables us to approximately linearize static non-linearities in feedback systems under heavy-tailed noise, and analyze the resulting error theoretically. The proposed method is applied to assessing wind power fluctuation to show the practical usefulness.