Dynamic spectrum leasing (DSL) is a promising method to improve the spectrum efficiency. A lot of work has been done about spectrum sharing issues with static models. In this paper, we want to study the behavior of the primary users (PUs) under uncertainty. For this objective, we propose a model with time-varying parameters. The model captures both technical and economic features. We study the long-term behavior of the PUs in the model with stochastic Cournot game theory and variational inequality theory. We show that the PUs can maximize their long-term utilities with a static supply of spectrum. In addition, we also propose two algorithms based on stochastic approximation theory. The convergence and performance of the algorithms are demonstrated theoretically and numerically. The proposed model and analytical framework can be used to study and predict the behaviors of PUs given a DSL system and provide design insights for the DSL system.