Abstract: Stochastic volatility models have proven relevant for various purposes, e.g. explaining the term structure of implied volatility or correlations in absolute returns. When used in conjunction with jump models, they provide powerful tools that have been applied successfully in recent years. Volatility is however but one aspect of risk: in the frame of jump processes, the (local) intensity of jumps is another one, which is equally important. It thus seems natural to supplement stochastic volatility models with “stochastic jump intensity” ones, where the local intensity of jumps evolves according to a random dynamics. The aim of this preliminary work is merely to set up such models as extensions of the celebrated CGMY process, leaving their exploration both for risk management and for option pricing to further studies.