Before revision: We propose a class of equity models whose volatility, L\'evy measure , and killing rate all have local stochastic state-dependence.
After revision: We propose a class of assets whose risk-neutral pricing dynamics are described by an exponential L\'evy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent L\'evy measure , and killing rate all have local stochastic state-dependence.