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Jump diffusion processes provide a means of modelling both small and large deviations in continuously evolving processes. Unfortunately, the calculus of jump diffusion processes makes it difficult to analyse non-linear models. This paper develops a method for approximating the transition densities of a class of time-inhomogeneous multivariate jump diffusions with state-dependent and/or stochastic intensity. By deriving a system of equations that govern the evolution of the moments of the process, we are able to approximate the transitional density through a density factorization that contrasts the dynamics of the jump diffusion with that of its jump free counterpart. Within this framework, we develop a class of quadratic jump diffusions for which we can calculate accurate approximations to the likelihood function. Subsequently, we analyse a number of non-linear jump diffusion models for Google equity volatility, alternating between various drift, diffusion, and jump mechanism specifications. In doing so we find evidence of both cyclical drift and a state-dependent jump mechanism.