  We introduce closed-form transition density expansions for multivariate
affine jump-diffusion processes. The expansions rely on a general approximation
theory which we develop in weighted Hilbert spaces for random variables which
possess all polynomial moments. We establish parametric conditions which
guarantee existence and differentiability of transition densities of affine
models and show how they naturally fit into the approximation framework.
Empirical applications in credit risk, likelihood inference, and option pricing
highlight the usefulness of our expansions. The approximations are extremely
fast to evaluate, and they perform very accurately and numerically stable.
