  In this paper we study time-inhomogeneous affine processes beyond the common
assumption of stochastic continuity. In this setting times of jumps can be both
inaccessible and predictable. To this end we develop a general theory of finite
dimensional affine semimartingales under very weak assumptions. We show that
the corresponding semimartingale characteristics have affine form and that the
conditional characteristic function can be represented with solutions to
measure differential equations of Riccati type. We prove existence of affine
Markov processes and affine semimartingales under mild conditions and elaborate
on examples and applications including affine processes in discrete time.
