set* in Section \[sec:characterization\] which is a key ingredient of our first main result, the characterization Theorem \[prop1\]. Section 4 discusses the relation between affine Markov processes and affine semimartingales as well as the important case of infinitely divisible processes. Section 5 is devoted to the existence of affine Markov processes and affine semimartingales under certain conditions on their good parameter set. Examples and applications are explained in Section 6 which concludes the paper with the introduction of a new *affine term-structure framework*. Details about measure differential equations that appear in the characterization and existence results instead of the ODEs appearing in [@DuffieFilipovicSchachermayer] and [@Filipovic05], are postponed to the appendix.

Preliminaries
=============

Affine Semimartingales
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Consider a filtered probability space $(\Omega,\cF,\bbF,P)$ with filtration $\bbF=(\cF_t)_{t \ge 0}$ satisfying the usual conditions. A stochastic process $X$ taking values in $\R^d$ is called *càdlàg* if all its paths are right-continuous with left limits. For a càdlàg process $X$ we define $X_-$ and $\Delta X$ by $$\begin{cases}X_{0-} &= X_0, \quad  X_{t-} = \lim_{s \uparrow t} X_s \quad \text{ for }t>0,\\
\Delta X_t &= X_t - X_{t-}  . \end{cases}$$ In particular, note that $\Delta X_0 = 0$ and that $X$ can be recovered from $X_-$ 