this section we derive the representation of affine semimartingales via their semimartingale characteristics as well as generalized measure Riccati equations for the coefficients $\phi$ and $\psi$. It turns out that the class of affine semimartingales substantially generalizes the class of stochastically continuous affine processes: first, jumps at fixed time points are allowed and second, the jump height may depend on the state of the process.

Throughout, we will use the short-hand notation $\alpha=(\alpha_0,\bar \alpha)$ for a generic $d+1$-dimensional vector $\alpha=(\alpha_0,\dots,\alpha_d)$. Moreover, we denote by $\cS^{d}_+$ the convex cone of symmetric positive semi-definite $d\times d$ matrices. Given characteristics $(B,C,\nu)$ of a semimartingale $X$, recall from [@JacodShiryaev Eq. II.1.23, Prop. II.2.6] that $C$ is always continuous and $B$ can be decomposed as $B = B^c + \sum \Delta B$. Furthermore, also a ‘continuous part’ $\nu^c$ of $\nu$ can be defined by $$\label{eq:cont_nu}
\begin{split}
\cJ &:= \set{(\omega,t): \nu(\omega,\set{t},D) > 0}\\
\nu^c(\omega,dt,dx) &:= \nu(\omega,dt,dx) \bI_{\cJ^\complement}(\omega,t).
\end{split}$$ Finally, if one chooses a ‘good version’ (as we always do) of the characteristics, then $$\label{eq:B_jumps}
\Delta B_t = \int_D h(x) \nu(\set{t}, dx),$$ where $h$ is the truncation function for the jumps; cf. [@JacodShiryaev Prop. II.2.9]. We introduce the following definition, which will be needed to formulate our 