main results.

\[def:good-parameter\] Let $A$ be a non-decreasing càdlàg function with continuous part $A^c$ and jump points $J^A\coloneqq\set{t\ge 0\vert \Delta A_t> 0}$. Let $(\gamma,\beta,\alpha,\mu)=\brac{\gamma_i,\beta_i,\alpha_i,\mu_i}_{i\in\setd}$ be functions such that $\gamma_0\colon \R_{\ge0}\times\cU\rightarrow\C$, $\bar{\gamma}\colon \R_{\ge0}\times\cU\rightarrow \C^d$, $\beta_i\colon\R_{\geq0}\rightarrow\R^d$, $\alpha_i\colon\R_{\ge0}\rightarrow\mathcal{S}^d$ and $\brac{\mu_i\brac{t,\cdot}}_{t\ge 0}$ are families of (possibly signed) Borel measures on $D\setminus\set{0}$. We call $(A,\gamma,\beta,\alpha,\mu)$ a *good parameter set* if for all $i\in\setd$,

(i) $\alpha_i$ and $\beta_i$ are locally integrable w.r.t. $A^c$,

(ii) for all compact sets $K\subset D\setminus\set{0}$, $\mu\brac{\cdot,K}$ is locally $A^c$-integrable.

(iii) $\gamma(t,u) = 0$ for all $(t,u) \in (\R_{\ge 0} \setminus J^A) \times \cU$.

\[prop1\] Let $X$ be a quasi-regular affine semimartingale satisfying the support condition \[def:full\_support\]. Then there exists a good parameter set $(A,\gamma,\beta, \alpha, \mu)$ such that the semimartingale characteristics $(B,C,\nu)$ of $X$ w.r.t. the truncation function $h$ satisfy, $\PP$-a.s. for any $t >0$,

\[char:affine\] $$\begin{aligned}
B^c_t(\omega) &= \int_0^t \big( \beta_0 (s)  + \sum_{i=1}^d X^i_{s-}(\omega) \beta_i(s)   \big) dA^c_s \label{eq:char_B}\\
C_t(\omega) &= \int_0^t \big( \alpha_0 (s) + \sum_{i=1}^d X^i_{s-}(\omega) \alpha_i (s)  \big) dA^c_s \label{eq:char_C}\\
\nu^c(\omega,ds,dx) &= \big( \mu_0 (s,dx) + \sum_{i=1}^d X^i_{s-}(\omega) \mu_i (s,dx)  \big) dA^c_s  \label{eq:char_nu}\\
\int_D \left(e^{\langle u,\xi \rangle}  - 1\right) \nu(\omega,\{t\},d\xi) &= \left(\exp\Big( \gamma_0(t,u) + \sum_{i=1}^d \langle X^i_{t-}(\omega), \bar \gamma_i (t,u) \rangle \Big) - 1\right).\label{eq:char_nu_dis}\end{aligned}$$

Moreover, for all 