for $(T,u) \in \Rplus \times \cU$ we define a complex-valued random measure on $[0,T]$ by $$\begin{aligned}
\label{eq:G}
G(dt,\omega,T,u) &:= \scal{\psi_t}{dB^c_t(\omega)} + \frac{1}{2}\scal{\psi_t}{dC_t(\omega)\psi_t} + \\
&+ \int_D \left(e^{\langle \psi_t, \xi \rangle} - 1 -\langle \psi_t,h(\xi) \rangle\right) \nu^c(\omega,dt,d\xi),\notag\end{aligned}$$ where we write $\psi_t := \psi_t(T,u)$ for short.

\[lem:characteristics\] Let $X$ be a quasi-regular affine semimartingale with a good version of its characteristics $(B,C,\nu)$, let $(T,u) \in (0,\infty) \times \cU$ and let $G(dt,\omega,T,u)$ be the complex-valued random measure defined in . It holds that $$\label{eq:measure_identity}
G(dt;\omega,T,u) + d\phi^c_{t}(T,u) + \scal{X_t(\omega)}{d\psi^c_{t}(T,u)} = 0, \qquad\PP-a.s,$$ as identity between measures on $[0,T]$.

For $(T,u) \in (0,\infty) \times \cU$ consider the process $$M^{u,T}_{t}:=\E\Big[e^{ \langle u , X_{T} \rangle}\big|\cF_{t}\Big]
    =\exp\left(\phi_t\left(T,u\right)+\scal{\psi_t\left(T,u\right)}{X_{t}}\right)\quad t \in [0,T),$$ which is a càdlàg martingale with the terminal value $M_{T}^{u,T} = \exp\left(\scal{u}{X_{T}}\right)$. To alleviate notation we consider $(T,u)$ fixed and write $$M_{t} = M^{u,T}_{t} = \exp\left(\phi_{t}+\scal{\psi_{t}}{X_{t}}\right),$$ with $\phi_{t} := \phi_t(T,u)$ and $\psi(t) := \psi_t(T,u)$. Applying the Itô-formula for semimartingales (cf. [@JacodShiryaev Prop.II.2.42]) to $M$ we obtain a decomposition $$M_t = L_t + F_t,$$ where $L$ is a local martingale and $F$ is the predictable finite variation process $$\begin{aligned}
F_t := &\int_0^t M_{s-} \left\{ d\phi^c_s + \scal{X_{s-}}{d\psi^c_s} + \scal{\psi_{s-}}{dB_s}  +\frac{1}{2}\scal{\psi_{s-}}{dC_s \psi_{s-}}  \right. \label{eq:prc-char}
\\
&+  \left.\int_D \left(e^{\Delta 