\phi_s + \langle \psi_s, X_{s-}+\xi\rangle - \langle \psi_{s-},X_{s-}\rangle } - 1  - \langle \psi_{s-},h(\xi) \rangle \right) \nu(\omega,ds,d\xi) \right\}.
\notag\end{aligned}$$ The jump part $\Delta F$ vanishes due to Lemma \[lem:affinejumps\] and , and we are left with the continuous part $$\begin{aligned}
F_t = F_t^c = &\int_0^t M_{s-} \left\{ d\phi^c_s + \scal{X_{s-}}{d\psi^c_s} + \scal{\psi_{s-}}{dB^c_s}  +\frac{1}{2}\scal{\psi_{s-}}{dC_s \psi_{s-}}  \right. \label{eq:prc-char}
\\
&+  \left.\int_D \left(e^{ \langle \psi_{s-},\xi\rangle } - 1  - \langle \psi_{s-},h(\xi) \rangle \right) \nu^c(\omega,ds,d\xi) \right\}.
\notag\end{aligned}$$ Recall that $M$ is a martingale, and hence $M \equiv L$ and $F \equiv 0$ on $[0,T]$, $\PP$-a.s. With , $F$ can be rewritten as $$F_t = \int_0^t M_{s-} \left\{ d\phi^c_{s} + \scal{X_{s-}}{d\psi^c_{s}} + G(ds;\omega,T,u) \right\}.$$ Since none of the measures appearing above charges points, the left limits $X_{s-}, \psi_{s-}$ can be substituted by right limits $X_s, \psi_s$. Moreover, $M_{s-}$ is nonzero everywhere and follows.

In order to make efficient use of the support condition \[def:full\_support\], we introduce the following convention: Given an affine semimartingale $X$, a tuple $\mathbf{X} = (X^0, \dotsc, X^d)$ represents $d+1$ *stochastically independent* copies of $X$. Formally, the tuple $\mathbf{X}$ can be realized on the product space $(\Omega^{(d+1)}, \cF^{\otimes (d+1)}, (\cF_t^{\otimes (d+1)})_{t \geq 0})$ equipped with the associated product measure. Moreover, for any 