\times (d+1)}$-valued process process $(\Theta_t)_{t \ge 0}$ defined in , we define $d+1$ independent copies of the complex-valued random measure $G(dt,\omega,T,u)$ from equation and denote them by $G_0,\dots,G_d$, respectively. With this notation and for any $(T,u) \in \Rplus \times \cU$, the $d+1$ corresponding equations can be written in matrix-vector form as $$\label{eq:phi_to_G}
\Theta_{t}(\omega) \cdot \begin{pmatrix} d\phi^c_t(T,u) \\ d\psi^{c,1}_{t}(T,u) \\ \vdots \\ d\psi^{c,d}_{t}(T,u) \end{pmatrix}
 =  - \begin{pmatrix} G_{0}(dt;\omega,T,u) \\ \vdots \\ G_{d}(dt;\omega,T,u)  \end{pmatrix}$$ which holds $\PP$-a.s. as an identity between complex-valued measures on $[0,T]$. The next Lemma gives a ’local’ version of the continuous part of Theorem \[prop1\].

\[lem:local\] Let $X$ be a quasi-regular affine semimartingale satisfying the support condition \[def:full\_support\] and let $\tau \in (0,\infty)$ be a deterministic timepoint. Then there exists an interval $I_\tau := (\tau, \tau+ \epsilon)$, where $\epsilon = \epsilon(\tau) > 0$, and good parameters $(A^c, \beta, \alpha, \mu)$ on $I_\tau$. With respect to these parameters, and with $F$ and $R$ as in , the measure Riccati equations and hold true for each $(T,u) \in \Rplus \times \cU$ and $t \in I_\tau \cap [0,T]$.

We emphasize that in this lemma the parameters $(A^c, \beta, \alpha, \mu)$ as well as the functions $F$ and $R$ may depend on 