Combining with , claim (i) follows. For (ii), let $t \in J^\nu$. Then, there exists an $u \in \cU$, such that the left hand side of is non-zero. Thus also the right hand side is non-zero and we conclude that either $\Delta \phi_t(t,u) \neq 0$ or $\Delta \psi_t(t,u) \neq 0$. It follows that $t \in J^{\phi,\psi}$ and hence that $J^\nu \subseteq J^{\phi,\psi}$. For the other direction let $t \in J^{\phi,\psi}$ and choose an $u \in \cU$ such that $\Delta \phi_t(t,u) \neq 0$ or $\Delta \psi_t(t,u) \neq 0$. Together with Condition \[def:full\_support\] on $X$ we conclude that the right hand side of is non-zero with strictly positive probability. The same must hold for the left hand side and we conclude that $t \in J^\nu$ and hence that $J^\nu = J^{\phi,\psi}$. For (iii) note that $\gamma$ has been defined in such a way that becomes . The jump equations are a direct consequence of . If $J^\nu \subset J^A$, then $\gamma(t,u) = 0$ whenever $t \not \in J^A$ and it follows that $\gamma$ is a good parameter.

We now focus on the continuous parts of the semimartingale characteristics, and make the following definition: For any affine semimartingale $X$ with characteristics $(B,C,\nu)$ and 