similar to . The rigorous formulation of the corresponding results will not be pursued here, and is left for future research.

The distribution of the jumps of the affine semimartingale occurring at fixed times $t$ can directly be characterized as follows.

\[lem:affinejumps\] Let $X$ be a quasi-regular affine semimartingale satisfying the support condition \[def:full\_support\] and with characteristics $(B,C,\nu)$.

(i) For any $(t,u) \in (0,\infty) \times \cU$, $$\label{eq:jump2}
        \int_{D} \left(e^{\scal u \xi} - 1\right) \nu(\omega; \{t\},d\xi) 
            = \exp\Big( - \Delta \phi_t(t,u) - \scal {\Delta \psi_t(t,u)} {X_{t-}}  \Big) - 1.$$

(ii) Set $$\label{eq:J}
    \begin{split}
    J^\nu &:= \set{t > 0: \PP(\nu(\omega,\set{t},D) >0) > 0}\\
    J^{\phi,\psi} &:= \set{t > 0: \exists\,u \in \cU \text{ such that } \Delta \phi_t(t,u) \neq 0 \text{ or } \Delta \psi_t(t,u) \neq 0}.
    \end{split}$$ Then $J^\nu = J^{\phi,\psi}$.

(iii) Set $\gamma_0(t,u) = -\Delta\phi_t(t,u)$ and $\bar\gamma(t,u) = - \Delta \psi_t(t,u)$. Then and hold true and $\gamma = (\gamma_0, \bar \gamma)$ is a good parameter in the sense of Definition \[def:good-parameter\] whenever $J^\nu \subset J^A$.

By definition, $\nu(\{t\},d\xi)$ is the dual predictable projection of $\delta_{\Delta X_t}(d\xi)$ such that (by Proposition II 1.17 in [@JacodShiryaev]) $$\begin{aligned}
 \int_{D} \left(e^{\scal u \xi} - 1\right) \nu(\omega; \{t\},d\xi) 
&= \cond{\left(e^{ \scal u {\Delta X_{t}}}  - 1\right)}{\cF_{t-}}.\end{aligned}$$ 