$(T,u) \in (0,\infty) \times \cU$, the functions $\phi$ and $\psi$ are absolutely continuous w.r.t $A$ and solve the following generalized measure Riccati equations: their continuous parts satisfy $$\begin{aligned}
\frac{d \phi^c_t(T,u)}{dA^c_t} &= - F(t,\psi_{t}(T,u)), 
\label{ric1}\\
\frac{d \psi^c_t(T,u)}{dA^c_t} &= - R(t,\psi_{t}(T,u)), 
\label{ric2}\end{aligned}$$ $dA^c$-a.e., where $$\label{eq:FR}
\begin{split}
F(s,u) &= \scal{\beta_0(s)}{u} + \half \scal{u}{\alpha_0(s) u} +   \int_D \Big( e^{\scal{x}{u}} -1 - \scal{h(x)}{u}\Big) \mu_0(s,dx)
\\
R_i(s,u) &= \scal{\beta_i(s)}{u} + \half \scal{u}{\alpha_i(s) u} + \int_D \Big( e^{\scal{x}{u}} -1 - \scal{h(x)}{u}\Big) \mu_i(s,dx),
\end{split}$$ while their jumps are given by $$\begin{aligned}
\label{eq:Deltaphipsi}
\begin{split}
\Delta \phi_{t}(T,u) &= - \gamma_0(t,\psi_t(T,u))   \\
\Delta \psi_{t}(T,u) &= - \bar \gamma(t,\psi_t(T,u)),
\end{split}\end{aligned}$$ and their terminal conditions are $$\phi_T\brac{T,u}=0 \quad\text{and} \quad\psi_T\brac{T,u}=u.\label{eq:initial-cond}$$

\[rem:parameter\_Set\] Note that the parameter set $(A, \gamma,\beta, \alpha, \mu)$ is not uniquely determined: indeed, consider some increasing function $A'$ such that $A \ll A'$ and write $g = \frac{dA}{dA'}$ for the Radon-Nikodym density of $A$ with respect to $A'$. It is easy to see that all statements of the theorem remain true for the alternative parameter set $(A', \gamma,g\beta, g \alpha, g \mu)$.

We expect that Theorem \[prop1\] can be extended to affine semimartingales with explosion or killing, by adding a ‘fourth characteristic’ (cf. [@schnurr2017fourth] and also [@cheridito2005equivalent]), which possesses an affine decomposition 