t$ and $u \in \cU$.

Fix $0 < s \le t$ and suppose that $\wt \phi_s(t,u)$ and $\wt \psi_s(t,u)$ are also continuous in $u \in \cU$ and satisfy . Write $p_s(t,u) := \wt \phi_s(t,u) - \phi_s(t,u)$ and $q_s(t,u) := \wt \phi_s(t,u) - \phi_s(t,u)$. Due to it must hold that $$p_s(t,u) + \scal{q_s(t,u)}{X_s} \quad \text{takes values in } \quad \set{2 \pi i k: k \in \NN}\quad a.s. \;\forall\, u \in \cU.$$ However, the set $\cU$ is simply connected, and hence its image under a continuous function must also be simply connected. It follows that $u \mapsto p_s(t,u) + \scal{q_s(t,u)}{X_s}$ is constant on $\cU$ and therefore equal to $p_s(t,0) + \scal{q_s(t,0)}{X_s} = 0$. Hence, $$p_s(t,u) + \scal{q_s(t,u)}{x} = 0,$$ for all $x \in \supp(X_s)$ and $u \in \cU$. Taking convex combinations, the equality can be extended for $x\in D$. Since $D$ has full linear span, we conclude that $p_s(t,u) = 0$ and $q_s(t,u) = 0$ for all $u \in \cU$, completing the proof.

\[def:quasi\_regular\] An affine semimartingale is called *quasi-regular*, if the following holds:

(i) The functions $\phi$ and $\psi$ are of finite variation in $s$ and càdlàg in both $s$ and $t$. More precisely, we assume that for all $(t,u) \in 