|\cF_{s-} \big]$. Equation  follows by evaluating at $s = t$ and noting that $\Delta \phi_t(t,u) = \phi_t(t,u) - \phi_{t-}(t,u) = -\phi_{t-}(t,u)$, and $\Delta \psi_t(t,u) = \psi_t(t,u) - \psi_{t-}(t,u) = u -\psi_{t-}(t,u)$, due to Lemma \[lem:flow\_cont\].

\[lem:double\_limit\] Let $X$ be a quasi-regular affine semimartingale satisfying the support condition \[def:full\_support\]. Then,

(i) for all $(s,u) \in \Rplus \times \cU$ the functions $$t \mapsto \phi_{s-}(t,u), \qquad t \mapsto \psi_{s-}(t,u)$$ are càdlàg on $[s, \infty)$.

(ii) The ‘double limits’ $\phi_{s-}(t\uminus,u)$ and $\psi_{s-}(t\uminus,u)$ are well-defined and independent of the order of limits, i.e., $$\lim_{\epsilon \downarrow 0} \psi_{s-}(t - \epsilon,u) = \lim_{\delta \downarrow 0} \psi_{s - \delta}(t-,u),$$ and similarly for $\phi$.

(iii) The semi-flow equations still hold when $s$ is replaced by $s-$ or $t$ is replaced by $t-$ (or both).

(iv) It holds that $$E\big[ e^{\scal{u}{X_{t-}}} |\cF_{s-} \big] = \exp\big( \phi_{s-}(t\uminus,u) + \scal{\psi_{s-}(t\uminus,u)}{X_{s-}} \big), \label{eq:Utcond3}$$ for all $0 < s \le t$ and $u \in \cU$.

(v) For all $u \in \cU$ and $0\leq s < t$ it holds that $$\begin{aligned}
    \label{eq:computeDeltaphipsi}
    \begin{split}
        \Delta \phi_s(t,u) & = \Delta \phi_s(s,\psi_s(t,u)), \\
        \Delta \psi_s(t,u) & = \Delta \psi_s(s,\psi_s(t,u)).
    \end{split}\end{aligned}$$

We show claims (i), (ii) and (iii) for $\psi$ only. The proof can easily be extended to $\phi$, 