e.g. by using the ‘Big flow’ argument of Remark \[rem:big\_flow\]. To show right continuity in (i), we write $$\begin{aligned}
\lim_{\epsilon \downarrow 0} \psi_{s-}(t+\epsilon,u) &= \lim_{\epsilon \downarrow 0} \psi_{s-}(t, \psi_t(t+\epsilon,u)) = \psi_{s-}\Big(t,  \lim_{\epsilon \downarrow 0} \psi_t(t+\epsilon,u)\Big) = \\ &= \psi_{s-}(t, \psi_t(t,u)) = \psi_{s-}(t,u).\end{aligned}$$ Here, we have used the flow property, the continuity of $\psi_{s-}(t,u)$ in $u$ and finally the right-continuity of $\psi_s(t,u)$ in $t$. As for the left limit, the equality $$\begin{aligned}
\lim_{\epsilon \downarrow 0} \psi_{s-}(t-\epsilon,u) &= \lim_{\epsilon \downarrow 0} \psi_{s-}(s, \psi_s(t-\epsilon,u)) = \psi_{s-}\Big(s,  \lim_{\epsilon \downarrow 0} \psi_t(t-\epsilon,u)\Big) = \\ &= \psi_{s-}(s, \psi_s(t-,u))\end{aligned}$$ shows that the left limit exists. Moreover, $$\psi_{s-}(s, \psi_s(t-,u)) =  \lim_{\delta \downarrow 0} \psi_{s-\delta}(s, \psi_s(t-,u)) = \lim_{\delta \downarrow 0} \psi_{s-\delta}(t-,u))$$ shows exchangeability of the limits in (ii). Claim (iii) follows from the semi-flow equations by taking left limits in $s$, left limits in $t$, or both. Similarly, claim (iv) follows from by taking left limits in $s$, or from by taking left limits in $t$.

For (v) we apply the semi-flow property for $r=s$ and obtain that $$\begin{aligned}
   \Delta \phi_s(t,u) &=  \phi_s(t,u) -  \phi_{s-}(t,u) 
        = \phi_s(s,\psi_s(t,u)) -  \phi_{s-}(s,\psi_s(t,u)) \end{aligned}$$ and the first part of follows. The second part follows analogously.

The characterization of affine semimartingales {#sec:characterization}
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In 