\mapsto \Psi_s(t,u)$ maps $\wh{\cU}$ to $\wh{\cU}$ and part (ii) is equivalent to $$\Psi_s(t,\wh{u}) = \Psi_s(r,\Psi_r(t,\wh{u})), \qquad \Psi_t(t,\wh{u}) = \wh{u},$$ for all $0 < s \le r \le t$ and $\wh{u} \in \wh{\cU}$.

\[lem:affine\_limits\] Let $X$ be a quasi-regular affine semimartingale. Then, $$\begin{aligned}
E\big[ e^{\scal{u}{X_{t-}}} |\cF_{s} \big] &= \exp\big( \phi_{s}(t\uminus,u) + \scal{\psi_{s}(t\uminus,u)}{X_{s}} \big), \quad \forall\,0 \le s < t, u \in \cU.\label{eq:Utcond1}\\
E\big[ e^{\scal{u}{X_{t}}} |\cF_{s-} \big] &= \exp\big( \phi_{s-}(t,u) + \scal{\psi_{s-}(t,u)}{X_{s-}} \big), \quad \forall\,0 < s \le t, u \in \cU \label{eq:Utcond2}.\end{aligned}$$ If in addition $X$ satisfies the support condition \[def:full\_support\], it also holds that $$\label{eq:jump}
E\big[ e^{\scal{u}{ \Delta X_{t}}} |\cF_{t-} \big] = \exp\big(- \Delta \phi_{t}(t,u) - \scal{\Delta \psi_{t}(t,u)}{X_{t-}} \big), \quad \forall\; (t,u) \in \Rplus \times \cU.$$

The first expression, , follows by taking left limits in $t$ on both sides of . On the right hand side, the limit is well-defined by the càdlàg property of $\phi$ and $\psi$ in $t$. On the left hand side, dominated convergence and the càdlàg property of $X$ yield . Equation follows from a similar argument, now taking left limits in $s$. Indeed, note that for any integrable random variable $Y$ martingale convergence yields that that $\lim_{\epsilon \downarrow 0} E\big[Y |\cF_{s-\epsilon} \big] = E\big[Y 