\Rplus \times \cU$ $$s \mapsto \phi_s(t,u) \quad \text{and} \quad s \mapsto \psi_s(t,u)$$ are càdlàg functions of finite variation on $[0,t]$, and for all $(s,u) \in \Rplus \times \cU$ $$t \mapsto \phi_s(t,u) \quad \text{and} \quad t \mapsto \psi_s(t,u)$$ are càdlàg functions on $[s, \infty)$.

(ii) For all $0 < s \le t$, the functions $$u \mapsto \phi_{s-}(t,u) \quad \text{and} \quad u \mapsto \psi_{s-}(t,u)$$ are continuous on $\cU$.

Definition \[def:quasi\_regular\] should be compared to the assumptions imposed in [@DuffieFilipovicSchachermayer] and [@Filipovic05]. In both papers technical ‘regularity conditions’ are defined. In [@DuffieFilipovicSchachermayer; @Filipovic05] $\phi$ and $\psi$ are automatically continuous in their first argument, due to the stochastic continuity of $X$. In addition they are assumed continuously differentiable from the right, with a derivative that is continuous in $u$. Thus, (i) and (ii) are clearly milder than the regularity assumptions in [@DuffieFilipovicSchachermayer] or [@Filipovic05].

First results on $\phi$ and $\psi$
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We proceed to show first analytic results on the functions $\phi$ and $\psi$ from .

\[lem:flow\_cont\] Let $X$ be an affine semimartingale satisfying the support condition \[def:full\_support\]. Then,

(i) the function $u \mapsto \phi_s(t,u)$ maps $\cU$ to $\C_{\le 0}$ and $u \mapsto \psi(t,u)$ maps $\cU$ to $\cU$, for all $0<s\le t$,

(ii) $\phi$ 