:= \set{u \in \C^d: \scal{\Re u}{x} \le 0  \text{ for all }  x \in D}.$$ For the canonical state space $\cU$ equals $\C_{\le 0}^m \times i\R^n$, where $\C_{\le 0} = \set{u \in \C: \Re u \le 0}$, which coincides with the definition used in [@DuffieFilipovicSchachermayer].[^2] We are now prepared to state the central definition of this paper.

\[def:affine\] Let $X$ be a càdlàg semimartingale, taking values in $D$. The process $X$ is called an *affine semimartingale*, if there exist $\C$ and $\C^d$-valued deterministic functions $\phi_s(t,u)$ and $\psi_s(t,u)$, continuous in $u \in \cU$ and with $\phi_s(t,0) = 0$ and $\psi_s(t,0)=0$, such that $$\begin{aligned}
\label{cond:affine}
E\big[ e^{\scal{u}{X_{t}}} |\cF_{s} \big] = \exp\big( \phi_s(t,u) + \scal{\psi_s(t,u)}{X_s} \big)\end{aligned}$$ for all $0 \le s \le t$ and $u \in \cU$. Moreover, $X$ is called *time-homogeneous*, if $\phi_s(t,u) = \phi_0(t-s,u)$ and $\psi_s(t,u)=\psi_0(t-s,u)$, again for all $0 \le s \le t$ and $u \in \cU$.

Note that the left-hand side of is always well-defined and bounded in absolute value by $1$, due to the definition of $\cU$.

Comparing Definition \[def:affine\] with the definition of an *affine process* in [@DuffieFilipovicSchachermayer] (which treats the time-homogeneous case) and [@Filipovic05] (which treats the time-inhomogeneous case), we have replaced the Markov assumption of 