[@DuffieFilipovicSchachermayer; @Filipovic05] with a semimartingale assumption. In view of [@DuffieFilipovicSchachermayer Thm. 2.12] this seems to slightly restrict the scope of the definition, since it excludes non-conservative processes. On the other hand, and this is the central point of our paper, we do not impose a stochastic continuity assumption on $X$, as has been done in [@DuffieFilipovicSchachermayer; @Filipovic05]. It turns out that omitting this assumption leads to a significantly larger class of stochastic processes and to a substantial extension of the results in [@DuffieFilipovicSchachermayer; @Filipovic05].

To continue, we introduce an important condition on the support of the process $X$. Recall that the support of a generic random variable $X$, is the smallest closed set $C$ such that $P(X\in C)=1$; we denote this set by $\supp(X)$. For a set $A$ we write $\conv (A)$ for its convex hull, i.e. the smallest convex set containing $A$.

\[def:full\_support\] We say that an affine semimartingale $X$ has *support of full convex span*, if $\conv(\supp(X_t)) = D$ for all $t >0$.

Under Condition \[def:full\_support\], $\phi$ and $\psi$ are uniquely specified:

\[lem:uniqueness\] Let $X$ be an affine semimartingale satisfying the support condition \[def:full\_support\]. Then $\phi_s(t,u)$ and $\psi_s(t,u)$ are uniquely specified by for all $0 < s \le 