bounded function satisfying $h(x)=x$ in a neighborhood of $0$. Then $\check X(h)=\sum_{s \le \cdot}( \Delta X_s - h(\Delta X_s) )$ and $X(h)=X-\check X(h)$ both define $d$-dimensional stochastic processes. Note that $\Delta X(h) = h(\Delta X)$, such that $X(h)$ has bounded jumps. The resulting process is a special semimartingale and we denote its canonical decomposition by $$X(h) = X_0 + B(h) + M(h),$$ with a predictable process of finite variation $B(h)$ and a local martingale $M(h)$. The *characteristics* of the semimartingale $X$ is the triplet $(B,C,\nu)$ where $B=B(h)$, $C=(C^{ij})$ with $C^{ij}=\scal{X^{i,c}}{X^{j,c}}$ and $\nu=\nu^X$ is the compensator of $\mu^X$ defined in Equation . For additional facts on semimartingales and stochastic analysis we refer to [@JacodShiryaev].

Let $D \subset \R^d$ be a closed convex cone of full dimension, i.e., a convex set, closed under multiplication with positive scalars, and with linear hull equal to $\R^d$. An important example is the set $\Rplus^m \times \R^n$ with $m+n=d$, which was used as the ‘canonical state-space’ for affine processes in [@DuffieFilipovicSchachermayer; @Filipovic05]. For $u,w$ in $\C^d$ we set $\scal{u}{w}=\sum_{i=1}^d u_i w_i$ and denote the real part of $u$ by $\Re u$. Moreover, we define the *complex dual cone* of the state space $D$ by $$\label{eq:defU}
\cU 