by taking right limits.

A *semimartingale* is a process $X$ with decomposition $X=X_0+N+M$ where $X_0$ is $\cF_0$-measurable, $N$ is càdlàg, adapted, has paths of finite variation over each finite interval with $N_0=0$ and $M$ is a local martingale starting in $0$. We will always consider a càdlàg version of the semimartingale $X$.

To the jumps of $X$ we associate an integer-valued random measure $\mu^X$ by $$\begin{aligned}
 \label{def:muX}
        \mu^X(dt,dx) = \sum_{s \ge 0} \ind{\Delta X_s \neq 0} \delta_{(s,\Delta X_s)}(dt,dx); 
    \end{aligned}$$ here $\delta_a$ is the Dirac measure at point $a$. We denote the compensator, or the dual predictable projection, of the random measure $\mu^X$ by $\nu$. This is the unique predictable random measure which renders stochastic integrals with respect to $\mu^X-\nu$ local martingales.

We briefly recall the well-known concept of *characteristics* of a semimartingale, cf. [@JacodShiryaev Ch. II]: a semimartingale $X$ with decomposition $X=X_0+N+M$ is called *special* if $N$ is predictable. In this case, the decomposition is unique, and we call it the *canonical decomposition*. The local martingale part $M$ can be decomposed in a continuous local martingale part, which we denote by $X^c$, and a purely discontinuous local martingale part, $X-X^c$. We fix a truncation function $h:\R^d\to \R^d$ which is a 