  We extend the class of semimartingales in a natural way. This allows us to
incorporate processes having paths that leave the state space R^d. In
particular Markov processes related to sub-Markovian kernels, but also
non-Markovian processes with path-dependent behavior. By carefully
distinguishing between two killing states, we are able to introduce a fourth
semimartingale characteristic which generalizes the fourth part of the L\'evy
quadruple. Using the probabilistic symbol, we analyze the close relationship
between the generators of certain Markov processes with killing and their (now
four) semimartingale characteristics.
